Results of Bessel Functions

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DLMF Formula Maple Mathematica Symbolic
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10.2.E1 (z)^(2)* diff(w, [z$(2)])+ z*diff(w, z)+((z)^(2)- (nu)^(2))* w = 0 (z)^(2)* D[w, {z, 2}]+ z*D[w, z]+((z)^(2)- (\[Nu])^(2))* w = 0 Failure Failure
Fail
11.31370849-11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
11.31370849-11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
-11.31370849-11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-11.31370849-11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-11.313708498984761, 11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-11.313708498984761, 11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[11.313708498984761, -11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[11.313708498984761, -11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.2.E2 BesselJ(nu, z)=((1)/(2)*z)^(nu)* sum((- 1)^(k)*(((1)/(4)*(z)^(2))^(k))/(factorial(k)*GAMMA(nu + k + 1)), k = 0..infinity) BesselJ[\[Nu], z]=(Divide[1,2]*z)^(\[Nu])* Sum[(- 1)^(k)*Divide[(Divide[1,4]*(z)^(2))^(k),(k)!*Gamma[\[Nu]+ k + 1]], {k, 0, Infinity}] Successful Successful - -
10.2.E3 BesselY(nu, z)=(BesselJ(nu, z)*cos(nu*Pi)- BesselJ(- nu, z))/(sin(nu*Pi)) BesselY[\[Nu], z]=Divide[BesselJ[\[Nu], z]*Cos[\[Nu]*Pi]- BesselJ[- \[Nu], z],Sin[\[Nu]*Pi]] Successful Successful - -
10.4#Ex1 BesselJ(- n, z)=(- 1)^(n)* BesselJ(n, z) BesselJ[- n, z]=(- 1)^(n)* BesselJ[n, z] Failure Failure Successful Successful
10.4#Ex2 BesselY(- n, z)=(- 1)^(n)* BesselY(n, z) BesselY[- n, z]=(- 1)^(n)* BesselY[n, z] Failure Failure Successful Successful
10.4#Ex3 HankelH1(- n, z)=(- 1)^(n)* HankelH1(n, z) HankelH1[- n, z]=(- 1)^(n)* HankelH1[n, z] Successful Failure - Successful
10.4#Ex4 HankelH2(- n, z)=(- 1)^(n)* HankelH2(n, z) HankelH2[- n, z]=(- 1)^(n)* HankelH2[n, z] Failure Failure Successful Successful
10.4#Ex5 HankelH1(nu, z)= BesselJ(nu, z)+ I*BesselY(nu, z) HankelH1[\[Nu], z]= BesselJ[\[Nu], z]+ I*BesselY[\[Nu], z] Failure Successful Successful -
10.4#Ex6 HankelH2(nu, z)= BesselJ(nu, z)- I*BesselY(nu, z) HankelH2[\[Nu], z]= BesselJ[\[Nu], z]- I*BesselY[\[Nu], z] Failure Successful Successful -
10.4#Ex7 BesselJ(nu, z)=(1)/(2)*(HankelH1(nu, z)+ HankelH2(nu, z)) BesselJ[\[Nu], z]=Divide[1,2]*(HankelH1[\[Nu], z]+ HankelH2[\[Nu], z]) Successful Successful - -
10.4#Ex8 BesselY(nu, z)=(1)/(2*I)*(HankelH1(nu, z)- HankelH2(nu, z)) BesselY[\[Nu], z]=Divide[1,2*I]*(HankelH1[\[Nu], z]- HankelH2[\[Nu], z]) Successful Successful - -
10.4.E5 BesselJ(nu, z)= csc(nu*Pi)*(BesselY(- nu, z)- BesselY(nu, z)*cos(nu*Pi)) BesselJ[\[Nu], z]= Csc[\[Nu]*Pi]*(BesselY[- \[Nu], z]- BesselY[\[Nu], z]*Cos[\[Nu]*Pi]) Successful Successful - -
10.4#Ex9 HankelH1(- nu, z)= exp(nu*Pi*I)*HankelH1(nu, z) HankelH1[- \[Nu], z]= Exp[\[Nu]*Pi*I]*HankelH1[\[Nu], z] Successful Failure - Successful
10.4#Ex10 HankelH2(- nu, z)= exp(- nu*Pi*I)*HankelH2(nu, z) HankelH2[- \[Nu], z]= Exp[- \[Nu]*Pi*I]*HankelH2[\[Nu], z] Successful Failure - Successful
10.4.E7 HankelH1(nu, z)= I*csc(nu*Pi)*(exp(- nu*Pi*I)*BesselJ(nu, z)- BesselJ(- nu, z)) HankelH1[\[Nu], z]= I*Csc[\[Nu]*Pi]*(Exp[- \[Nu]*Pi*I]*BesselJ[\[Nu], z]- BesselJ[- \[Nu], z]) Successful Successful - -
10.4.E7 I*csc(nu*Pi)*(exp(- nu*Pi*I)*BesselJ(nu, z)- BesselJ(- nu, z))= csc(nu*Pi)*(BesselY(- nu, z)- exp(- nu*Pi*I)*BesselY(nu, z)) I*Csc[\[Nu]*Pi]*(Exp[- \[Nu]*Pi*I]*BesselJ[\[Nu], z]- BesselJ[- \[Nu], z])= Csc[\[Nu]*Pi]*(BesselY[- \[Nu], z]- Exp[- \[Nu]*Pi*I]*BesselY[\[Nu], z]) Failure Successful Successful -
10.4.E8 HankelH2(nu, z)= I*csc(nu*Pi)*(BesselJ(- nu, z)- exp(nu*Pi*I)*BesselJ(nu, z)) HankelH2[\[Nu], z]= I*Csc[\[Nu]*Pi]*(BesselJ[- \[Nu], z]- Exp[\[Nu]*Pi*I]*BesselJ[\[Nu], z]) Successful Successful - -
10.4.E8 I*csc(nu*Pi)*(BesselJ(- nu, z)- exp(nu*Pi*I)*BesselJ(nu, z))= csc(nu*Pi)*(BesselY(- nu, z)- exp(nu*Pi*I)*BesselY(nu, z)) I*Csc[\[Nu]*Pi]*(BesselJ[- \[Nu], z]- Exp[\[Nu]*Pi*I]*BesselJ[\[Nu], z])= Csc[\[Nu]*Pi]*(BesselY[- \[Nu], z]- Exp[\[Nu]*Pi*I]*BesselY[\[Nu], z]) Failure Successful Successful -
10.5.E1 (BesselJ(nu, z))*diff(BesselJ(- nu, z), z)-diff(BesselJ(nu, z), z)*(BesselJ(- nu, z))= BesselJ(nu + 1, z)*BesselJ(- nu, z)+ BesselJ(nu, z)*BesselJ(- nu - 1, z) Wronskian[{BesselJ[\[Nu], z], BesselJ[- \[Nu], z]}, z]= BesselJ[\[Nu]+ 1, z]*BesselJ[- \[Nu], z]+ BesselJ[\[Nu], z]*BesselJ[- \[Nu]- 1, z] Successful Successful - -
10.5.E1 BesselJ(nu + 1, z)*BesselJ(- nu, z)+ BesselJ(nu, z)*BesselJ(- nu - 1, z)= - 2*sin(nu*Pi)/(Pi*z) BesselJ[\[Nu]+ 1, z]*BesselJ[- \[Nu], z]+ BesselJ[\[Nu], z]*BesselJ[- \[Nu]- 1, z]= - 2*Sin[\[Nu]*Pi]/(Pi*z) Failure Successful Successful -
10.5.E2 (BesselJ(nu, z))*diff(BesselY(nu, z), z)-diff(BesselJ(nu, z), z)*(BesselY(nu, z))= BesselJ(nu + 1, z)*BesselY(nu, z)- BesselJ(nu, z)*BesselY(nu + 1, z) Wronskian[{BesselJ[\[Nu], z], BesselY[\[Nu], z]}, z]= BesselJ[\[Nu]+ 1, z]*BesselY[\[Nu], z]- BesselJ[\[Nu], z]*BesselY[\[Nu]+ 1, z] Successful Successful - -
10.5.E2 BesselJ(nu + 1, z)*BesselY(nu, z)- BesselJ(nu, z)*BesselY(nu + 1, z)= 2/(Pi*z) BesselJ[\[Nu]+ 1, z]*BesselY[\[Nu], z]- BesselJ[\[Nu], z]*BesselY[\[Nu]+ 1, z]= 2/(Pi*z) Failure Successful Successful -
10.5.E3 BesselJ(nu + 1, z)*HankelH1(nu, z)- BesselJ(nu, z)*HankelH1(nu + 1, z)= 2*I/(Pi*z) BesselJ[\[Nu]+ 1, z]*HankelH1[\[Nu], z]- BesselJ[\[Nu], z]*HankelH1[\[Nu]+ 1, z]= 2*I/(Pi*z) Failure Successful Successful -
10.5.E4 BesselJ(nu + 1, z)*HankelH2(nu, z)- BesselJ(nu, z)*HankelH2(nu + 1, z)= - 2*I/(Pi*z) BesselJ[\[Nu]+ 1, z]*HankelH2[\[Nu], z]- BesselJ[\[Nu], z]*HankelH2[\[Nu]+ 1, z]= - 2*I/(Pi*z) Failure Successful Successful -
10.5.E5 (HankelH1(nu, z))*diff(HankelH2(nu, z), z)-diff(HankelH1(nu, z), z)*(HankelH2(nu, z))= HankelH1(nu + 1, z)*HankelH2(nu, z)- HankelH1(nu, z)*HankelH2(nu + 1, z) Wronskian[{HankelH1[\[Nu], z], HankelH2[\[Nu], z]}, z]= HankelH1[\[Nu]+ 1, z]*HankelH2[\[Nu], z]- HankelH1[\[Nu], z]*HankelH2[\[Nu]+ 1, z] Successful Successful - -
10.5.E5 HankelH1(nu + 1, z)*HankelH2(nu, z)- HankelH1(nu, z)*HankelH2(nu + 1, z)= - 4*I/(Pi*z) HankelH1[\[Nu]+ 1, z]*HankelH2[\[Nu], z]- HankelH1[\[Nu], z]*HankelH2[\[Nu]+ 1, z]= - 4*I/(Pi*z) Failure Successful Successful -
10.6#Ex11 p[nu]= BesselJ(nu, a)*BesselY(nu, b)- BesselJ(nu, b)*BesselY(nu, a) Subscript[p, \[Nu]]= BesselJ[\[Nu], a]*BesselY[\[Nu], b]- BesselJ[\[Nu], b]*BesselY[\[Nu], a] Failure Failure
Fail
1.414213562+1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), p[nu] = 2^(1/2)+I*2^(1/2)}
1.414213562-1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), p[nu] = 2^(1/2)-I*2^(1/2)}
-1.414213562-1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), p[nu] = -2^(1/2)-I*2^(1/2)}
-1.414213562+1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), p[nu] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Skip
10.6#Ex12 q[nu]= BesselJ(nu, a)*subs( temp=b, diff( BesselY(nu, temp), temp$(1) ) )- subs( temp=b, diff( BesselJ(nu, temp), temp$(1) ) )*BesselY(nu, a) Subscript[q, \[Nu]]= BesselJ[\[Nu], a]*(D[BesselY[\[Nu], temp], {temp, 1}]/.temp-> b)- (D[BesselJ[\[Nu], temp], {temp, 1}]/.temp-> b)*BesselY[\[Nu], a] Failure Failure
Fail
1.189134483+1.639292641*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), q[nu] = 2^(1/2)+I*2^(1/2)}
1.189134483-1.189134483*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), q[nu] = 2^(1/2)-I*2^(1/2)}
-1.639292641-1.189134483*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), q[nu] = -2^(1/2)-I*2^(1/2)}
-1.639292641+1.639292641*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), q[nu] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Skip
10.6#Ex13 r[nu]= subs( temp=a, diff( BesselJ(nu, temp), temp$(1) ) )*BesselY(nu, b)- BesselJ(nu, b)*subs( temp=a, diff( BesselY(nu, temp), temp$(1) ) ) Subscript[r, \[Nu]]= (D[BesselJ[\[Nu], temp], {temp, 1}]/.temp-> a)*BesselY[\[Nu], b]- BesselJ[\[Nu], b]*(D[BesselY[\[Nu], temp], {temp, 1}]/.temp-> a) Failure Failure
Fail
1.639292641+1.189134483*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), r[nu] = 2^(1/2)+I*2^(1/2)}
1.639292641-1.639292641*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), r[nu] = 2^(1/2)-I*2^(1/2)}
-1.189134483-1.639292641*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), r[nu] = -2^(1/2)-I*2^(1/2)}
-1.189134483+1.189134483*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), r[nu] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Skip
10.6#Ex14 s[nu]= subs( temp=a, diff( BesselJ(nu, temp), temp$(1) ) )*subs( temp=b, diff( BesselY(nu, temp), temp$(1) ) )- subs( temp=b, diff( BesselJ(nu, temp), temp$(1) ) )*subs( temp=a, diff( BesselY(nu, temp), temp$(1) ) ) Subscript[s, \[Nu]]= (D[BesselJ[\[Nu], temp], {temp, 1}]/.temp-> a)*(D[BesselY[\[Nu], temp], {temp, 1}]/.temp-> b)- (D[BesselJ[\[Nu], temp], {temp, 1}]/.temp-> b)*(D[BesselY[\[Nu], temp], {temp, 1}]/.temp-> a) Failure Failure
Fail
1.414213562+1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), s[nu] = 2^(1/2)+I*2^(1/2)}
1.414213562-1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), s[nu] = 2^(1/2)-I*2^(1/2)}
-1.414213562-1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), s[nu] = -2^(1/2)-I*2^(1/2)}
-1.414213562+1.414213562*I <- {a = 2^(1/2)+I*2^(1/2), b = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), s[nu] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Skip
10.8.E1 BesselY(n, z)= -(((1)/(2)*z)^(- n))/(Pi)*sum((factorial(n - k - 1))/(factorial(k))*((1)/(4)*(z)^(2))^(k), k = 0..n - 1)+(2)/(Pi)*ln((1)/(2)*z)*BesselJ(n, z)-(((1)/(2)*z)^(n))/(Pi)*sum((Psi(k + 1)+ Psi(n + k + 1))*((-(1)/(4)*(z)^(2))^(k))/(factorial(k)*factorial(n + k)), k = 0..infinity) BesselY[n, z]= -Divide[(Divide[1,2]*z)^(- n),Pi]*Sum[Divide[(n - k - 1)!,(k)!]*(Divide[1,4]*(z)^(2))^(k), {k, 0, n - 1}]+Divide[2,Pi]*Log[Divide[1,2]*z]*BesselJ[n, z]-Divide[(Divide[1,2]*z)^(n),Pi]*Sum[(PolyGamma[k + 1]+ PolyGamma[n + k + 1])*Divide[(-Divide[1,4]*(z)^(2))^(k),(k)!*(n + k)!], {k, 0, Infinity}] Error Failure - Successful
10.9.E1 BesselJ(0, z)=(1)/(Pi)*int(cos(z*sin(theta)), theta = 0..Pi) BesselJ[0, z]=Divide[1,Pi]*Integrate[Cos[z*Sin[\[Theta]]], {\[Theta], 0, Pi}] Successful Failure - Successful
10.9.E1 (1)/(Pi)*int(cos(z*sin(theta)), theta = 0..Pi)=(1)/(Pi)*int(cos(z*cos(theta)), theta = 0..Pi) Divide[1,Pi]*Integrate[Cos[z*Sin[\[Theta]]], {\[Theta], 0, Pi}]=Divide[1,Pi]*Integrate[Cos[z*Cos[\[Theta]]], {\[Theta], 0, Pi}] Successful Failure - Successful
10.9.E2 BesselJ(n, z)=(1)/(Pi)*int(cos(z*sin(theta)- n*theta), theta = 0..Pi) BesselJ[n, z]=Divide[1,Pi]*Integrate[Cos[z*Sin[\[Theta]]- n*\[Theta]], {\[Theta], 0, Pi}] Failure Failure Skip Successful
10.9.E2 (1)/(Pi)*int(cos(z*sin(theta)- n*theta), theta = 0..Pi)=((I)^(- n))/(Pi)*int(exp(I*z*cos(theta))*cos(n*theta), theta = 0..Pi) Divide[1,Pi]*Integrate[Cos[z*Sin[\[Theta]]- n*\[Theta]], {\[Theta], 0, Pi}]=Divide[(I)^(- n),Pi]*Integrate[Exp[I*z*Cos[\[Theta]]]*Cos[n*\[Theta]], {\[Theta], 0, Pi}] Failure Failure Skip Successful
10.9.E3 BesselY(0, z)=(4)/((Pi)^(2))*int(cos(z*cos(theta))*(gamma + ln(2*z*(sin(theta))^(2))), theta = 0..(1)/(2)*Pi) BesselY[0, z]=Divide[4,(Pi)^(2)]*Integrate[Cos[z*Cos[\[Theta]]]*(EulerGamma + Log[2*z*(Sin[\[Theta]])^(2)]), {\[Theta], 0, Divide[1,2]*Pi}] Failure Failure Skip Successful
10.9.E4 BesselJ(nu, z)=(((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int(cos(z*cos(theta))*(sin(theta))^(2*nu), theta = 0..Pi) BesselJ[\[Nu], z]=Divide[(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[Cos[z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*\[Nu]), {\[Theta], 0, Pi}] Failure Failure Skip Successful
10.9.E4 (((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int(cos(z*cos(theta))*(sin(theta))^(2*nu), theta = 0..Pi)=(2*((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int((1 - (t)^(2))^(nu -(1)/(2))* cos(z*t), t = 0..1) Divide[(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[Cos[z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*\[Nu]), {\[Theta], 0, Pi}]=Divide[2*(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[(1 - (t)^(2))^(\[Nu]-Divide[1,2])* Cos[z*t], {t, 0, 1}] Failure Failure Skip Skip
10.9.E5 BesselY(nu, z)=(2*((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*(int((1 - (t)^(2))^(nu -(1)/(2))* sin(z*t), t = 0..1)- int(exp(- z*t)*(1 + (t)^(2))^(nu -(1)/(2)), t = 0..infinity)) BesselY[\[Nu], z]=Divide[2*(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*(Integrate[(1 - (t)^(2))^(\[Nu]-Divide[1,2])* Sin[z*t], {t, 0, 1}]- Integrate[Exp[- z*t]*(1 + (t)^(2))^(\[Nu]-Divide[1,2]), {t, 0, Infinity}]) Successful Failure - Skip
10.9.E6 BesselJ(nu, z)=(1)/(Pi)*int(cos(z*sin(theta)- nu*theta), theta = 0..Pi)-(sin(nu*Pi))/(Pi)*int(exp(- z*sinh(t)- nu*t), t = 0..infinity) BesselJ[\[Nu], z]=Divide[1,Pi]*Integrate[Cos[z*Sin[\[Theta]]- \[Nu]*\[Theta]], {\[Theta], 0, Pi}]-Divide[Sin[\[Nu]*Pi],Pi]*Integrate[Exp[- z*Sinh[t]- \[Nu]*t], {t, 0, Infinity}] Failure Failure Skip Error
10.9.E7 BesselY(nu, z)=(1)/(Pi)*int(sin(z*sin(theta)- nu*theta), theta = 0..Pi)-(1)/(Pi)*int((exp(nu*t)+ exp(- nu*t)*cos(nu*Pi))* exp(- z*sinh(t)), t = 0..infinity) BesselY[\[Nu], z]=Divide[1,Pi]*Integrate[Sin[z*Sin[\[Theta]]- \[Nu]*\[Theta]], {\[Theta], 0, Pi}]-Divide[1,Pi]*Integrate[(Exp[\[Nu]*t]+ Exp[- \[Nu]*t]*Cos[\[Nu]*Pi])* Exp[- z*Sinh[t]], {t, 0, Infinity}] Failure Failure Skip Error
10.9#Ex1 BesselJ(nu, x)=(2)/(Pi)*int(sin(x*cosh(t)-(1)/(2)*nu*Pi)*cosh(nu*t), t = 0..infinity) BesselJ[\[Nu], x]=Divide[2,Pi]*Integrate[Sin[x*Cosh[t]-Divide[1,2]*\[Nu]*Pi]*Cosh[\[Nu]*t], {t, 0, Infinity}] Failure Failure Skip Error
10.9#Ex2 BesselY(nu, x)= -(2)/(Pi)*int(cos(x*cosh(t)-(1)/(2)*nu*Pi)*cosh(nu*t), t = 0..infinity) BesselY[\[Nu], x]= -Divide[2,Pi]*Integrate[Cos[x*Cosh[t]-Divide[1,2]*\[Nu]*Pi]*Cosh[\[Nu]*t], {t, 0, Infinity}] Failure Failure Skip Error
10.9#Ex3 BesselJ(0, x)=(2)/(Pi)*int(sin(x*cosh(t)), t = 0..infinity) BesselJ[0, x]=Divide[2,Pi]*Integrate[Sin[x*Cosh[t]], {t, 0, Infinity}] Failure Failure Skip Error
10.9#Ex4 BesselY(0, x)= -(2)/(Pi)*int(cos(x*cosh(t)), t = 0..infinity) BesselY[0, x]= -Divide[2,Pi]*Integrate[Cos[x*Cosh[t]], {t, 0, Infinity}] Failure Failure Skip Error
10.9.E10 HankelH1(nu, z)=(exp(-(1)/(2)*nu*Pi*I))/(Pi*I)*int(exp(I*z*cosh(t)- nu*t), t = - infinity..infinity) HankelH1[\[Nu], z]=Divide[Exp[-Divide[1,2]*\[Nu]*Pi*I],Pi*I]*Integrate[Exp[I*z*Cosh[t]- \[Nu]*t], {t, - Infinity, Infinity}] Failure Failure Skip Error
10.9.E11 HankelH2(nu, z)= -(exp((1)/(2)*nu*Pi*I))/(Pi*I)*int(exp(- I*z*cosh(t)- nu*t), t = - infinity..infinity) HankelH2[\[Nu], z]= -Divide[Exp[Divide[1,2]*\[Nu]*Pi*I],Pi*I]*Integrate[Exp[- I*z*Cosh[t]- \[Nu]*t], {t, - Infinity, Infinity}] Failure Failure Skip Error
10.9#Ex5 BesselJ(nu, x)=(2*((1)/(2)*x)^(- nu))/((Pi)^((1)/(2))* GAMMA((1)/(2)- nu))*int((sin(x*t))/(((t)^(2)- 1)^(nu +(1)/(2))), t = 1..infinity) BesselJ[\[Nu], x]=Divide[2*(Divide[1,2]*x)^(- \[Nu]),(Pi)^(Divide[1,2])* Gamma[Divide[1,2]- \[Nu]]]*Integrate[Divide[Sin[x*t],((t)^(2)- 1)^(\[Nu]+Divide[1,2])], {t, 1, Infinity}] Successful Failure - Successful
10.9#Ex6 BesselY(nu, x)= -(2*((1)/(2)*x)^(- nu))/((Pi)^((1)/(2))* GAMMA((1)/(2)- nu))*int((cos(x*t))/(((t)^(2)- 1)^(nu +(1)/(2))), t = 1..infinity) BesselY[\[Nu], x]= -Divide[2*(Divide[1,2]*x)^(- \[Nu]),(Pi)^(Divide[1,2])* Gamma[Divide[1,2]- \[Nu]]]*Integrate[Divide[Cos[x*t],((t)^(2)- 1)^(\[Nu]+Divide[1,2])], {t, 1, Infinity}] Failure Failure Skip Error
10.9.E13 ((z + zeta)/(z - zeta))^((1)/(2)*nu)* BesselJ(nu, ((z)^(2)- (zeta)^(2))^((1)/(2)))=(1)/(Pi)*int(exp(zeta*cos(theta))*cos(z*sin(theta)- nu*theta), theta = 0..Pi)-(sin(nu*Pi))/(Pi)*int(exp(- zeta*cosh(t)- z*sinh(t)- nu*t), t = 0..infinity) (Divide[z + \[zeta],z - \[zeta]])^(Divide[1,2]*\[Nu])* BesselJ[\[Nu], ((z)^(2)- (\[zeta])^(2))^(Divide[1,2])]=Divide[1,Pi]*Integrate[Exp[\[zeta]*Cos[\[Theta]]]*Cos[z*Sin[\[Theta]]- \[Nu]*\[Theta]], {\[Theta], 0, Pi}]-Divide[Sin[\[Nu]*Pi],Pi]*Integrate[Exp[- \[zeta]*Cosh[t]- z*Sinh[t]- \[Nu]*t], {t, 0, Infinity}] Failure Failure Skip Error
10.9.E14 ((z + zeta)/(z - zeta))^((1)/(2)*nu)* BesselY(nu, ((z)^(2)- (zeta)^(2))^((1)/(2)))=(1)/(Pi)*int(exp(zeta*cos(theta))*sin(z*sin(theta)- nu*theta), theta = 0..Pi)-(1)/(Pi)*int((exp(nu*t + zeta*cosh(t))+ exp(- nu*t - zeta*cosh(t))*cos(nu*Pi))* exp(- z*sinh(t)), t = 0..infinity) (Divide[z + \[zeta],z - \[zeta]])^(Divide[1,2]*\[Nu])* BesselY[\[Nu], ((z)^(2)- (\[zeta])^(2))^(Divide[1,2])]=Divide[1,Pi]*Integrate[Exp[\[zeta]*Cos[\[Theta]]]*Sin[z*Sin[\[Theta]]- \[Nu]*\[Theta]], {\[Theta], 0, Pi}]-Divide[1,Pi]*Integrate[(Exp[\[Nu]*t + \[zeta]*Cosh[t]]+ Exp[- \[Nu]*t - \[zeta]*Cosh[t]]*Cos[\[Nu]*Pi])* Exp[- z*Sinh[t]], {t, 0, Infinity}] Failure Failure Skip Error
10.9.E15 ((z + zeta)/(z - zeta))^((1)/(2)*nu)* HankelH1(nu, ((z)^(2)- (zeta)^(2))^((1)/(2)))=(1)/(Pi*I)*exp(-(1)/(2)*nu*Pi*I)*int(exp(I*z*cosh(t)+ I*zeta*sinh(t)- nu*t), t = - infinity..infinity) (Divide[z + \[zeta],z - \[zeta]])^(Divide[1,2]*\[Nu])* HankelH1[\[Nu], ((z)^(2)- (\[zeta])^(2))^(Divide[1,2])]=Divide[1,Pi*I]*Exp[-Divide[1,2]*\[Nu]*Pi*I]*Integrate[Exp[I*z*Cosh[t]+ I*\[zeta]*Sinh[t]- \[Nu]*t], {t, - Infinity, Infinity}] Failure Failure Skip Error
10.9.E16 ((z + zeta)/(z - zeta))^((1)/(2)*nu)* HankelH2(nu, ((z)^(2)- (zeta)^(2))^((1)/(2)))= -(1)/(Pi*I)*exp((1)/(2)*nu*Pi*I)*int(exp(- I*z*cosh(t)- I*zeta*sinh(t)- nu*t), t = - infinity..infinity) (Divide[z + \[zeta],z - \[zeta]])^(Divide[1,2]*\[Nu])* HankelH2[\[Nu], ((z)^(2)- (\[zeta])^(2))^(Divide[1,2])]= -Divide[1,Pi*I]*Exp[Divide[1,2]*\[Nu]*Pi*I]*Integrate[Exp[- I*z*Cosh[t]- I*\[zeta]*Sinh[t]- \[Nu]*t], {t, - Infinity, Infinity}] Failure Failure Skip Error
10.9.E17 BesselJ(nu, z)=(1)/(2*Pi*I)*int(exp(z*sinh(t)- nu*t), t = infinity - Pi*I..infinity + Pi*I) BesselJ[\[Nu], z]=Divide[1,2*Pi*I]*Integrate[Exp[z*Sinh[t]- \[Nu]*t], {t, Infinity - Pi*I, Infinity + Pi*I}] Failure Failure Skip
Fail
Complex[0.342503927390088, -0.08973210023585859] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.3263028372306598, 4.480608248698951] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-26.41355287980499, 14.935276359740396] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.2663767645899945, 0.9702347233898156] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.9#Ex7 HankelH1(nu, z)=(1)/(Pi*I)*int(exp(z*sinh(t)- nu*t), t = - infinity..infinity + Pi*I) HankelH1[\[Nu], z]=Divide[1,Pi*I]*Integrate[Exp[z*Sinh[t]- \[Nu]*t], {t, - Infinity, Infinity + Pi*I}] Failure Failure Skip Error
10.9#Ex8 HankelH2(nu, z)= -(1)/(Pi*I)*int(exp(z*sinh(t)- nu*t), t = - infinity..infinity - Pi*I) HankelH2[\[Nu], z]= -Divide[1,Pi*I]*Integrate[Exp[z*Sinh[t]- \[Nu]*t], {t, - Infinity, Infinity - Pi*I}] Failure Failure Skip Error
10.9.E19 BesselJ(nu, z)=(((1)/(2)*z)^(nu))/(2*Pi*I)*int(exp(t -((z)^(2))/(4*t))*(1)/((t)^(nu + 1)), t = - infinity..(0 +)) BesselJ[\[Nu], z]=Divide[(Divide[1,2]*z)^(\[Nu]),2*Pi*I]*Integrate[Exp[t -Divide[(z)^(2),4*t]]*Divide[1,(t)^(\[Nu]+ 1)], {t, - Infinity, (0 +)}] Error Failure - Error
10.9.E20 BesselJ(nu, z)=(GAMMA((1)/(2)- nu)*((1)/(2)*z)^(nu))/((Pi)^((3)/(2))* I)*int(cos(z*t)*((t)^(2)- 1)^(nu -(1)/(2)), t = 0..(1 +)) BesselJ[\[Nu], z]=Divide[Gamma[Divide[1,2]- \[Nu]]*(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[3,2])* I]*Integrate[Cos[z*t]*((t)^(2)- 1)^(\[Nu]-Divide[1,2]), {t, 0, (1 +)}] Error Failure - Error
10.9#Ex9 HankelH1(nu, z)=(GAMMA((1)/(2)- nu)*((1)/(2)*z)^(nu))/((Pi)^((3)/(2))* I)*int(exp(I*z*t)*((t)^(2)- 1)^(nu -(1)/(2)), t = 1 + I*infinity..(1 +)) HankelH1[\[Nu], z]=Divide[Gamma[Divide[1,2]- \[Nu]]*(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[3,2])* I]*Integrate[Exp[I*z*t]*((t)^(2)- 1)^(\[Nu]-Divide[1,2]), {t, 1 + I*Infinity, (1 +)}] Error Failure - Error
10.9#Ex10 HankelH2(nu, z)=(GAMMA((1)/(2)- nu)*((1)/(2)*z)^(nu))/((Pi)^((3)/(2))* I)*int(exp(- I*z*t)*((t)^(2)- 1)^(nu -(1)/(2)), t = 1 - I*infinity..(1 +)) HankelH2[\[Nu], z]=Divide[Gamma[Divide[1,2]- \[Nu]]*(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[3,2])* I]*Integrate[Exp[- I*z*t]*((t)^(2)- 1)^(\[Nu]-Divide[1,2]), {t, 1 - I*Infinity, (1 +)}] Error Failure - Error
10.9.E22 BesselJ(nu, x)=(1)/(2*Pi*I)*int((GAMMA(- t)*((1)/(2)*x)^(nu + 2*t))/(GAMMA(nu + t + 1)), t = - I*infinity..I*infinity) BesselJ[\[Nu], x]=Divide[1,2*Pi*I]*Integrate[Divide[Gamma[- t]*(Divide[1,2]*x)^(\[Nu]+ 2*t),Gamma[\[Nu]+ t + 1]], {t, - I*Infinity, I*Infinity}] Failure Failure Skip Error
10.9.E23 BesselJ(nu, z)=(1)/(2*Pi*I)*int((GAMMA(t))/(GAMMA(nu - t + 1))*((1)/(2)*z)^(nu - 2*t), t = - infinity - I*c..- infinity + I*c) BesselJ[\[Nu], z]=Divide[1,2*Pi*I]*Integrate[Divide[Gamma[t],Gamma[\[Nu]- t + 1]]*(Divide[1,2]*z)^(\[Nu]- 2*t), {t, - Infinity - I*c, - Infinity + I*c}] Failure Failure Skip
Fail
Complex[0.342503927390088, -0.08973210023585859] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.3263028372306598, 4.480608248698951] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-26.41355287980499, 14.935276359740396] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.2663767645899945, 0.9702347233898156] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.9.E24 HankelH1(nu, z)= -(exp(-(1)/(2)*nu*Pi*I))/(2*(Pi)^(2))* int(GAMMA(t)*GAMMA(t - nu)*(-(1)/(2)*I*z)^(nu - 2*t), t = c - I*infinity..c + I*infinity) HankelH1[\[Nu], z]= -Divide[Exp[-Divide[1,2]*\[Nu]*Pi*I],2*(Pi)^(2)]* Integrate[Gamma[t]*Gamma[t - \[Nu]]*(-Divide[1,2]*I*z)^(\[Nu]- 2*t), {t, c - I*Infinity, c + I*Infinity}] Failure Failure Skip Error
10.9.E25 HankelH2(nu, z)=(exp((1)/(2)*nu*Pi*I))/(2*(Pi)^(2))*int(GAMMA(t)*GAMMA(t - nu)*((1)/(2)*I*z)^(nu - 2*t), t = c - I*infinity..c + I*infinity) HankelH2[\[Nu], z]=Divide[Exp[Divide[1,2]*\[Nu]*Pi*I],2*(Pi)^(2)]*Integrate[Gamma[t]*Gamma[t - \[Nu]]*(Divide[1,2]*I*z)^(\[Nu]- 2*t), {t, c - I*Infinity, c + I*Infinity}] Failure Failure Skip Error
10.9.E26 BesselJ(mu, z)*BesselJ(nu, z)=(2)/(Pi)*int(BesselJ(mu + nu, 2*z*cos(theta))*cos((mu - nu)* theta), theta = 0..Pi/ 2) BesselJ[\[Mu], z]*BesselJ[\[Nu], z]=Divide[2,Pi]*Integrate[BesselJ[\[Mu]+ \[Nu], 2*z*Cos[\[Theta]]]*Cos[(\[Mu]- \[Nu])* \[Theta]], {\[Theta], 0, Pi/ 2}] Failure Failure Skip Skip
10.9.E27 BesselJ(nu, z)*BesselJ(nu, zeta)=(2)/(Pi)*int(BesselJ(2*nu, 2*(z*zeta)^((1)/(2))* sin(theta))*cos((z - zeta)* cos(theta)), theta = 0..Pi/ 2) BesselJ[\[Nu], z]*BesselJ[\[Nu], \[zeta]]=Divide[2,Pi]*Integrate[BesselJ[2*\[Nu], 2*(z*\[zeta])^(Divide[1,2])* Sin[\[Theta]]]*Cos[(z - \[zeta])* Cos[\[Theta]]], {\[Theta], 0, Pi/ 2}] Failure Failure Skip Error
10.9.E28 BesselJ(nu, z)*BesselJ(nu, zeta)=(1)/(2*Pi*I)*int(* exp((1)/(2)*t -((z)^(2)+ (zeta)^(2))/(2*t))*BesselI(nu, (z*zeta)/(t))*(1)/(t), t = c - I*infinity..c + I*infinity) BesselJ[\[Nu], z]*BesselJ[\[Nu], \[zeta]]=Divide[1,2*Pi*I]*Integrate[* Exp[Divide[1,2]*t -Divide[(z)^(2)+ (\[zeta])^(2),2*t]]*BesselI[\[Nu], Divide[z*\[zeta],t]]*Divide[1,t], {t, c - I*Infinity, c + I*Infinity}] Error Failure - Error
10.9.E29 BesselJ(mu, x)*BesselJ(nu, x)=(1)/(2*Pi*I)*int((GAMMA(- t)*GAMMA(2*t + mu + nu + 1)*((1)/(2)*x)^(mu + nu + 2*t))/(GAMMA(t + mu + 1)*GAMMA(t + nu + 1)*GAMMA(t + mu + nu + 1)), t = - I*infinity..I*infinity) BesselJ[\[Mu], x]*BesselJ[\[Nu], x]=Divide[1,2*Pi*I]*Integrate[Divide[Gamma[- t]*Gamma[2*t + \[Mu]+ \[Nu]+ 1]*(Divide[1,2]*x)^(\[Mu]+ \[Nu]+ 2*t),Gamma[t + \[Mu]+ 1]*Gamma[t + \[Nu]+ 1]*Gamma[t + \[Mu]+ \[Nu]+ 1]], {t, - I*Infinity, I*Infinity}] Failure Failure Skip Error
10.9.E30 (BesselJ(nu, z))^(2)+ (BesselY(nu, z))^(2)=(8)/((Pi)^(2))*int(cosh(2*nu*t)*BesselK(0, 2*z*sinh(t)), t = 0..infinity) (BesselJ[\[Nu], z])^(2)+ (BesselY[\[Nu], z])^(2)=Divide[8,(Pi)^(2)]*Integrate[Cosh[2*\[Nu]*t]*BesselK[0, 2*z*Sinh[t]], {t, 0, Infinity}] Failure Failure Skip Error
10.11.E1 BesselJ(nu, z*exp(m*Pi*I))= exp(m*nu*Pi*I)*BesselJ(nu, z) BesselJ[\[Nu], z*Exp[m*Pi*I]]= Exp[m*\[Nu]*Pi*I]*BesselJ[\[Nu], z] Failure Failure
Fail
-.3975453294+30.10329939*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}
.3425382206-.8976707513e-1*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}
-.3996358010+30.09969700*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}
-1.326778468-4.481046040*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2}
... skip entries to safe data
Fail
Complex[-0.3975452718986143, 30.10329943602099] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.34253822069590145, -0.08976707542141499] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.3996357209647074, 30.09969706489566] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-397.25907376207414, -7.293217978872368] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.11.E2 BesselY(nu, z*exp(m*Pi*I))= exp(- m*nu*Pi*I)*BesselY(nu, z)+ 2*I*sin(m*nu*Pi)*cot(nu*Pi)*BesselJ(nu, z) BesselY[\[Nu], z*Exp[m*Pi*I]]= Exp[- m*\[Nu]*Pi*I]*BesselY[\[Nu], z]+ 2*I*Sin[m*\[Nu]*Pi]*Cot[\[Nu]*Pi]*BesselJ[\[Nu], z] Failure Failure
Fail
59.96664792+53.22883098*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}
-5005.486114+1251.725768*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}
10758.9952-438485.5093*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}
-4.21339-169.756927*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2}
... skip entries to safe data
Fail
Complex[59.966647971782265, 53.22883092179323] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-5005.486119861246, 1251.7257744468322] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[10758.99323773, -438485.5113105696] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[9.175550013151128, -396.6330304507175] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.11.E3 sin(nu*Pi)*HankelH1(nu, z*exp(m*Pi*I))= - sin((m - 1)* nu*Pi)*HankelH1(nu, z)- exp(- nu*Pi*I)*sin(m*nu*Pi)*HankelH2(nu, z) Sin[\[Nu]*Pi]*HankelH1[\[Nu], z*Exp[m*Pi*I]]= - Sin[(m - 1)* \[Nu]*Pi]*HankelH1[\[Nu], z]- Exp[- \[Nu]*Pi*I]*Sin[m*\[Nu]*Pi]*HankelH2[\[Nu], z] Failure Failure
Fail
3216.976842-3084.273397*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}
-5364.683403+219295.3867*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}
-17847467.19-5404443.822*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}
-7000.832672-1549.801603*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2}
... skip entries to safe data
Fail
Complex[3216.976837863537, -3084.273404768022] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-5364.6831188620945, 219295.38712307377] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.7847467293085534*^7, -5404443.760123314] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[4.353713736263555, -84.22475855786759] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.11.E4 sin(nu*Pi)*HankelH2(nu, z*exp(m*Pi*I))= exp(nu*Pi*I)*sin(m*nu*Pi)*HankelH1(nu, z)+ sin((m + 1)* nu*Pi)*HankelH2(nu, z) Sin[\[Nu]*Pi]*HankelH2[\[Nu], z*Exp[m*Pi*I]]= Exp[\[Nu]*Pi*I]*Sin[m*\[Nu]*Pi]*HankelH1[\[Nu], z]+ Sin[(m + 1)* \[Nu]*Pi]*HankelH2[\[Nu], z] Failure Failure
Fail
-2503.040664+625.9436263*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}
5334.577216-219295.7816*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}
17848181.49+5401985.686*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}
7008.154936+1947.107340*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2}
... skip entries to safe data
Fail
Complex[-2503.040666874715, 625.9436301275297] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[5334.576217054554, -219295.782577897] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.7848181319058534*^7, 5401985.77292121] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[32720.882533131233, -8309.4971554526] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.11#Ex1 HankelH1(nu, z*exp(Pi*I))= - exp(- nu*Pi*I)*HankelH2(nu, z) HankelH1[\[Nu], z*Exp[Pi*I]]= - Exp[- \[Nu]*Pi*I]*HankelH2[\[Nu], z] Failure Failure
Fail
-53.62637626+90.06994733*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
168.4301368-8.694434719*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
-.6260433097+1.882332034*I <- {nu = 2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
1.189100470-.3259126629*I <- {nu = 2^(1/2)-I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-53.62637619369183, 90.06994740780324] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.6260433113566327, 1.8823320342787686] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.1891004703150516, 0.325912661920741] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[168.43013693288603, 8.694434886635074] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.11#Ex2 HankelH2(nu, z*exp(- Pi*I))= - exp(nu*Pi*I)*HankelH1(nu, z) HankelH2[\[Nu], z*Exp[- Pi*I]]= - Exp[\[Nu]*Pi*I]*HankelH1[\[Nu], z] Failure Failure
Fail
-.6260433097-1.882332034*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
1.189100470+.3259126629*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-53.62637626-90.06994733*I <- {nu = 2^(1/2)-I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
168.4301368+8.694434719*I <- {nu = 2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.6260433113566327, -1.8823320342787686] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-53.62637619369183, -90.06994740780324] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[168.43013693288603, -8.694434886635074] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.1891004703150516, -0.325912661920741] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.11.E6 BesselY(n, z*exp(m*Pi*I))=(- 1)^(m*n)*(BesselY(n, z)+ 2*I*m*BesselJ(n, z)) BesselY[n, z*Exp[m*Pi*I]]=(- 1)^(m*n)*(BesselY[n, z]+ 2*I*m*BesselJ[n, z]) Failure Failure -
Fail
Complex[-1.199101748008134, 3.9883106077057144] <- {Rule[m, 1], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.9168980103888886, -0.6611177226809124] <- {Rule[m, 1], Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.5768397662292734, -0.34244579398718544] <- {Rule[m, 1], Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.199101748008134, -3.9883106077057144] <- {Rule[m, 2], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.11.E7 HankelH1(n, z*exp(m*Pi*I))=(- 1)^(m*n - 1)*((m - 1)*HankelH1(n, z)+ m*HankelH2(n, z)) HankelH1[n, z*Exp[m*Pi*I]]=(- 1)^(m*n - 1)*((m - 1)*HankelH1[n, z]+ m*HankelH2[n, z]) Failure Failure
Fail
-3.988310607-1.199101751*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 1}
.6611177206+1.916898011*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 2}
.3424457937-.5768397666*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 3}
3.988310606+1.199101748*I <- {z = 2^(1/2)+I*2^(1/2), m = 2, n = 1}
... skip entries to safe data
Fail
Complex[-3.988310607705715, -1.1991017480081343] <- {Rule[m, 1], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.6611177226809126, 1.9168980103888886] <- {Rule[m, 1], Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.34244579398718544, -0.5768397662292732] <- {Rule[m, 1], Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[3.988310607705715, 1.1991017480081343] <- {Rule[m, 2], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.11.E8 HankelH2(n, z*exp(m*Pi*I))=(- 1)^(m*n)*(m*HankelH1(n, z)+(m + 1)*HankelH2(n, z)) HankelH2[n, z*Exp[m*Pi*I]]=(- 1)^(m*n)*(m*HankelH1[n, z]+(m + 1)*HankelH2[n, z]) Failure Failure
Fail
3.988310606+1.199101748*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 1}
-.6611177221-1.916898010*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 2}
-.3424457926+.5768397669*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 3}
-3.988310606-1.199101746*I <- {z = 2^(1/2)+I*2^(1/2), m = 2, n = 1}
... skip entries to safe data
Fail
Complex[3.988310607705715, 1.1991017480081343] <- {Rule[m, 1], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.6611177226809126, -1.9168980103888886] <- {Rule[m, 1], Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.34244579398718566, 0.5768397662292732] <- {Rule[m, 1], Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-3.988310607705715, -1.1991017480081343] <- {Rule[m, 2], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.12.E1 exp((1)/(2)*z*(t - (t)^(- 1)))= sum((t)^(m)* BesselJ(m, z), m = - infinity..infinity) Exp[Divide[1,2]*z*(t - (t)^(- 1))]= Sum[(t)^(m)* BesselJ[m, z], {m, - Infinity, Infinity}] Failure Successful Skip -
10.12#Ex1 cos(z*sin(theta))= BesselJ(0, z)+ 2*sum(BesselJ(2*k, z)*cos(2*k*theta), k = 1..infinity) Cos[z*Sin[\[Theta]]]= BesselJ[0, z]+ 2*Sum[BesselJ[2*k, z]*Cos[2*k*\[Theta]], {k, 1, Infinity}] Failure Successful Skip -
10.12#Ex2 sin(z*sin(theta))= 2*sum(BesselJ(2*k + 1, z)*sin((2*k + 1)* theta), k = 0..infinity) Sin[z*Sin[\[Theta]]]= 2*Sum[BesselJ[2*k + 1, z]*Sin[(2*k + 1)* \[Theta]], {k, 0, Infinity}] Failure Successful Skip -
10.12#Ex3 cos(z*cos(theta))= BesselJ(0, z)+ 2*sum((- 1)^(k)* BesselJ(2*k, z)*cos(2*k*theta), k = 1..infinity) Cos[z*Cos[\[Theta]]]= BesselJ[0, z]+ 2*Sum[(- 1)^(k)* BesselJ[2*k, z]*Cos[2*k*\[Theta]], {k, 1, Infinity}] Failure Successful Skip -
10.12#Ex4 sin(z*cos(theta))= 2*sum((- 1)^(k)* BesselJ(2*k + 1, z)*cos((2*k + 1)* theta), k = 0..infinity) Sin[z*Cos[\[Theta]]]= 2*Sum[(- 1)^(k)* BesselJ[2*k + 1, z]*Cos[(2*k + 1)* \[Theta]], {k, 0, Infinity}] Failure Successful Skip -
10.14#Ex1 abs(BesselJ(nu, x))< = 1 Abs[BesselJ[\[Nu], x]]< = 1 Failure Failure
Fail
1.148999867 <= 1. <- {nu = 2^(1/2)+I*2^(1/2), x = 3}
1.148999867 <= 1. <- {nu = 2^(1/2)-I*2^(1/2), x = 3}
14.70966635 <= 1. <- {nu = -2^(1/2)-I*2^(1/2), x = 1}
6.163423173 <= 1. <- {nu = -2^(1/2)-I*2^(1/2), x = 2}
... skip entries to safe data
Successful
10.14#Ex2 abs(BesselJ(nu, x))< = (2)^(-(1)/(2)) Abs[BesselJ[\[Nu], x]]< = (2)^(-Divide[1,2]) Failure Failure
Fail
.9422017731 <= .7071067810 <- {nu = 2^(1/2)+I*2^(1/2), x = 2}
1.148999867 <= .7071067810 <- {nu = 2^(1/2)+I*2^(1/2), x = 3}
.9422017731 <= .7071067810 <- {nu = 2^(1/2)-I*2^(1/2), x = 2}
1.148999867 <= .7071067810 <- {nu = 2^(1/2)-I*2^(1/2), x = 3}
... skip entries to safe data
Successful
10.14.E2 0 < BesselJ(nu, nu) 0 < BesselJ[\[Nu], \[Nu]] Failure Failure Successful Successful
10.14.E2 BesselJ(nu, nu)<((2)^((1)/(3)))/((3)^((2)/(3))* GAMMA((2)/(3))*(nu)^((1)/(3))) BesselJ[\[Nu], \[Nu]]<Divide[(2)^(Divide[1,3]),(3)^(Divide[2,3])* Gamma[Divide[2,3]]*(\[Nu])^(Divide[1,3])] Failure Failure Successful Successful
10.14.E3 abs(BesselJ(n, z))< = exp(abs(Im(z))) Abs[BesselJ[n, z]]< = Exp[Abs[Im[z]]] Failure Failure Successful Successful
10.14.E4 abs(BesselJ(nu, z))< =((abs((1)/(2)*z))^(nu)* exp(abs(Im(z))))/(GAMMA(nu + 1)) Abs[BesselJ[\[Nu], z]]< =Divide[(Abs[Divide[1,2]*z])^(\[Nu])* Exp[Abs[Im[z]]],Gamma[\[Nu]+ 1]] Failure Failure Successful Successful
10.14.E5 abs(BesselJ(nu, nu*x))< =((x)^(nu)* exp(nu*(1 - (x)^(2))^((1)/(2))))/((1 +(1 - (x)^(2))^((1)/(2)))^(nu)) Abs[BesselJ[\[Nu], \[Nu]*x]]< =Divide[(x)^(\[Nu])* Exp[\[Nu]*(1 - (x)^(2))^(Divide[1,2])],(1 +(1 - (x)^(2))^(Divide[1,2]))^(\[Nu])] Failure Failure Skip Successful
10.14.E6 abs(subs( temp=nu*x, diff( BesselJ(nu, temp), temp$(1) ) ))< =((1 + (x)^(2))^((1)/(4)))/(x*(2*Pi*nu)^((1)/(2)))*((x)^(nu)* exp(nu*(1 - (x)^(2))^((1)/(2))))/((1 +(1 - (x)^(2))^((1)/(2)))^(nu)) Abs[D[BesselJ[\[Nu], temp], {temp, 1}]/.temp-> \[Nu]*x]< =Divide[(1 + (x)^(2))^(Divide[1,4]),x*(2*Pi*\[Nu])^(Divide[1,2])]*Divide[(x)^(\[Nu])* Exp[\[Nu]*(1 - (x)^(2))^(Divide[1,2])],(1 +(1 - (x)^(2))^(Divide[1,2]))^(\[Nu])] Failure Failure Skip Successful
10.14.E7 1 < =(BesselJ(nu, nu*x))/((x)^(nu)* BesselJ(nu, nu)) 1 < =Divide[BesselJ[\[Nu], \[Nu]*x],(x)^(\[Nu])* BesselJ[\[Nu], \[Nu]]] Failure Failure Skip Successful
10.14.E7 (BesselJ(nu, nu*x))/((x)^(nu)* BesselJ(nu, nu))< = exp(nu*(1 - x)) Divide[BesselJ[\[Nu], \[Nu]*x],(x)^(\[Nu])* BesselJ[\[Nu], \[Nu]]]< = Exp[\[Nu]*(1 - x)] Failure Failure Skip Successful
10.14.E8 abs(BesselJ(n, n*z))< =(abs((z)^(n)* exp(n*(1 - (z)^(2))^((1)/(2)))))/((abs(1 +(1 - (z)^(2))^((1)/(2))))^(n)) Abs[BesselJ[n, n*z]]< =Divide[Abs[(z)^(n)* Exp[n*(1 - (z)^(2))^(Divide[1,2])]],(Abs[1 +(1 - (z)^(2))^(Divide[1,2])])^(n)] Failure Failure Successful Successful
10.14.E9 abs(BesselJ(n, n*z))< = 1 Abs[BesselJ[n, n*z]]< = 1 Failure Failure
Fail
1.041167208 <= 1. <- {z = 2^(1/2)+I*2^(1/2), n = 1}
2.428697298 <= 1. <- {z = 2^(1/2)+I*2^(1/2), n = 2}
6.705297847 <= 1. <- {z = 2^(1/2)+I*2^(1/2), n = 3}
1.041167208 <= 1. <- {z = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
Successful
10.15.E1 diff(BesselJ(+ nu, z), nu)= + BesselJ(+ nu, z)*ln((1)/(2)*z)-((1)/(2)*z)^(+ nu)* sum((- 1)^(k)*(Psi(k + 1 + nu))/(GAMMA(k + 1 + nu))*(((1)/(4)*(z)^(2))^(k))/(factorial(k)), k = 0..infinity) D[BesselJ[+ \[Nu], z], \[Nu]]= + BesselJ[+ \[Nu], z]*Log[Divide[1,2]*z]-(Divide[1,2]*z)^(+ \[Nu])* Sum[(- 1)^(k)*Divide[PolyGamma[k + 1 + \[Nu]],Gamma[k + 1 + \[Nu]]]*Divide[(Divide[1,4]*(z)^(2))^(k),(k)!], {k, 0, Infinity}] Failure Failure Skip Successful
10.15.E1 diff(BesselJ(- nu, z), nu)= - BesselJ(- nu, z)*ln((1)/(2)*z)+((1)/(2)*z)^(- nu)* sum((- 1)^(k)*(Psi(k + 1 - nu))/(GAMMA(k + 1 - nu))*(((1)/(4)*(z)^(2))^(k))/(factorial(k)), k = 0..infinity) D[BesselJ[- \[Nu], z], \[Nu]]= - BesselJ[- \[Nu], z]*Log[Divide[1,2]*z]+(Divide[1,2]*z)^(- \[Nu])* Sum[(- 1)^(k)*Divide[PolyGamma[k + 1 - \[Nu]],Gamma[k + 1 - \[Nu]]]*Divide[(Divide[1,4]*(z)^(2))^(k),(k)!], {k, 0, Infinity}] Failure Failure Skip Successful
10.15.E2 diff(BesselY(nu, z), nu)= cot(nu*Pi)*(diff(BesselJ(nu, z), nu)- Pi*BesselY(nu, z))- csc(nu*Pi)*diff(BesselJ(- nu, z), nu)- Pi*BesselJ(nu, z) D[BesselY[\[Nu], z], \[Nu]]= Cot[\[Nu]*Pi]*(D[BesselJ[\[Nu], z], \[Nu]]- Pi*BesselY[\[Nu], z])- Csc[\[Nu]*Pi]*D[BesselJ[- \[Nu], z], \[Nu]]- Pi*BesselJ[\[Nu], z] Successful Failure - Successful
10.16#Ex1 BesselJ((1)/(2), z)= BesselY(-(1)/(2), z) BesselJ[Divide[1,2], z]= BesselY[-Divide[1,2], z] Successful Successful - -
10.16#Ex1 BesselY(-(1)/(2), z)=((2)/(Pi*z))^((1)/(2))* sin(z) BesselY[-Divide[1,2], z]=(Divide[2,Pi*z])^(Divide[1,2])* Sin[z] Failure Failure Successful Successful
10.16#Ex2 BesselJ(-(1)/(2), z)= - BesselY((1)/(2), z) BesselJ[-Divide[1,2], z]= - BesselY[Divide[1,2], z] Successful Successful - -
10.16#Ex2 - BesselY((1)/(2), z)=((2)/(Pi*z))^((1)/(2))* cos(z) - BesselY[Divide[1,2], z]=(Divide[2,Pi*z])^(Divide[1,2])* Cos[z] Failure Failure Successful Successful
10.16#Ex3 HankelH1((1)/(2), z)= - I*HankelH1(-(1)/(2), z) HankelH1[Divide[1,2], z]= - I*HankelH1[-Divide[1,2], z] Successful Successful - -
10.16#Ex3 - I*HankelH1(-(1)/(2), z)= - I*((2)/(Pi*z))^((1)/(2))* exp(I*z) - I*HankelH1[-Divide[1,2], z]= - I*(Divide[2,Pi*z])^(Divide[1,2])* Exp[I*z] Failure Failure Successful Successful
10.16#Ex4 HankelH2((1)/(2), z)= I*HankelH2(-(1)/(2), z) HankelH2[Divide[1,2], z]= I*HankelH2[-Divide[1,2], z] Successful Successful - -
10.16#Ex4 I*HankelH2(-(1)/(2), z)= I*((2)/(Pi*z))^((1)/(2))* exp(- I*z) I*HankelH2[-Divide[1,2], z]= I*(Divide[2,Pi*z])^(Divide[1,2])* Exp[- I*z] Failure Failure Successful Successful
10.16.E5 BesselJ(nu, z)=(((1)/(2)*z)^(nu)* exp(- I*z))/(GAMMA(nu + 1))*KummerM(nu +(1)/(2), 2*nu + 1, + 2*I*z) BesselJ[\[Nu], z]=Divide[(Divide[1,2]*z)^(\[Nu])* Exp[- I*z],Gamma[\[Nu]+ 1]]*Hypergeometric1F1[\[Nu]+Divide[1,2], 2*\[Nu]+ 1, + 2*I*z] Failure Successful Successful -
10.16.E5 BesselJ(nu, z)=(((1)/(2)*z)^(nu)* exp(+ I*z))/(GAMMA(nu + 1))*KummerM(nu +(1)/(2), 2*nu + 1, - 2*I*z) BesselJ[\[Nu], z]=Divide[(Divide[1,2]*z)^(\[Nu])* Exp[+ I*z],Gamma[\[Nu]+ 1]]*Hypergeometric1F1[\[Nu]+Divide[1,2], 2*\[Nu]+ 1, - 2*I*z] Failure Successful Successful -
10.16.E7 BesselJ(nu, z)=(exp(-(2*nu + 1)* Pi*I/ 4))/((2)^(2*nu)* GAMMA(nu + 1))*(2*z)^(-(1)/(2))* WhittakerM(0, nu, + 2*I*z) BesselJ[\[Nu], z]=Divide[Exp[-(2*\[Nu]+ 1)* Pi*I/ 4],(2)^(2*\[Nu])* Gamma[\[Nu]+ 1]]*(2*z)^(-Divide[1,2])* WhittakerM[0, \[Nu], + 2*I*z] Failure Failure
Fail
1.497834930+.9457736110*I <- {z = -2^(1/2)+I*2^(1/2), nu = 1/4}
Fail
Complex[1.4978349305286898, 0.945773612365157] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Rational[1, 4]]}
10.16.E7 BesselJ(nu, z)=(exp(+(2*nu + 1)* Pi*I/ 4))/((2)^(2*nu)* GAMMA(nu + 1))*(2*z)^(-(1)/(2))* WhittakerM(0, nu, - 2*I*z) BesselJ[\[Nu], z]=Divide[Exp[+(2*\[Nu]+ 1)* Pi*I/ 4],(2)^(2*\[Nu])* Gamma[\[Nu]+ 1]]*(2*z)^(-Divide[1,2])* WhittakerM[0, \[Nu], - 2*I*z] Failure Failure
Fail
1.497834930-.9457736110*I <- {z = -2^(1/2)-I*2^(1/2), nu = 1/4}
Fail
Complex[1.4978349305286898, -0.945773612365157] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Rational[1, 4]]}
10.16.E9 BesselJ(nu, z)=(((1)/(2)*z)^(nu))/(GAMMA(nu + 1))*hypergeom([-], [nu + 1], -(1)/(4)*(z)^(2)) BesselJ[\[Nu], z]=Divide[(Divide[1,2]*z)^(\[Nu]),Gamma[\[Nu]+ 1]]*HypergeometricPFQ[{-}, {\[Nu]+ 1}, -Divide[1,4]*(z)^(2)] Error Failure - Error
10.17.E7 (z)^((1)/(2))= exp((1)/(2)*ln(abs(z))+(1)/(2)*I*argument(z)) (z)^(Divide[1,2])= Exp[Divide[1,2]*Log[Abs[z]]+Divide[1,2]*I*Arg[z]] Failure Failure Successful Successful
10.17.E16 (exp(z)/(2*Pi))*GAMMA(p)*GAMMA(1-p,z)=(exp(z))/(2*Pi)*GAMMA(p)*GAMMA(1 - p, z) Error Successful Error - -
10.17.E17 (R[ell])^(+)*(nu , z)=(- 1)^(ell)* 2*cos(nu*Pi)(sum((+ I)^(k)*(a[k]*(nu))/((z)^(k))*(exp(- 2*I*z)/(2*Pi))*GAMMA(ell - k)*GAMMA(1-ell - k,- 2*I*z), k = 0..m - 1)+ R(R[m , ell])^(+)*(nu , z)) Error Error Error - -
10.17.E17 (R[ell])^(-)*(nu , z)=(- 1)^(ell)* 2*cos(nu*Pi)(sum((- I)^(k)*(a[k]*(nu))/((z)^(k))*(exp(+ 2*I*z)/(2*Pi))*GAMMA(ell - k)*GAMMA(1-ell - k,+ 2*I*z), k = 0..m - 1)+ R(R[m , ell])^(-)*(nu , z)) Error Error Error - -
10.19#Ex11 LegendreP(0, a)= 1 LegendreP[0, 0, 3, a]= 1 Successful Successful - -
10.19#Ex12 LegendreP(1, a)= -(1)/(5)*a LegendreP[1, 0, 3, a]= -Divide[1,5]*a Failure Failure
Fail
1.697056274+1.697056274*I <- {a = 2^(1/2)+I*2^(1/2)}
1.697056274-1.697056274*I <- {a = 2^(1/2)-I*2^(1/2)}
-1.697056274-1.697056274*I <- {a = -2^(1/2)-I*2^(1/2)}
-1.697056274+1.697056274*I <- {a = -2^(1/2)+I*2^(1/2)}
Fail
Complex[1.697056274847714, 1.697056274847714] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.697056274847714, -1.697056274847714] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.697056274847714, -1.697056274847714] <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.697056274847714, 1.697056274847714] <- {Rule[a, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
10.19#Ex13 LegendreP(2, a)= -(9)/(100)*(a)^(5)+(3)/(35)*(a)^(2) LegendreP[2, 0, 3, a]= -Divide[9,100]*(a)^(5)+Divide[3,35]*(a)^(2) Failure Failure
Fail
-2.536467527+3.620675327*I <- {a = 2^(1/2)+I*2^(1/2)}
-2.536467527-3.620675327*I <- {a = 2^(1/2)-I*2^(1/2)}
1.536467527+7.693610381*I <- {a = -2^(1/2)-I*2^(1/2)}
1.536467527-7.693610381*I <- {a = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-2.5364675298172568, 3.6206753273256007] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-2.5364675298172568, -3.6206753273256007] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.5364675298172568, 7.693610386960114] <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.5364675298172568, -7.693610386960114] <- {Rule[a, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
10.19#Ex14 LegendreP(3, a)=(957)/(7000)*(a)^(6)-(173)/(3150)*(a)^(3)-(1)/(225) LegendreP[3, 0, 3, a]=Divide[957,7000]*(a)^(6)-Divide[173,3150]*(a)^(3)-Divide[1,225] Failure Failure
Fail
-16.56968955+21.08120757*I <- {a = 2^(1/2)+I*2^(1/2)}
-16.56968955-21.08120757*I <- {a = 2^(1/2)-I*2^(1/2)}
16.57857843-3.581779026*I <- {a = -2^(1/2)-I*2^(1/2)}
16.57857843+3.581779026*I <- {a = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-16.569689549881762, 21.081207592921206] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-16.569689549881762, -21.081207592921206] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[16.578578438770652, -3.5817790214926353] <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[16.578578438770652, 3.5817790214926353] <- {Rule[a, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
10.19#Ex15 LegendreP(4, a)=(27)/(20000)*(a)^(10)-(23573)/(147000)*(a)^(7)+(5903)/(138600)*(a)^(4)+(947)/(346500)*a LegendreP[4, 0, 3, a]=Divide[27,20000]*(a)^(10)-Divide[23573,147000]*(a)^(7)+Divide[5903,138600]*(a)^(4)+Divide[947,346500]*a Failure Failure
Fail
-54.43324245-30.90044469*I <- {a = 2^(1/2)+I*2^(1/2)}
-54.43324245+30.90044469*I <- {a = 2^(1/2)-I*2^(1/2)}
-83.45387141-1.864355291*I <- {a = -2^(1/2)-I*2^(1/2)}
-83.45387141+1.864355291*I <- {a = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-54.4332424913453, -30.900444725067786] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-54.4332424913453, 30.900444725067786] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-83.4538715057687, -1.8643552749322136] <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-83.4538715057687, 1.8643552749322136] <- {Rule[a, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
10.19#Ex16 LegendreQ(0, a)=(3)/(10)*(a)^(2) LegendreQ[0, 0, 3, a]=Divide[3,10]*(a)^(2) Failure Failure
Fail
.3205774699-1.577984704*I <- {a = 2^(1/2)+I*2^(1/2)}
.3205774699+1.577984704*I <- {a = 2^(1/2)-I*2^(1/2)}
-.3205774699-.8220152937*I <- {a = -2^(1/2)-I*2^(1/2)}
-.3205774699+.8220152937*I <- {a = -2^(1/2)+I*2^(1/2)}
Fail
Complex[0.3205774698651409, -1.5779847052119538] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.3205774698651409, 1.5779847052119538] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.3205774698651409, -0.8220152947880462] <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.3205774698651409, 0.8220152947880462] <- {Rule[a, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
10.19#Ex17 LegendreQ(1, a)= -(17)/(70)*(a)^(3)+(1)/(70) LegendreQ[1, 0, 3, a]= -Divide[17,70]*(a)^(3)+Divide[1,70] Failure Failure
Fail
-1.400177072+1.292621369*I <- {a = 2^(1/2)+I*2^(1/2)}
-1.400177072-1.292621369*I <- {a = 2^(1/2)-I*2^(1/2)}
1.347437848-1.454993551*I <- {a = -2^(1/2)-I*2^(1/2)}
1.347437848+1.454993551*I <- {a = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-1.4001770727218448, 1.2926213697851998] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.4001770727218448, -1.2926213697851998] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.3474378484601681, -1.4549935513968135] <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.3474378484601681, 1.4549935513968135] <- {Rule[a, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
10.19#Ex18 LegendreQ(2, a)= -(9)/(1000)*(a)^(7)+(611)/(3150)*(a)^(4)-(37)/(3150)*a LegendreQ[2, 0, 3, a]= -Divide[9,1000]*(a)^(7)+Divide[611,3150]*(a)^(4)-Divide[37,3150]*a Failure Failure
Fail
3.920989620-.8068387848*I <- {a = 2^(1/2)+I*2^(1/2)}
3.920989620+.8068387848*I <- {a = 2^(1/2)-I*2^(1/2)}
2.285994500+.8068387848*I <- {a = -2^(1/2)-I*2^(1/2)}
2.285994500-.8068387848*I <- {a = -2^(1/2)+I*2^(1/2)}
Fail
Complex[3.920989625597779, -0.8068387862904207] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[3.920989625597779, 0.8068387862904207] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.285994501386349, 0.8068387862904207] <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.285994501386349, -0.8068387862904207] <- {Rule[a, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
10.19#Ex19 LegendreQ(3, a)=(549)/(28000)*(a)^(8)-(110767)/(693000)*(a)^(5)+(79)/(12375)*(a)^(2) LegendreQ[3, 0, 3, a]=Divide[549,28000]*(a)^(8)-Divide[110767,693000]*(a)^(5)+Divide[79,12375]*(a)^(2) Failure Failure
Fail
-8.639472248-3.641292303*I <- {a = 2^(1/2)+I*2^(1/2)}
-8.639472248+3.641292303*I <- {a = 2^(1/2)-I*2^(1/2)}
-1.406078034+3.592101911*I <- {a = -2^(1/2)-I*2^(1/2)}
-1.406078034-3.592101911*I <- {a = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-8.639472261933392, -3.641292307775128] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-8.639472261933392, 3.641292307775128] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4060780379389062, 3.592101916219356] <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.4060780379389062, -3.592101916219356] <- {Rule[a, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
10.20.E1 (diff(zeta, z))^(2)=(1 - (z)^(2))/(zeta*(z)^(2)) (D[\[zeta], z])^(2)=Divide[1 - (z)^(2),\[zeta]*(z)^(2)] Failure Failure
Fail
.4419417384-.2651650430*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)}
.2651650430+.4419417384*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)}
-.4419417384+.2651650430*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)}
-.2651650430-.4419417384*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Error
10.20.E2 (2)/(3)*(zeta)^((3)/(2))= int((sqrt(1 - (t)^(2)))/(t), t = z..1) Divide[2,3]*(\[zeta])^(Divide[3,2])= Integrate[Divide[Sqrt[1 - (t)^(2)],t], {t, z, 1}] Error Failure - Error
10.20.E2 int((sqrt(1 - (t)^(2)))/(t), t = z..1)= ln((1 +sqrt(1 - (z)^(2)))/(z))-sqrt(1 - (z)^(2)) Integrate[Divide[Sqrt[1 - (t)^(2)],t], {t, z, 1}]= Log[Divide[1 +Sqrt[1 - (z)^(2)],z]]-Sqrt[1 - (z)^(2)] Error Failure - Error
10.20.E3 (2)/(3)*(- zeta)^((3)/(2))= int((sqrt((t)^(2)- 1))/(t), t = 1..z) Divide[2,3]*(- \[zeta])^(Divide[3,2])= Integrate[Divide[Sqrt[(t)^(2)- 1],t], {t, 1, z}] Failure Failure Skip Error
10.20.E3 int((sqrt((t)^(2)- 1))/(t), t = 1..z)=sqrt((z)^(2)- 1)- arcsec(z) Integrate[Divide[Sqrt[(t)^(2)- 1],t], {t, 1, z}]=Sqrt[(z)^(2)- 1]- ArcSec[z] Failure Failure Skip Error
10.20.E17 z = +(tau*coth(tau)- (tau)^(2))^((1)/(2))+ I*((tau)^(2)- tau*tanh(tau))^((1)/(2)) z = +(\[Tau]*Coth[\[Tau]]- (\[Tau])^(2))^(Divide[1,2])+ I*((\[Tau])^(2)- \[Tau]*Tanh[\[Tau]])^(Divide[1,2]) Failure Failure Skip Successful
10.20.E17 z = -(tau*coth(tau)- (tau)^(2))^((1)/(2))- I*((tau)^(2)- tau*tanh(tau))^((1)/(2)) z = -(\[Tau]*Coth[\[Tau]]- (\[Tau])^(2))^(Divide[1,2])- I*((\[Tau])^(2)- \[Tau]*Tanh[\[Tau]])^(Divide[1,2]) Failure Failure Skip Successful
10.21.E11 2*(rho[nu])^(2)*diff(rho[nu], t)*diff(rho[nu], [t$(3)])- 3*(rho[nu])^(2)*(diff(rho[nu], [t$(2)]))^(2)- 4*(Pi)^(2)* (rho[nu])^(2)*(diff(rho[nu], t))^(2)(4*rho(rho[nu])^(2)+ 1 - 4*(nu)^(2))*(diff(rho[nu], t))^(4)= 0 2*(Subscript[\[Rho], \[Nu]])^(2)*D[Subscript[\[Rho], \[Nu]], t]*D[Subscript[\[Rho], \[Nu]], {t, 3}]- 3*(Subscript[\[Rho], \[Nu]])^(2)*(D[Subscript[\[Rho], \[Nu]], {t, 2}])^(2)- 4*(Pi)^(2)* (Subscript[\[Rho], \[Nu]])^(2)*(D[Subscript[\[Rho], \[Nu]], t])^(2)(4*\[Rho](Subscript[\[Rho], \[Nu]])^(2)+ 1 - 4*(\[Nu])^(2))*(D[Subscript[\[Rho], \[Nu]], t])^(4)= 0 Successful Successful - -
10.21.E17 diff(c, nu)= 2*c*int(BesselK(0, 2*c*sinh(t))*exp(- 2*nu*t), t = 0..infinity) D[c, \[Nu]]= 2*c*Integrate[BesselK[0, 2*c*Sinh[t]]*Exp[- 2*\[Nu]*t], {t, 0, Infinity}] Failure Failure Skip Error
10.21.E46 a =(1)/(2)*ln(3) a =Divide[1,2]*Log[3] Failure Failure
Fail
.8649074175+1.414213562*I <- {a = 2^(1/2)+I*2^(1/2)}
.8649074175-1.414213562*I <- {a = 2^(1/2)-I*2^(1/2)}
-1.963519706-1.414213562*I <- {a = -2^(1/2)-I*2^(1/2)}
-1.963519706+1.414213562*I <- {a = -2^(1/2)+I*2^(1/2)}
Fail
Complex[0.8649074180390403, 1.4142135623730951] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.8649074180390403, -1.4142135623730951] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.96351970670715, -1.4142135623730951] <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.96351970670715, 1.4142135623730951] <- {Rule[a, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
10.22.E8 int(BesselJ(nu, t), t = 0..x)= 2*sum(BesselJ(nu + 2*k + 1, x), k = 0..infinity) Integrate[BesselJ[\[Nu], t], {t, 0, x}]= 2*Sum[BesselJ[\[Nu]+ 2*k + 1, x], {k, 0, Infinity}] Failure Failure Skip Skip
10.22.E9 int(BesselJ(2*n, t), t = 0..x)= int(BesselJ(0, t), t = 0..x)- 2*sum(BesselJ(2*k + 1, x), k = 0..n - 1), int(BesselJ(2*n + 1, t), t = 0..x) Integrate[BesselJ[2*n, t], {t, 0, x}]= Integrate[BesselJ[0, t], {t, 0, x}]- 2*Sum[BesselJ[2*k + 1, x], {k, 0, n - 1}], Integrate[BesselJ[2*n + 1, t], {t, 0, x}] Error Failure - Error
10.22.E9 int(BesselJ(0, t), t = 0..x)- 2*sum(BesselJ(2*k + 1, x), k = 0..n - 1), int(BesselJ(2*n + 1, t), t = 0..x)= 1 - BesselJ(0, x)- 2*sum(BesselJ(2*k, x), k = 1..n) Integrate[BesselJ[0, t], {t, 0, x}]- 2*Sum[BesselJ[2*k + 1, x], {k, 0, n - 1}], Integrate[BesselJ[2*n + 1, t], {t, 0, x}]= 1 - BesselJ[0, x]- 2*Sum[BesselJ[2*k, x], {k, 1, n}] Error Failure - Error
10.22.E10 int((t)^(mu)* BesselJ(nu, t), t = 0..x)= (x)^(mu)*(GAMMA((1)/(2)*nu +(1)/(2)*mu +(1)/(2)))/(GAMMA((1)/(2)*nu -(1)/(2)*mu +(1)/(2)))* sum(((nu + 2*k + 1)* GAMMA((1)/(2)*nu -(1)/(2)*mu +(1)/(2)+ k))/(GAMMA((1)/(2)*nu +(1)/(2)*mu +(3)/(2)+ k))*BesselJ(nu + 2*k + 1, x), k = 0..infinity) Integrate[(t)^(\[Mu])* BesselJ[\[Nu], t], {t, 0, x}]= (x)^(\[Mu])*Divide[Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2]],Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]]]* Sum[Divide[(\[Nu]+ 2*k + 1)* Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]+ k],Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[3,2]+ k]]*BesselJ[\[Nu]+ 2*k + 1, x], {k, 0, Infinity}] Failure Failure Skip Successful
10.22.E11 int((1 - BesselJ(0, t))/(t), t = 0..x)=(1)/(2)*sum((Psi(k + 1)- Psi(1))/(factorial(k))*((1)/(2)*x)^(k)* BesselJ(k, x), k = 1..infinity) Integrate[Divide[1 - BesselJ[0, t],t], {t, 0, x}]=Divide[1,2]*Sum[Divide[PolyGamma[k + 1]- PolyGamma[1],(k)!]*(Divide[1,2]*x)^(k)* BesselJ[k, x], {k, 1, Infinity}] Failure Failure Skip Skip
10.22.E13 int(BesselJ(2*nu, 2*z*cos(theta))*cos(2*mu*theta), theta = 0..(1)/(2)*Pi)=(1)/(2)*Pi*BesselJ(nu + mu, z)*BesselJ(nu - mu, z) Integrate[BesselJ[2*\[Nu], 2*z*Cos[\[Theta]]]*Cos[2*\[Mu]*\[Theta]], {\[Theta], 0, Divide[1,2]*Pi}]=Divide[1,2]*Pi*BesselJ[\[Nu]+ \[Mu], z]*BesselJ[\[Nu]- \[Mu], z] Failure Failure Skip Skip
10.22.E14 int(BesselJ(2*nu, 2*z*sin(theta))*cos(2*mu*theta), theta = 0..Pi)= Pi*cos(mu*Pi)*BesselJ(nu + mu, z)*BesselJ(nu - mu, z) Integrate[BesselJ[2*\[Nu], 2*z*Sin[\[Theta]]]*Cos[2*\[Mu]*\[Theta]], {\[Theta], 0, Pi}]= Pi*Cos[\[Mu]*Pi]*BesselJ[\[Nu]+ \[Mu], z]*BesselJ[\[Nu]- \[Mu], z] Failure Failure Skip Skip
10.22.E15 int(BesselJ(2*nu, 2*z*sin(theta))*sin(2*mu*theta), theta = 0..Pi)= Pi*sin(mu*Pi)*BesselJ(nu + mu, z)*BesselJ(nu - mu, z) Integrate[BesselJ[2*\[Nu], 2*z*Sin[\[Theta]]]*Sin[2*\[Mu]*\[Theta]], {\[Theta], 0, Pi}]= Pi*Sin[\[Mu]*Pi]*BesselJ[\[Nu]+ \[Mu], z]*BesselJ[\[Nu]- \[Mu], z] Failure Failure Skip Skip
10.22.E16 int(BesselJ(0, 2*z*sin(theta))*cos(2*n*theta), theta = 0..(1)/(2)*Pi)=(1)/(2)*Pi*(BesselJ(n, z))^(2) Integrate[BesselJ[0, 2*z*Sin[\[Theta]]]*Cos[2*n*\[Theta]], {\[Theta], 0, Divide[1,2]*Pi}]=Divide[1,2]*Pi*(BesselJ[n, z])^(2) Failure Failure Skip Successful
10.22.E17 int(BesselY(2*nu, 2*z*cos(theta))*cos(2*mu*theta), theta = 0..(1)/(2)*Pi)=(1)/(2)*Pi*cot(2*nu*Pi)*BesselJ(nu + mu, z)*BesselJ(nu - mu, z)-(1)/(2)*Pi*csc(2*nu*Pi)*BesselJ(mu - nu, z)*BesselJ(- mu - nu, z) Integrate[BesselY[2*\[Nu], 2*z*Cos[\[Theta]]]*Cos[2*\[Mu]*\[Theta]], {\[Theta], 0, Divide[1,2]*Pi}]=Divide[1,2]*Pi*Cot[2*\[Nu]*Pi]*BesselJ[\[Nu]+ \[Mu], z]*BesselJ[\[Nu]- \[Mu], z]-Divide[1,2]*Pi*Csc[2*\[Nu]*Pi]*BesselJ[\[Mu]- \[Nu], z]*BesselJ[- \[Mu]- \[Nu], z] Failure Failure Skip Successful
10.22.E18 int(BesselY(0, 2*z*sin(theta))*cos(2*n*theta), theta = 0..(1)/(2)*Pi)=(1)/(2)*Pi*BesselJ(n, z)*BesselY(n, z) Integrate[BesselY[0, 2*z*Sin[\[Theta]]]*Cos[2*n*\[Theta]], {\[Theta], 0, Divide[1,2]*Pi}]=Divide[1,2]*Pi*BesselJ[n, z]*BesselY[n, z] Failure Failure Skip Successful
10.22.E19 int(BesselJ(mu, z*sin(theta))*(sin(theta))^(mu + 1)*(cos(theta))^(2*nu + 1), theta = 0..(1)/(2)*Pi)= (2)^(nu)* GAMMA(nu + 1)*(z)^(- nu - 1)* BesselJ(mu + nu + 1, z) Integrate[BesselJ[\[Mu], z*Sin[\[Theta]]]*(Sin[\[Theta]])^(\[Mu]+ 1)*(Cos[\[Theta]])^(2*\[Nu]+ 1), {\[Theta], 0, Divide[1,2]*Pi}]= (2)^(\[Nu])* Gamma[\[Nu]+ 1]*(z)^(- \[Nu]- 1)* BesselJ[\[Mu]+ \[Nu]+ 1, z] Successful Failure - Error
10.22.E20 int(BesselJ(mu, z*sin(theta))*(sin(theta))^(mu)*(cos(theta))^(2*mu), theta = 0..(1)/(2)*Pi)= (Pi)^((1)/(2))* (2)^(mu - 1)* (z)^(- mu)* GAMMA(mu +(1)/(2))*(BesselJ(mu, (1)/(2)*z))^(2) Integrate[BesselJ[\[Mu], z*Sin[\[Theta]]]*(Sin[\[Theta]])^(\[Mu])*(Cos[\[Theta]])^(2*\[Mu]), {\[Theta], 0, Divide[1,2]*Pi}]= (Pi)^(Divide[1,2])* (2)^(\[Mu]- 1)* (z)^(- \[Mu])* Gamma[\[Mu]+Divide[1,2]]*(BesselJ[\[Mu], Divide[1,2]*z])^(2) Successful Failure - Error
10.22.E21 int(BesselY(mu, z*sin(theta))*(sin(theta))^(mu)*(cos(theta))^(2*mu), theta = 0..(1)/(2)*Pi)= (Pi)^((1)/(2))* (2)^(mu - 1)* (z)^(- mu)* GAMMA(mu +(1)/(2))*BesselJ(mu, (1)/(2)*z)*BesselY(mu, (1)/(2)*z) Integrate[BesselY[\[Mu], z*Sin[\[Theta]]]*(Sin[\[Theta]])^(\[Mu])*(Cos[\[Theta]])^(2*\[Mu]), {\[Theta], 0, Divide[1,2]*Pi}]= (Pi)^(Divide[1,2])* (2)^(\[Mu]- 1)* (z)^(- \[Mu])* Gamma[\[Mu]+Divide[1,2]]*BesselJ[\[Mu], Divide[1,2]*z]*BesselY[\[Mu], Divide[1,2]*z] Successful Failure - Error
10.22.E22 int(BesselJ(mu, z*(sin(theta))^(2))*BesselJ(nu, z*(cos(theta))^(2))*(sin(theta))^(2*mu + 1)*(cos(theta))^(2*nu + 1), theta = 0..(1)/(2)*Pi)=(GAMMA(mu +(1)/(2))*GAMMA(nu +(1)/(2))*BesselJ(mu + nu +(1)/(2), z))/((8*Pi*z)^((1)/(2))* GAMMA(mu + nu + 1)) Integrate[BesselJ[\[Mu], z*(Sin[\[Theta]])^(2)]*BesselJ[\[Nu], z*(Cos[\[Theta]])^(2)]*(Sin[\[Theta]])^(2*\[Mu]+ 1)*(Cos[\[Theta]])^(2*\[Nu]+ 1), {\[Theta], 0, Divide[1,2]*Pi}]=Divide[Gamma[\[Mu]+Divide[1,2]]*Gamma[\[Nu]+Divide[1,2]]*BesselJ[\[Mu]+ \[Nu]+Divide[1,2], z],(8*Pi*z)^(Divide[1,2])* Gamma[\[Mu]+ \[Nu]+ 1]] Error Failure - Error
10.22.E23 int(BesselJ(mu, z*(sin(theta))^(2))*BesselJ(nu, z*(cos(theta))^(2))*(sin(theta))^(2*alpha - 1)* sec(theta), theta = 0..(1)/(2)*Pi)=((mu + nu + alpha)* GAMMA(mu + alpha)*(2)^(alpha - 1))/(nu*GAMMA(mu + 1)*(z)^(alpha))*BesselJ(mu + nu + alpha, z) Integrate[BesselJ[\[Mu], z*(Sin[\[Theta]])^(2)]*BesselJ[\[Nu], z*(Cos[\[Theta]])^(2)]*(Sin[\[Theta]])^(2*\[Alpha]- 1)* Sec[\[Theta]], {\[Theta], 0, Divide[1,2]*Pi}]=Divide[(\[Mu]+ \[Nu]+ \[Alpha])* Gamma[\[Mu]+ \[Alpha]]*(2)^(\[Alpha]- 1),\[Nu]*Gamma[\[Mu]+ 1]*(z)^(\[Alpha])]*BesselJ[\[Mu]+ \[Nu]+ \[Alpha], z] Failure Failure Skip Error
10.22.E24 int(BesselJ(mu, z*(sin(theta))^(2))*BesselJ(nu, z*(cos(theta))^(2))*cot(theta), theta = 0..(1)/(2)*Pi)=(1)/(2)*(mu)^(- 1)* BesselJ(mu + nu, z) Integrate[BesselJ[\[Mu], z*(Sin[\[Theta]])^(2)]*BesselJ[\[Nu], z*(Cos[\[Theta]])^(2)]*Cot[\[Theta]], {\[Theta], 0, Divide[1,2]*Pi}]=Divide[1,2]*(\[Mu])^(- 1)* BesselJ[\[Mu]+ \[Nu], z] Failure Failure Skip Error
10.22.E25 int(BesselJ(mu, z*sin(theta))*BesselI(nu, z*cos(theta))*(tan(theta))^(mu + 1), theta = 0..(1)/(2)*Pi)=(GAMMA((1)/(2)*nu -(1)/(2)*mu)*((1)/(2)*z)^(mu))/(2*GAMMA((1)/(2)*nu +(1)/(2)*mu + 1))*BesselJ(nu, z) Integrate[BesselJ[\[Mu], z*Sin[\[Theta]]]*BesselI[\[Nu], z*Cos[\[Theta]]]*(Tan[\[Theta]])^(\[Mu]+ 1), {\[Theta], 0, Divide[1,2]*Pi}]=Divide[Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]]*(Divide[1,2]*z)^(\[Mu]),2*Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+ 1]]*BesselJ[\[Nu], z] Failure Failure Skip Error
10.22.E27 int(t*(BesselJ(nu - 1, t))^(2), t = 0..x)= 2*sum((nu + 2*k)* (BesselJ(nu + 2*k, x))^(2), k = 0..infinity) Integrate[t*(BesselJ[\[Nu]- 1, t])^(2), {t, 0, x}]= 2*Sum[(\[Nu]+ 2*k)* (BesselJ[\[Nu]+ 2*k, x])^(2), {k, 0, Infinity}] Failure Failure Skip Successful
10.22.E28 int(t*((BesselJ(nu - 1, t))^(2)- (BesselJ(nu + 1, t))^(2)), t = 0..x)= 2*nu*(BesselJ(nu, x))^(2) Integrate[t*((BesselJ[\[Nu]- 1, t])^(2)- (BesselJ[\[Nu]+ 1, t])^(2)), {t, 0, x}]= 2*\[Nu]*(BesselJ[\[Nu], x])^(2) Successful Failure - Skip
10.22.E29 int(t*(BesselJ(0, t))^(2), t = 0..x)=(1)/(2)*(x)^(2)*((BesselJ(0, x))^(2)+ (BesselJ(1, x))^(2)) Integrate[t*(BesselJ[0, t])^(2), {t, 0, x}]=Divide[1,2]*(x)^(2)*((BesselJ[0, x])^(2)+ (BesselJ[1, x])^(2)) Successful Successful - -
10.22.E30 int(BesselJ(n, t)*BesselJ(n + 1, t), t = 0..x)=(1)/(2)*(1 - (BesselJ(0, x))^(2))- sum((BesselJ(k, x))^(2), k = 1..n) Integrate[BesselJ[n, t]*BesselJ[n + 1, t], {t, 0, x}]=Divide[1,2]*(1 - (BesselJ[0, x])^(2))- Sum[(BesselJ[k, x])^(2), {k, 1, n}] Failure Failure Skip Successful
10.22.E30 (1)/(2)*(1 - (BesselJ(0, x))^(2))- sum((BesselJ(k, x))^(2), k = 1..n)= sum((BesselJ(k, x))^(2), k = n + 1..infinity) Divide[1,2]*(1 - (BesselJ[0, x])^(2))- Sum[(BesselJ[k, x])^(2), {k, 1, n}]= Sum[(BesselJ[k, x])^(2), {k, n + 1, Infinity}] Failure Failure Skip Successful
10.22.E31 int(BesselJ(mu, t)*BesselJ(nu, x - t), t = 0..x)= 2*sum((- 1)^(k)* BesselJ(mu + nu + 2*k + 1, x), k = 0..infinity) Integrate[BesselJ[\[Mu], t]*BesselJ[\[Nu], x - t], {t, 0, x}]= 2*Sum[(- 1)^(k)* BesselJ[\[Mu]+ \[Nu]+ 2*k + 1, x], {k, 0, Infinity}] Failure Failure Skip
Fail
Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[μ, Rational[-1, 2]], Rule[ν, Rational[-1, 2]], Rule[Integrate[Times[BesselJ[μ, t], BesselJ[ν, Plus[Times[-1, t], x]]], {t, 0, x}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], BesselJ[Plus[1, Times[2, k], μ, ν], x]], {k, 0, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.4142135623730951, 4.242640687119286] <- {Rule[μ, Rational[-1, 2]], Rule[ν, Rational[-1, 2]], Rule[Integrate[Times[BesselJ[μ, t], BesselJ[ν, Plus[Times[-1, t], x]]], {t, 0, x}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], BesselJ[Plus[1, Times[2, k], μ, ν], x]], {k, 0, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[4.242640687119286, 4.242640687119286] <- {Rule[μ, Rational[-1, 2]], Rule[ν, Rational[-1, 2]], Rule[Integrate[Times[BesselJ[μ, t], BesselJ[ν, Plus[Times[-1, t], x]]], {t, 0, x}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], BesselJ[Plus[1, Times[2, k], μ, ν], x]], {k, 0, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[4.242640687119286, -1.4142135623730951] <- {Rule[μ, Rational[-1, 2]], Rule[ν, Rational[-1, 2]], Rule[Integrate[Times[BesselJ[μ, t], BesselJ[ν, Plus[Times[-1, t], x]]], {t, 0, x}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], BesselJ[Plus[1, Times[2, k], μ, ν], x]], {k, 0, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.22.E32 int(BesselJ(nu, t)*BesselJ(1 - nu, x - t), t = 0..x)= BesselJ(0, x)- cos(x) Integrate[BesselJ[\[Nu], t]*BesselJ[1 - \[Nu], x - t], {t, 0, x}]= BesselJ[0, x]- Cos[x] Failure Failure Skip Successful
10.22.E33 int(BesselJ(nu, t)*BesselJ(- nu, x - t), t = 0..x)= sin(x) Integrate[BesselJ[\[Nu], t]*BesselJ[- \[Nu], x - t], {t, 0, x}]= Sin[x] Failure Failure Skip Error
10.22.E34 int((t)^(- 1)* BesselJ(mu, t)*BesselJ(nu, x - t), t = 0..x)=(BesselJ(mu + nu, x))/(mu) Integrate[(t)^(- 1)* BesselJ[\[Mu], t]*BesselJ[\[Nu], x - t], {t, 0, x}]=Divide[BesselJ[\[Mu]+ \[Nu], x],\[Mu]] Failure Failure - -
10.22.E35 int((BesselJ(mu, t)*BesselJ(nu, x - t))/(t*(x - t)), t = 0..x)=((mu + nu)* BesselJ(mu + nu, x))/(mu*nu*x) Integrate[Divide[BesselJ[\[Mu], t]*BesselJ[\[Nu], x - t],t*(x - t)], {t, 0, x}]=Divide[(\[Mu]+ \[Nu])* BesselJ[\[Mu]+ \[Nu], x],\[Mu]*\[Nu]*x] Failure Failure Skip Error
10.22.E36 (1)/(GAMMA(alpha))*int((x - t)^(alpha - 1)* BesselJ(nu, t), t = 0..x)= (2)^(alpha)* sum((alpha[k])/(factorial(k))*BesselJ(nu + alpha + 2*k, x), k = 0..infinity) Divide[1,Gamma[\[Alpha]]]*Integrate[(x - t)^(\[Alpha]- 1)* BesselJ[\[Nu], t], {t, 0, x}]= (2)^(\[Alpha])* Sum[Divide[Subscript[\[Alpha], k],(k)!]*BesselJ[\[Nu]+ \[Alpha]+ 2*k, x], {k, 0, Infinity}] Failure Failure Skip Successful
10.22.E37 int(t*BesselJ(nu, j[nu , ell]*t)*BesselJ(nu, j[nu , m]*t), t = 0..1)=(1)/(2)*(subs( temp=j[nu , ell], diff( BesselJ(nu, temp), temp$(1) ) ))^(2)* KroneckerDelta[ell, m] Integrate[t*BesselJ[\[Nu], Subscript[j, \[Nu], \[ScriptL]]*t]*BesselJ[\[Nu], Subscript[j, \[Nu], m]*t], {t, 0, 1}]=Divide[1,2]*(((D[BesselJ[\[Nu], temp], {temp, 1}]/.temp-> Subscript[j, \[Nu], \[ScriptL]])))^(2)* KroneckerDelta[\[ScriptL], m] Failure Failure Skip Successful
10.22.E38 int(t*BesselJ(nu, alpha[ell]*t)*BesselJ(nu, alpha[m]*t), t = 0..1)((BesselJ(nu, alpha[ell]))^(2))/(2*alpha(alpha[ell])^(2))*KroneckerDelta[ell, m] Integrate[t*BesselJ[\[Nu], Subscript[\[Alpha], \[ScriptL]]*t]*BesselJ[\[Nu], Subscript[\[Alpha], m]*t], {t, 0, 1}]Divide[(BesselJ[\[Nu], Subscript[\[Alpha], \[ScriptL]]])^(2),2*\[Alpha](Subscript[\[Alpha], \[ScriptL]])^(2)]*KroneckerDelta[\[ScriptL], m] Failure Failure Skip Successful
10.22.E39 int((BesselJ(0, t))/(t), t = x..infinity)+ gamma + ln((1)/(2)*x)= int((1 - BesselJ(0, t))/(t), t = 0..x) Integrate[Divide[BesselJ[0, t],t], {t, x, Infinity}]+ EulerGamma + Log[Divide[1,2]*x]= Integrate[Divide[1 - BesselJ[0, t],t], {t, 0, x}] Successful Failure - Successful
10.22.E39 int((1 - BesselJ(0, t))/(t), t = 0..x)= sum((- 1)^(k - 1)*(((1)/(2)*x)^(2*k))/(2*k*(factorial(k))^(2)), k = 1..infinity) Integrate[Divide[1 - BesselJ[0, t],t], {t, 0, x}]= Sum[(- 1)^(k - 1)*Divide[(Divide[1,2]*x)^(2*k),2*k*((k)!)^(2)], {k, 1, Infinity}] Successful Failure - Successful
10.22.E40 int((BesselY(0, t))/(t), t = x..infinity)= -(1)/(Pi)*(ln((1)/(2)*x)+ gamma)^(2)+(Pi)/(6)+(2)/(Pi)*sum((- 1)^(k)*(Psi(k + 1)+(1)/(2*k)- ln((1)/(2)*x))*(((1)/(2)*x)^(2*k))/(2*k*(factorial(k))^(2)), k = 1..infinity) Integrate[Divide[BesselY[0, t],t], {t, x, Infinity}]= -Divide[1,Pi]*(Log[Divide[1,2]*x]+ EulerGamma)^(2)+Divide[Pi,6]+Divide[2,Pi]*Sum[(- 1)^(k)*(PolyGamma[k + 1]+Divide[1,2*k]- Log[Divide[1,2]*x])*Divide[(Divide[1,2]*x)^(2*k),2*k*((k)!)^(2)], {k, 1, Infinity}] Failure Failure Skip Error
10.22.E41 int(BesselJ(nu, t), t = 0..infinity)= 1 Integrate[BesselJ[\[Nu], t], {t, 0, Infinity}]= 1 Successful Failure - Successful
10.22.E42 int(BesselY(nu, t), t = 0..infinity)= - tan((1)/(2)*nu*Pi) Integrate[BesselY[\[Nu], t], {t, 0, Infinity}]= - Tan[Divide[1,2]*\[Nu]*Pi] Successful Failure - Error
10.22.E43 int((t)^(mu)* BesselJ(nu, t), t = 0..infinity)= (2)^(mu)*(GAMMA((1)/(2)*nu +(1)/(2)*mu +(1)/(2)))/(GAMMA((1)/(2)*nu -(1)/(2)*mu +(1)/(2))) Integrate[(t)^(\[Mu])* BesselJ[\[Nu], t], {t, 0, Infinity}]= (2)^(\[Mu])*Divide[Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]+Divide[1,2]],Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]]] Successful Failure - Successful
10.22.E44 int((t)^(mu)* BesselY(nu, t), t = 0..infinity)=((2)^(mu))/(Pi)*GAMMA((1)/(2)*mu +(1)/(2)*nu +(1)/(2))*GAMMA((1)/(2)*mu -(1)/(2)*nu +(1)/(2))*sin((1)/(2)*mu -(1)/(2)*nu)*Pi Integrate[(t)^(\[Mu])* BesselY[\[Nu], t], {t, 0, Infinity}]=Divide[(2)^(\[Mu]),Pi]*Gamma[Divide[1,2]*\[Mu]+Divide[1,2]*\[Nu]+Divide[1,2]]*Gamma[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]+Divide[1,2]]*Sin[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]]*Pi Failure Failure Skip Error
10.22.E45 int((1 - BesselJ(0, t))/((t)^(mu)), t = 0..infinity)= -(Pi*sec((1)/(2)*mu*Pi))/((2)^(mu)* (GAMMA((1)/(2)*mu +(1)/(2)))^(2)) Integrate[Divide[1 - BesselJ[0, t],(t)^(\[Mu])], {t, 0, Infinity}]= -Divide[Pi*Sec[Divide[1,2]*\[Mu]*Pi],(2)^(\[Mu])* (Gamma[Divide[1,2]*\[Mu]+Divide[1,2]])^(2)] Failure Failure Skip Error
10.22.E46 int(((t)^(nu + 1)* BesselJ(nu, a*t))/(((t)^(2)+ (b)^(2))^(mu + 1)), t = 0..infinity)=((a)^(mu)* (b)^(nu - mu))/((2)^(mu)* GAMMA(mu + 1))*BesselK(nu - mu, a*b) Integrate[Divide[(t)^(\[Nu]+ 1)* BesselJ[\[Nu], a*t],((t)^(2)+ (b)^(2))^(\[Mu]+ 1)], {t, 0, Infinity}]=Divide[(a)^(\[Mu])* (b)^(\[Nu]- \[Mu]),(2)^(\[Mu])* Gamma[\[Mu]+ 1]]*BesselK[\[Nu]- \[Mu], a*b] Failure Failure Skip Error
10.22.E47 int(((t)^(nu)* BesselY(nu, a*t))/((t)^(2)+ (b)^(2)), t = 0..infinity)= - (b)^(nu - 1)* BesselK(nu, a*b) Integrate[Divide[(t)^(\[Nu])* BesselY[\[Nu], a*t],(t)^(2)+ (b)^(2)], {t, 0, Infinity}]= - (b)^(\[Nu]- 1)* BesselK[\[Nu], a*b] Failure Failure Skip Error
10.22.E48 int(BesselJ(mu, x*cosh(phi))*(cosh(phi))^(1 - mu)*(sinh(phi))^(2*nu + 1), phi = 0..infinity)= (2)^(nu)* GAMMA(nu + 1)*(x)^(- nu - 1)* BesselJ(mu - nu - 1, x) Integrate[BesselJ[\[Mu], x*Cosh[\[Phi]]]*(Cosh[\[Phi]])^(1 - \[Mu])*(Sinh[\[Phi]])^(2*\[Nu]+ 1), {\[Phi], 0, Infinity}]= (2)^(\[Nu])* Gamma[\[Nu]+ 1]*(x)^(- \[Nu]- 1)* BesselJ[\[Mu]- \[Nu]- 1, x] Failure Failure Skip Error
10.22.E49 int((t)^(mu - 1)* exp(- a*t)*BesselJ(nu, b*t), t = 0..infinity)=(((1)/(2)*b)^(nu))/((a)^(mu + nu))*GAMMA(mu + nu)* hypergeom([(mu + nu)/(2), (mu + nu + 1)/(2)], [nu + 1], -((b)^(2))/((a)^(2)))/GAMMA(nu + 1) Integrate[(t)^(\[Mu]- 1)* Exp[- a*t]*BesselJ[\[Nu], b*t], {t, 0, Infinity}]=Divide[(Divide[1,2]*b)^(\[Nu]),(a)^(\[Mu]+ \[Nu])]*Gamma[\[Mu]+ \[Nu]]* Hypergeometric2F1Regularized[Divide[\[Mu]+ \[Nu],2], Divide[\[Mu]+ \[Nu]+ 1,2], \[Nu]+ 1, -Divide[(b)^(2),(a)^(2)]] Failure Failure Skip Error
10.22.E50 int((t)^(mu - 1)* exp(- a*t)*BesselY(nu, b*t), t = 0..infinity)= cot(nu*Pi)*(((1)/(2)*b)^(nu)* GAMMA(mu + nu))/(((a)^(2)+ (b)^(2))^((1)/(2)*(mu + nu)))* hypergeom([(mu + nu)/(2), (1 - mu + nu)/(2)], [nu + 1], ((b)^(2))/((a)^(2)+ (b)^(2)))/GAMMA(nu + 1)- csc(nu*Pi)*(((1)/(2)*b)^(- nu)* GAMMA(mu - nu))/(((a)^(2)+ (b)^(2))^((1)/(2)*(mu - nu)))* hypergeom([(mu - nu)/(2), (1 - mu - nu)/(2)], [1 - nu], ((b)^(2))/((a)^(2)+ (b)^(2)))/GAMMA(1 - nu) Integrate[(t)^(\[Mu]- 1)* Exp[- a*t]*BesselY[\[Nu], b*t], {t, 0, Infinity}]= Cot[\[Nu]*Pi]*Divide[(Divide[1,2]*b)^(\[Nu])* Gamma[\[Mu]+ \[Nu]],((a)^(2)+ (b)^(2))^(Divide[1,2]*(\[Mu]+ \[Nu]))]* Hypergeometric2F1Regularized[Divide[\[Mu]+ \[Nu],2], Divide[1 - \[Mu]+ \[Nu],2], \[Nu]+ 1, Divide[(b)^(2),(a)^(2)+ (b)^(2)]]- Csc[\[Nu]*Pi]*Divide[(Divide[1,2]*b)^(- \[Nu])* Gamma[\[Mu]- \[Nu]],((a)^(2)+ (b)^(2))^(Divide[1,2]*(\[Mu]- \[Nu]))]* Hypergeometric2F1Regularized[Divide[\[Mu]- \[Nu],2], Divide[1 - \[Mu]- \[Nu],2], 1 - \[Nu], Divide[(b)^(2),(a)^(2)+ (b)^(2)]] Failure Failure Skip Error
10.22.E51 int(BesselJ(nu, b*t)*exp(- (p)^(2)* (t)^(2))*(t)^(nu + 1), t = 0..infinity)=((b)^(nu))/((2*(p)^(2))^(nu + 1))*exp(-((b)^(2))/(4*(p)^(2))) Integrate[BesselJ[\[Nu], b*t]*Exp[- (p)^(2)* (t)^(2)]*(t)^(\[Nu]+ 1), {t, 0, Infinity}]=Divide[(b)^(\[Nu]),(2*(p)^(2))^(\[Nu]+ 1)]*Exp[-Divide[(b)^(2),4*(p)^(2)]] Successful Failure - Error
10.22.E52 int(BesselJ(nu, b*t)*exp(- (p)^(2)* (t)^(2)), t = 0..infinity)=(sqrt(Pi))/(2*p)*exp(-((b)^(2))/(8*(p)^(2)))*BesselI((nu)/(2), ((b)^(2))/(8*(p)^(2))) Integrate[BesselJ[\[Nu], b*t]*Exp[- (p)^(2)* (t)^(2)], {t, 0, Infinity}]=Divide[Sqrt[Pi],2*p]*Exp[-Divide[(b)^(2),8*(p)^(2)]]*BesselI[Divide[\[Nu],2], Divide[(b)^(2),8*(p)^(2)]] Failure Failure Skip Error
10.22.E53 int(BesselY(2*nu, b*t)*exp(- (p)^(2)* (t)^(2)), t = 0..infinity)= -(sqrt(Pi))/(2*p)*exp(-((b)^(2))/(8*(p)^(2)))*(BesselI(nu, ((b)^(2))/(8*(p)^(2)))*tan(nu*Pi)+(1)/(Pi)*BesselK(nu, ((b)^(2))/(8*(p)^(2)))*sec(nu*Pi)) Integrate[BesselY[2*\[Nu], b*t]*Exp[- (p)^(2)* (t)^(2)], {t, 0, Infinity}]= -Divide[Sqrt[Pi],2*p]*Exp[-Divide[(b)^(2),8*(p)^(2)]]*(BesselI[\[Nu], Divide[(b)^(2),8*(p)^(2)]]*Tan[\[Nu]*Pi]+Divide[1,Pi]*BesselK[\[Nu], Divide[(b)^(2),8*(p)^(2)]]*Sec[\[Nu]*Pi]) Failure Failure Skip Error
10.22.E54 int(BesselJ(nu, b*t)*exp(- (p)^(2)* (t)^(2))*(t)^(mu - 1), t = 0..infinity)=(((1)/(2)*b/ p)^(nu)* GAMMA((1)/(2)*nu +(1)/(2)*mu))/(2*(p)^(mu))*exp(-((b)^(2))/(4*(p)^(2)))* KummerM((1)/(2)*nu -(1)/(2)*mu + 1, nu + 1, ((b)^(2))/(4*(p)^(2)))/GAMMA(nu + 1) Integrate[BesselJ[\[Nu], b*t]*Exp[- (p)^(2)* (t)^(2)]*(t)^(\[Mu]- 1), {t, 0, Infinity}]=Divide[(Divide[1,2]*b/ p)^(\[Nu])* Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]],2*(p)^(\[Mu])]*Exp[-Divide[(b)^(2),4*(p)^(2)]]* Hypergeometric1F1Regularized[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+ 1, \[Nu]+ 1, Divide[(b)^(2),4*(p)^(2)]] Failure Failure Skip Error
10.22.E55 int((t)^(- 1)* BesselJ(nu + 2*ell + 1, t)*BesselJ(nu + 2*m + 1, t), t = 0..infinity)=(KroneckerDelta[ell, m])/(2*(2*ell + nu + 1)) Integrate[(t)^(- 1)* BesselJ[\[Nu]+ 2*\[ScriptL]+ 1, t]*BesselJ[\[Nu]+ 2*m + 1, t], {t, 0, Infinity}]=Divide[KroneckerDelta[\[ScriptL], m],2*(2*\[ScriptL]+ \[Nu]+ 1)] Failure Failure Skip Successful
10.22.E56 int((BesselJ(mu, a*t)*BesselJ(nu, b*t))/((t)^(lambda)), t = 0..infinity)=((a)^(mu)* GAMMA((1)/(2)*nu +(1)/(2)*mu -(1)/(2)*lambda +(1)/(2)))/((2)^(lambda)* (b)^(mu - lambda + 1)* GAMMA((1)/(2)*nu -(1)/(2)*mu +(1)/(2)*lambda +(1)/(2)))* hypergeom([(1)/(2)*(mu + nu - lambda + 1), (1)/(2)*(mu - nu - lambda + 1)], [mu + 1], ((a)^(2))/((b)^(2)))/GAMMA(mu + 1) Integrate[Divide[BesselJ[\[Mu], a*t]*BesselJ[\[Nu], b*t],(t)^(\[Lambda])], {t, 0, Infinity}]=Divide[(a)^(\[Mu])* Gamma[Divide[1,2]*\[Nu]+Divide[1,2]*\[Mu]-Divide[1,2]*\[Lambda]+Divide[1,2]],(2)^(\[Lambda])* (b)^(\[Mu]- \[Lambda]+ 1)* Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]*\[Lambda]+Divide[1,2]]]* Hypergeometric2F1Regularized[Divide[1,2]*(\[Mu]+ \[Nu]- \[Lambda]+ 1), Divide[1,2]*(\[Mu]- \[Nu]- \[Lambda]+ 1), \[Mu]+ 1, Divide[(a)^(2),(b)^(2)]] Failure Failure Skip Error
10.22.E57 int((BesselJ(mu, a*t)*BesselJ(nu, a*t))/((t)^(lambda)), t = 0..infinity)=(((1)/(2)*a)^(lambda - 1)* GAMMA((1)/(2)*mu +(1)/(2)*nu -(1)/(2)*lambda +(1)/(2))*GAMMA(lambda))/(2*GAMMA((1)/(2)*lambda +(1)/(2)*nu -(1)/(2)*mu +(1)/(2))*GAMMA((1)/(2)*lambda +(1)/(2)*mu -(1)/(2)*nu +(1)/(2))*GAMMA((1)/(2)*lambda +(1)/(2)*mu +(1)/(2)*nu +(1)/(2))) Integrate[Divide[BesselJ[\[Mu], a*t]*BesselJ[\[Nu], a*t],(t)^(\[Lambda])], {t, 0, Infinity}]=Divide[(Divide[1,2]*a)^(\[Lambda]- 1)* Gamma[Divide[1,2]*\[Mu]+Divide[1,2]*\[Nu]-Divide[1,2]*\[Lambda]+Divide[1,2]]*Gamma[\[Lambda]],2*Gamma[Divide[1,2]*\[Lambda]+Divide[1,2]*\[Nu]-Divide[1,2]*\[Mu]+Divide[1,2]]*Gamma[Divide[1,2]*\[Lambda]+Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]+Divide[1,2]]*Gamma[Divide[1,2]*\[Lambda]+Divide[1,2]*\[Mu]+Divide[1,2]*\[Nu]+Divide[1,2]]] Failure Failure Skip Error
10.22.E58 int((BesselJ(nu, a*t)*BesselJ(nu, b*t))/((t)^(lambda)), t = 0..infinity)=((a*b)^(nu)* GAMMA(nu -(1)/(2)*lambda +(1)/(2)))/((2)^(lambda)*((a)^(2)+ (b)^(2))^(nu -(1)/(2)*lambda +(1)/(2))* GAMMA((1)/(2)*lambda +(1)/(2)))*hypergeom([(2*nu + 1 - lambda)/(4), (2*nu + 3 - lambda)/(4)], [nu + 1], (4*(a)^(2)* (b)^(2))/(((a)^(2)+ (b)^(2))^(2)))/GAMMA(nu + 1) Integrate[Divide[BesselJ[\[Nu], a*t]*BesselJ[\[Nu], b*t],(t)^(\[Lambda])], {t, 0, Infinity}]=Divide[(a*b)^(\[Nu])* Gamma[\[Nu]-Divide[1,2]*\[Lambda]+Divide[1,2]],(2)^(\[Lambda])*((a)^(2)+ (b)^(2))^(\[Nu]-Divide[1,2]*\[Lambda]+Divide[1,2])* Gamma[Divide[1,2]*\[Lambda]+Divide[1,2]]]*Hypergeometric2F1Regularized[Divide[2*\[Nu]+ 1 - \[Lambda],4], Divide[2*\[Nu]+ 3 - \[Lambda],4], \[Nu]+ 1, Divide[4*(a)^(2)* (b)^(2),((a)^(2)+ (b)^(2))^(2)]] Failure Failure Skip Error
10.22.E59 int(exp(I*b*t)*BesselJ(mu, a*t), t = 0..infinity)= Integrate[Exp[I*b*t]*BesselJ[\[Mu], a*t], {t, 0, Infinity}]= Error Failure - -
10.22.E59 Error Failure - Error
10.22.E60 int(exp(I*b*t)*BesselY(0, a*t), t = 0..infinity)= Integrate[Exp[I*b*t]*BesselY[0, a*t], {t, 0, Infinity}]= Error Failure - -
10.22.E60 Error Failure - Error
10.22.E61 int((t)^(- 1)* exp(I*b*t)*BesselJ(mu, a*t), t = 0..infinity)= Integrate[(t)^(- 1)* Exp[I*b*t]*BesselJ[\[Mu], a*t], {t, 0, Infinity}]= Error Failure - -
10.22.E61 Error Failure - Error
10.22.E62 int((t)^(mu - nu + 1)* BesselJ(mu, a*t)*BesselJ(nu, b*t), t = 0..infinity)= Integrate[(t)^(\[Mu]- \[Nu]+ 1)* BesselJ[\[Mu], a*t]*BesselJ[\[Nu], b*t], {t, 0, Infinity}]= Error Failure - -
10.22.E62 Error Failure - Error
10.22.E63 int(BesselJ(mu, a*t)*BesselJ(mu - 1, b*t), t = 0..infinity)= Integrate[BesselJ[\[Mu], a*t]*BesselJ[\[Mu]- 1, b*t], {t, 0, Infinity}]= Error Failure - -
10.22.E63 Error Failure - Error
10.22.E63 b < a ,(2*b)^(- 1), b < a ,(2*b)^(- 1), Error Failure - Error
10.22.E64 int(BesselJ(mu + 2*n + 1, a*t)*BesselJ(mu, b*t), t = 0..infinity)= Integrate[BesselJ[\[Mu]+ 2*n + 1, a*t]*BesselJ[\[Mu], b*t], {t, 0, Infinity}]= Error Failure - -
10.22.E64 Error Failure - Error
10.22.E64 b < a ,(- 1)^(n)/(2*a), b < a ,(- 1)^(n)/(2*a), Error Failure - Error
10.22.E65 int(BesselJ(0, a*t)*(BesselJ(0, b*t)- BesselJ(0, c*t))*(1)/(t), t = 0..infinity)= Integrate[BesselJ[0, a*t]*(BesselJ[0, b*t]- BesselJ[0, c*t])*Divide[1,t], {t, 0, Infinity}]= Failure Failure Error -
10.22.E65 Failure Failure Error -
10.22.E65 b < a , 0 b < a , 0 Error Failure - -
10.22.E65 a , 0 < c a , 0 < c Error Failure - Error
10.22.E65 c < = a , ln(c/ a), c < = a , Log[c/ a], Failure Failure Skip Successful
10.22.E66 int(exp(- a*t)*BesselJ(nu, b*t)*BesselJ(nu, c*t), t = 0..infinity)=(1)/(Pi*(b*c)^((1)/(2)))* LegendreQ(nu -(1)/(2), ((a)^(2)+ (b)^(2)+ (c)^(2))/(2*b*c)) Integrate[Exp[- a*t]*BesselJ[\[Nu], b*t]*BesselJ[\[Nu], c*t], {t, 0, Infinity}]=Divide[1,Pi*(b*c)^(Divide[1,2])]* LegendreQ[\[Nu]-Divide[1,2], 0, 3, Divide[(a)^(2)+ (b)^(2)+ (c)^(2),2*b*c]] Failure Failure Skip Error
10.22.E67 int(t*exp(- (p)^(2)* (t)^(2))*BesselJ(nu, a*t)*BesselJ(nu, b*t), t = 0..infinity)=(1)/(2*(p)^(2))*exp(-((a)^(2)+ (b)^(2))/(4*(p)^(2)))*BesselI(nu, (a*b)/(2*(p)^(2))) Integrate[t*Exp[- (p)^(2)* (t)^(2)]*BesselJ[\[Nu], a*t]*BesselJ[\[Nu], b*t], {t, 0, Infinity}]=Divide[1,2*(p)^(2)]*Exp[-Divide[(a)^(2)+ (b)^(2),4*(p)^(2)]]*BesselI[\[Nu], Divide[a*b,2*(p)^(2)]] Failure Failure Skip Error
10.22.E68 int(t*exp(- (p)^(2)* (t)^(2))*BesselJ(0, a*t)*BesselY(0, a*t), t = 0..infinity)= -(1)/(2*Pi*(p)^(2))*exp(-((a)^(2))/(2*(p)^(2)))*BesselK(0, ((a)^(2))/(2*(p)^(2))) Integrate[t*Exp[- (p)^(2)* (t)^(2)]*BesselJ[0, a*t]*BesselY[0, a*t], {t, 0, Infinity}]= -Divide[1,2*Pi*(p)^(2)]*Exp[-Divide[(a)^(2),2*(p)^(2)]]*BesselK[0, Divide[(a)^(2),2*(p)^(2)]] Failure Failure Skip Error
10.22.E70 int(BesselY(nu, a*t)*BesselJ(nu + 1, b*t)*(t)/((t)^(2)- (z)^(2)), t = 0..infinity)=(1)/(2)*Pi*BesselJ(nu + 1, b*z)*HankelH1(nu, a*z) Integrate[BesselY[\[Nu], a*t]*BesselJ[\[Nu]+ 1, b*t]*Divide[t,(t)^(2)- (z)^(2)], {t, 0, Infinity}]=Divide[1,2]*Pi*BesselJ[\[Nu]+ 1, b*z]*HankelH1[\[Nu], a*z] Failure Failure Skip Error
10.22.E71 int(BesselJ(mu, a*t)*BesselJ(nu, b*t)*BesselJ(nu, c*t)*(t)^(1 - mu), t = 0..infinity)=((b*c)^(mu - 1)*(sin(phi))^(mu -(1)/(2)))/((2*Pi)^((1)/(2))* (a)^(mu))*LegendreP(nu -(1)/(2), (1)/(2)- mu, cos(phi)) Integrate[BesselJ[\[Mu], a*t]*BesselJ[\[Nu], b*t]*BesselJ[\[Nu], c*t]*(t)^(1 - \[Mu]), {t, 0, Infinity}]=Divide[(b*c)^(\[Mu]- 1)*(Sin[\[Phi]])^(\[Mu]-Divide[1,2]),(2*Pi)^(Divide[1,2])* (a)^(\[Mu])]*LegendreP[\[Nu]-Divide[1,2], Divide[1,2]- \[Mu], Cos[\[Phi]]] Failure Failure Skip Skip
10.22.E75 int(BesselY(nu, a*t)*BesselJ(nu, b*t)*BesselJ(nu, c*t)*(t)^(1 + nu), t = 0..infinity)= Integrate[BesselY[\[Nu], a*t]*BesselJ[\[Nu], b*t]*BesselJ[\[Nu], c*t]*(t)^(1 + \[Nu]), {t, 0, Infinity}]= Error Failure - -
10.22.E75 Error Failure - Error
10.22.E75 a <abs(b - c), 0 , a <Abs[b - c], 0 , Error Failure - -
10.23.E3 (BesselJ(0, z))^(2)+ 2*sum((BesselJ(k, z))^(2), k = 1..infinity)= 1 (BesselJ[0, z])^(2)+ 2*Sum[(BesselJ[k, z])^(2), {k, 1, Infinity}]= 1 Failure Successful Skip -
10.23.E4 sum((- 1)^(k)* BesselJ(k, z)*BesselJ(2*n - k, z), k = 0..2*n)+ 2*sum(BesselJ(k, z)*BesselJ(2*n + k, z), k = 1..infinity)= 0 Sum[(- 1)^(k)* BesselJ[k, z]*BesselJ[2*n - k, z], {k, 0, 2*n}]+ 2*Sum[BesselJ[k, z]*BesselJ[2*n + k, z], {k, 1, Infinity}]= 0 Failure Failure Skip
Fail
Complex[4.242640687119286, 4.242640687119286] <- {Rule[n, 3], Rule[Sum[Times[BesselJ[k, z], BesselJ[Plus[k, Times[2, n]], z]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[DifferenceRoot[Function[{, }, {Equal[Plus[Times[-1, Plus[-5, Times[-2, ], Times[2, n]], Power[z, 2], []], Times[Plus[-20, Times[-48, ], Times[-36, Power[, 2]], Times[-8, Power[, 3]], Times[48, n], Times[72, , n], Times[24, Power[, 2], n], Times[-16, Power[n, 2]], Times[-16, , Power[n, 2]], Times[-7, Power[z, 2]], Times[-2, , Power[z, 2]], Times[2, n, Power[z, 2]]], [Plus[1, ]]], Times[-2, Plus[-3, Times[-2, ], Times[2, n]], Plus[22, Times[24, ], Times[6, Power[, 2]], Times[-24, n], Times[-12, , n], Times[8, Power[n, 2]], Times[-1, Power[z, 2]]], [Plus[2, ]]], Times[2, Plus[-5, Times[-2, ], Times[2, n]], Plus[22, Times[24, ], Times[6, Power[, 2]], Times[-24, n], Times[-12, , n], Times[8, Power[n, 2]], Times[-1, Power[z, 2]]], [Plus[3, ]]], Times[Plus[108, Times[144, ], Times[60, Power[, 2]], Times[8, Power[, 3]], Times[-144, n], Times[-120, , n], Times[-24, Power[, 2], n], Times[48, Power[n, 2]], Times[16, , Power[n, 2]], Power[z, 2], Times[2, , Power[z, 2]], Times[-2, n, Power[z, 2]]], [Plus[4, ]]], Times[Plus[-3, Times[-2, ], Times[2, n]], Power[z, 2], [Plus[5, ]]]], 0], Equal[[0], 0], Equal[[1], Times[BesselJ[0, z], BesselJ[Times[2, n], z]]], Equal[[2], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]]]], Equal[[3], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]], Times[Power[z, -2], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]]]], Equal[[4], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]], Times[Power[z, -2], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]], Times[Power[z, -2], Plus[Times[z, BesselJ[1, z]], Times[-4, Power[z, -1], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]]]], Plus[Times[-1, z, BesselJ[Plus[-1, Times[2, n]], z]], Times[-4, Power[z, -1], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]], Times[4, n, Power[z, -1], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]]]]]]}]][Plus[1, Times[2, n]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[4.242640687119286, 1.4142135623730951] <- {Rule[n, 3], Rule[Sum[Times[BesselJ[k, z], BesselJ[Plus[k, Times[2, n]], z]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[DifferenceRoot[Function[{, }, {Equal[Plus[Times[-1, Plus[-5, Times[-2, ], Times[2, n]], Power[z, 2], []], Times[Plus[-20, Times[-48, ], Times[-36, Power[, 2]], Times[-8, Power[, 3]], Times[48, n], Times[72, , n], Times[24, Power[, 2], n], Times[-16, Power[n, 2]], Times[-16, , Power[n, 2]], Times[-7, Power[z, 2]], Times[-2, , Power[z, 2]], Times[2, n, Power[z, 2]]], [Plus[1, ]]], Times[-2, Plus[-3, Times[-2, ], Times[2, n]], Plus[22, Times[24, ], Times[6, Power[, 2]], Times[-24, n], Times[-12, , n], Times[8, Power[n, 2]], Times[-1, Power[z, 2]]], [Plus[2, ]]], Times[2, Plus[-5, Times[-2, ], Times[2, n]], Plus[22, Times[24, ], Times[6, Power[, 2]], Times[-24, n], Times[-12, , n], Times[8, Power[n, 2]], Times[-1, Power[z, 2]]], [Plus[3, ]]], Times[Plus[108, Times[144, ], Times[60, Power[, 2]], Times[8, Power[, 3]], Times[-144, n], Times[-120, , n], Times[-24, Power[, 2], n], Times[48, Power[n, 2]], Times[16, , Power[n, 2]], Power[z, 2], Times[2, , Power[z, 2]], Times[-2, n, Power[z, 2]]], [Plus[4, ]]], Times[Plus[-3, Times[-2, ], Times[2, n]], Power[z, 2], [Plus[5, ]]]], 0], Equal[[0], 0], Equal[[1], Times[BesselJ[0, z], BesselJ[Times[2, n], z]]], Equal[[2], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]]]], Equal[[3], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]], Times[Power[z, -2], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]]]], Equal[[4], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]], Times[Power[z, -2], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]], Times[Power[z, -2], Plus[Times[z, BesselJ[1, z]], Times[-4, Power[z, -1], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]]]], Plus[Times[-1, z, BesselJ[Plus[-1, Times[2, n]], z]], Times[-4, Power[z, -1], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]], Times[4, n, Power[z, -1], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]]]]]]}]][Plus[1, Times[2, n]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[n, 3], Rule[Sum[Times[BesselJ[k, z], BesselJ[Plus[k, Times[2, n]], z]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[DifferenceRoot[Function[{, }, {Equal[Plus[Times[-1, Plus[-5, Times[-2, ], Times[2, n]], Power[z, 2], []], Times[Plus[-20, Times[-48, ], Times[-36, Power[, 2]], Times[-8, Power[, 3]], Times[48, n], Times[72, , n], Times[24, Power[, 2], n], Times[-16, Power[n, 2]], Times[-16, , Power[n, 2]], Times[-7, Power[z, 2]], Times[-2, , Power[z, 2]], Times[2, n, Power[z, 2]]], [Plus[1, ]]], Times[-2, Plus[-3, Times[-2, ], Times[2, n]], Plus[22, Times[24, ], Times[6, Power[, 2]], Times[-24, n], Times[-12, , n], Times[8, Power[n, 2]], Times[-1, Power[z, 2]]], [Plus[2, ]]], Times[2, Plus[-5, Times[-2, ], Times[2, n]], Plus[22, Times[24, ], Times[6, Power[, 2]], Times[-24, n], Times[-12, , n], Times[8, Power[n, 2]], Times[-1, Power[z, 2]]], [Plus[3, ]]], Times[Plus[108, Times[144, ], Times[60, Power[, 2]], Times[8, Power[, 3]], Times[-144, n], Times[-120, , n], Times[-24, Power[, 2], n], Times[48, Power[n, 2]], Times[16, , Power[n, 2]], Power[z, 2], Times[2, , Power[z, 2]], Times[-2, n, Power[z, 2]]], [Plus[4, ]]], Times[Plus[-3, Times[-2, ], Times[2, n]], Power[z, 2], [Plus[5, ]]]], 0], Equal[[0], 0], Equal[[1], Times[BesselJ[0, z], BesselJ[Times[2, n], z]]], Equal[[2], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]]]], Equal[[3], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]], Times[Power[z, -2], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]]]], Equal[[4], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]], Times[Power[z, -2], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]], Times[Power[z, -2], Plus[Times[z, BesselJ[1, z]], Times[-4, Power[z, -1], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]]]], Plus[Times[-1, z, BesselJ[Plus[-1, Times[2, n]], z]], Times[-4, Power[z, -1], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]], Times[4, n, Power[z, -1], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]]]]]]}]][Plus[1, Times[2, n]]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.4142135623730951, 4.242640687119286] <- {Rule[n, 3], Rule[Sum[Times[BesselJ[k, z], BesselJ[Plus[k, Times[2, n]], z]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[DifferenceRoot[Function[{, }, {Equal[Plus[Times[-1, Plus[-5, Times[-2, ], Times[2, n]], Power[z, 2], []], Times[Plus[-20, Times[-48, ], Times[-36, Power[, 2]], Times[-8, Power[, 3]], Times[48, n], Times[72, , n], Times[24, Power[, 2], n], Times[-16, Power[n, 2]], Times[-16, , Power[n, 2]], Times[-7, Power[z, 2]], Times[-2, , Power[z, 2]], Times[2, n, Power[z, 2]]], [Plus[1, ]]], Times[-2, Plus[-3, Times[-2, ], Times[2, n]], Plus[22, Times[24, ], Times[6, Power[, 2]], Times[-24, n], Times[-12, , n], Times[8, Power[n, 2]], Times[-1, Power[z, 2]]], [Plus[2, ]]], Times[2, Plus[-5, Times[-2, ], Times[2, n]], Plus[22, Times[24, ], Times[6, Power[, 2]], Times[-24, n], Times[-12, , n], Times[8, Power[n, 2]], Times[-1, Power[z, 2]]], [Plus[3, ]]], Times[Plus[108, Times[144, ], Times[60, Power[, 2]], Times[8, Power[, 3]], Times[-144, n], Times[-120, , n], Times[-24, Power[, 2], n], Times[48, Power[n, 2]], Times[16, , Power[n, 2]], Power[z, 2], Times[2, , Power[z, 2]], Times[-2, n, Power[z, 2]]], [Plus[4, ]]], Times[Plus[-3, Times[-2, ], Times[2, n]], Power[z, 2], [Plus[5, ]]]], 0], Equal[[0], 0], Equal[[1], Times[BesselJ[0, z], BesselJ[Times[2, n], z]]], Equal[[2], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]]]], Equal[[3], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]], Times[Power[z, -2], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]]]], Equal[[4], Plus[Times[BesselJ[0, z], BesselJ[Times[2, n], z]], Times[-1, BesselJ[1, z], BesselJ[Plus[-1, Times[2, n]], z]], Times[Power[z, -2], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]], Times[Power[z, -2], Plus[Times[z, BesselJ[1, z]], Times[-4, Power[z, -1], Plus[Times[-1, z, BesselJ[0, z]], Times[2, BesselJ[1, z]]]]], Plus[Times[-1, z, BesselJ[Plus[-1, Times[2, n]], z]], Times[-4, Power[z, -1], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]], Times[4, n, Power[z, -1], Plus[Times[-1, z, BesselJ[Times[2, n], z]], Times[-2, BesselJ[Plus[-1, Times[2, n]], z]], Times[4, n, BesselJ[Plus[-1, Times[2, n]], z]]]]]]]]}]][Plus[1, Times[2, n]]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.23.E5 sum(BesselJ(k, z)*BesselJ(n - k, z), k = 0..n)+ 2*sum((- 1)^(k)* BesselJ(k, z)*BesselJ(n + k, z), k = 1..infinity)= BesselJ(n, 2*z) Sum[BesselJ[k, z]*BesselJ[n - k, z], {k, 0, n}]+ 2*Sum[(- 1)^(k)* BesselJ[k, z]*BesselJ[n + k, z], {k, 1, Infinity}]= BesselJ[n, 2*z] Failure Failure Skip Skip
10.23#Ex1 w =sqrt((u)^(2)+ (v)^(2)- 2*u*v*cos(alpha)) w =Sqrt[(u)^(2)+ (v)^(2)- 2*u*v*Cos[\[Alpha]]] Failure Failure
Fail
.7483404465-2.552464124*I <- {alpha = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2)}
.7483404465-5.380891248*I <- {alpha = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2)}
-2.080086678-5.380891248*I <- {alpha = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), w = -2^(1/2)-I*2^(1/2)}
-2.080086678-2.552464124*I <- {alpha = 2^(1/2)+I*2^(1/2), u = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), w = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Skip
10.23#Ex2 u - v*cos(alpha)= w*cos(chi) u - v*Cos[\[Alpha]]= w*Cos[\[Chi]] Failure Failure Skip Skip
10.23#Ex3 v*sin(alpha)= w*sin(chi) v*Sin[\[Alpha]]= w*Sin[\[Chi]] Failure Failure
Fail
-.853510328+6.085461480*I <- {alpha = 2^(1/2)+I*2^(1/2), chi = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2)}
5.231951152+6.938971808*I <- {alpha = 2^(1/2)+I*2^(1/2), chi = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), w = -2^(1/2)-I*2^(1/2)}
6.085461480+.853510328*I <- {alpha = 2^(1/2)+I*2^(1/2), chi = 2^(1/2)+I*2^(1/2), v = 2^(1/2)+I*2^(1/2), w = -2^(1/2)+I*2^(1/2)}
.853510328-6.085461480*I <- {alpha = 2^(1/2)+I*2^(1/2), chi = 2^(1/2)+I*2^(1/2), v = 2^(1/2)-I*2^(1/2), w = 2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.853510326577255, 0.853510326577255] <- {Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[5.231951158402261, 6.938971811556771] <- {Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[6.085461484979516, 6.085461484979516] <- {Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.853510326577255, -0.853510326577255] <- {Rule[v, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[α, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[χ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.23.E9 exp(I*v*cos(alpha))=(GAMMA(nu))/(((1)/(2)*v)^(nu))* sum((nu + k)* (I)^(k)* BesselJ(nu + k, v)*GegenbauerC(k, nu, cos(alpha)), k = 0..infinity) Exp[I*v*Cos[\[Alpha]]]=Divide[Gamma[\[Nu]],(Divide[1,2]*v)^(\[Nu])]* Sum[(\[Nu]+ k)* (I)^(k)* BesselJ[\[Nu]+ k, v]*GegenbauerC[k, \[Nu], Cos[\[Alpha]]], {k, 0, Infinity}] Error Failure - Skip
10.23.E15 ((1)/(2)*z)^(nu)= sum(((nu + 2*k)* GAMMA(nu + k))/(factorial(k))*BesselJ(nu + 2*k, z), k = 0..infinity) (Divide[1,2]*z)^(\[Nu])= Sum[Divide[(\[Nu]+ 2*k)* Gamma[\[Nu]+ k],(k)!]*BesselJ[\[Nu]+ 2*k, z], {k, 0, Infinity}] Failure Successful Skip -
10.23.E16 BesselY(0, z)=(2)/(Pi)*(ln((1)/(2)*z)+ gamma)* BesselJ(0, z)-(4)/(Pi)*sum((- 1)^(k)*(BesselJ(2*k, z))/(k), k = 1..infinity) BesselY[0, z]=Divide[2,Pi]*(Log[Divide[1,2]*z]+ EulerGamma)* BesselJ[0, z]-Divide[4,Pi]*Sum[(- 1)^(k)*Divide[BesselJ[2*k, z],k], {k, 1, Infinity}] Failure Successful Skip -
10.23.E17 BesselY(n, z)= -(factorial(n)*((1)/(2)*z)^(- n))/(Pi)*sum((((1)/(2)*z)^(k)* BesselJ(k, z))/(factorial(k)*(n - k)), k = 0..n - 1)+(2)/(Pi)*(ln((1)/(2)*z)- Psi(n + 1))* BesselJ(n, z)-(2)/(Pi)*sum((- 1)^(k)*((n + 2*k)* BesselJ(n + 2*k, z))/(k*(n + k)), k = 1..infinity) BesselY[n, z]= -Divide[(n)!*(Divide[1,2]*z)^(- n),Pi]*Sum[Divide[(Divide[1,2]*z)^(k)* BesselJ[k, z],(k)!*(n - k)], {k, 0, n - 1}]+Divide[2,Pi]*(Log[Divide[1,2]*z]- PolyGamma[n + 1])* BesselJ[n, z]-Divide[2,Pi]*Sum[(- 1)^(k)*Divide[(n + 2*k)* BesselJ[n + 2*k, z],k*(n + k)], {k, 1, Infinity}] Failure Failure Skip Successful
10.24.E1 (x)^(2)* diff(w, [x$(2)])+ x*diff(w, x)+((x)^(2)+ (nu)^(2))* w = 0 (x)^(2)* D[w, {x, 2}]+ x*D[w, x]+((x)^(2)+ (\[Nu])^(2))* w = 0 Failure Failure
Fail
-4.242640683+7.071067807*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 1}
.2828427124e-8+11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 2}
7.071067813+18.38477630*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 3}
7.071067807+4.242640683*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Fail
Complex[-4.242640687119286, 7.0710678118654755] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.0, 11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[7.0710678118654755, 18.38477631085024] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[7.0710678118654755, -4.242640687119286] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.24#Ex1 sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x))= sech((1)/(2)*Pi*nu)*Re(BesselJ(I*nu, x)) Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]]= Sech[Divide[1,2]*Pi*\[Nu]]*Re[BesselJ[I*\[Nu], x]] Successful Successful - -
10.24#Ex2 sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x))= sech((1)/(2)*Pi*nu)*Re(BesselY(I*nu, x)) Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]]= Sech[Divide[1,2]*Pi*\[Nu]]*Re[BesselY[I*\[Nu], x]] Successful Successful - -
10.24.E3 GAMMA(1 + I*nu)=((Pi*nu)/(sinh(Pi*nu)))^((1)/(2))* exp(I*gamma[nu]) Gamma[1 + I*\[Nu]]=(Divide[Pi*\[Nu],Sinh[Pi*\[Nu]]])^(Divide[1,2])* Exp[I*Subscript[\[Gamma], \[Nu]]] Failure Failure
Fail
-.7864250629e-1-.1325922997*I <- {nu = 2^(1/2)+I*2^(1/2), gamma[nu] = 2^(1/2)+I*2^(1/2)}
1.283131241-.7318661334*I <- {nu = 2^(1/2)+I*2^(1/2), gamma[nu] = 2^(1/2)-I*2^(1/2)}
-1.737329775+.6511700055e-1*I <- {nu = 2^(1/2)+I*2^(1/2), gamma[nu] = -2^(1/2)-I*2^(1/2)}
-.2571691098-.8548601655e-1*I <- {nu = 2^(1/2)+I*2^(1/2), gamma[nu] = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Successful
10.24#Ex3 sech((1/2)*Pi*(- nu))*Re(BesselJ(I*(- nu), x))= sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)) Sech[1/2 Pi - \[Nu]] Re[BesselJ[I - \[Nu], x]]= Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]] Failure Failure
Fail
-.4072055387-.5224985000*I <- {nu = 2^(1/2)+I*2^(1/2), x = 1}
-.9080795132-1.165185978*I <- {nu = 2^(1/2)+I*2^(1/2), x = 2}
-.3702824234-.4751212654*I <- {nu = 2^(1/2)+I*2^(1/2), x = 3}
.4072055387-.5224985000*I <- {nu = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Fail
Complex[-5.717128116473797, -5.753336678220267] <- {Rule[x, 1], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-3.18027157410826, -3.3753924459653097] <- {Rule[x, 2], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.3142029624806735, -0.37398699406023267] <- {Rule[x, 3], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[335.05864396153805, -329.61926001758485] <- {Rule[x, 1], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.24#Ex4 sech((1/2)*Pi*(- nu))*Re(BesselY(I*(- nu), x))= sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x)) Sech[1/2 Pi - \[Nu]] Re[BesselY[I - \[Nu], x]]= Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]] Failure Failure
Fail
1.996706293+2.562037949*I <- {nu = 2^(1/2)+I*2^(1/2), x = 1}
.9116896387e-1+.1169818245*I <- {nu = 2^(1/2)+I*2^(1/2), x = 2}
-.4855048259-.6229668292*I <- {nu = 2^(1/2)+I*2^(1/2), x = 3}
-1.996706293+2.562037949*I <- {nu = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Fail
Complex[3.2898455559215325, 3.8112807679993184] <- {Rule[x, 1], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.6593720309671711, -1.6388399049382785] <- {Rule[x, 2], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-2.5986068001960096, -2.7005355423537045] <- {Rule[x, 3], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[104.18379750674467, -102.47171282147995] <- {Rule[x, 1], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.24.E9 sech((1/2)*Pi*(0))*Re(BesselY(I*(0), x))= BesselY(0, x) Sech[1/2 Pi 0] Re[BesselY[I 0, x]]= BesselY[0, x] Failure Failure Successful Successful
10.25.E1 (z)^(2)* diff(w, [z$(2)])+ z*diff(w, z)-((z)^(2)+ (nu)^(2))* w = 0 (z)^(2)* D[w, {z, 2}]+ z*D[w, z]-((z)^(2)+ (\[Nu])^(2))* w = 0 Failure Failure
Fail
11.31370849-11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
11.31370849-11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-11.31370849-11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-11.31370849-11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
... skip entries to safe data
Fail
Complex[11.313708498984761, -11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[11.313708498984761, -11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-11.313708498984761, 11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-11.313708498984761, 11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.25.E2 BesselI(nu, z)=((1)/(2)*z)^(nu)* sum((((1)/(4)*(z)^(2))^(k))/(factorial(k)*GAMMA(nu + k + 1)), k = 0..infinity) BesselI[\[Nu], z]=(Divide[1,2]*z)^(\[Nu])* Sum[Divide[(Divide[1,4]*(z)^(2))^(k),(k)!*Gamma[\[Nu]+ k + 1]], {k, 0, Infinity}] Successful Successful - -
10.27.E1 BesselI(- n, z)= BesselI(n, z) BesselI[- n, z]= BesselI[n, z] Failure Failure Successful Successful
10.27.E2 BesselI(- nu, z)= BesselI(nu, z)+(2/ Pi)* sin(nu*Pi)*BesselK(nu, z) BesselI[- \[Nu], z]= BesselI[\[Nu], z]+(2/ Pi)* Sin[\[Nu]*Pi]*BesselK[\[Nu], z] Successful Successful - -
10.27.E3 BesselK(- nu, z)= BesselK(nu, z) BesselK[- \[Nu], z]= BesselK[\[Nu], z] Successful Successful - -
10.27.E4 BesselK(nu, z)=(1)/(2)*Pi*(BesselI(- nu, z)- BesselI(nu, z))/(sin(nu*Pi)) BesselK[\[Nu], z]=Divide[1,2]*Pi*Divide[BesselI[- \[Nu], z]- BesselI[\[Nu], z],Sin[\[Nu]*Pi]] Successful Successful - -
10.27.E6 BesselI(nu, z)= exp(- nu*Pi*I/ 2)*BesselJ(nu, z*exp(+ Pi*I/ 2)) BesselI[\[Nu], z]= Exp[- \[Nu]*Pi*I/ 2]*BesselJ[\[Nu], z*Exp[+ Pi*I/ 2]] Failure Failure Error Error
10.27.E6 BesselI(nu, z)= exp(+ nu*Pi*I/ 2)*BesselJ(nu, z*exp(- Pi*I/ 2)) BesselI[\[Nu], z]= Exp[+ \[Nu]*Pi*I/ 2]*BesselJ[\[Nu], z*Exp[- Pi*I/ 2]] Failure Failure Error Error
10.27.E7 BesselI(nu, z)=(1)/(2)*exp(- nu*Pi*I/ 2)*(HankelH1(nu, z*exp(+ Pi*I/ 2))+ HankelH2(nu, z*exp(+ Pi*I/ 2))) BesselI[\[Nu], z]=Divide[1,2]*Exp[- \[Nu]*Pi*I/ 2]*(HankelH1[\[Nu], z*Exp[+ Pi*I/ 2]]+ HankelH2[\[Nu], z*Exp[+ Pi*I/ 2]]) Failure Failure Error Error
10.27.E7 BesselI(nu, z)=(1)/(2)*exp(+ nu*Pi*I/ 2)*(HankelH1(nu, z*exp(- Pi*I/ 2))+ HankelH2(nu, z*exp(- Pi*I/ 2))) BesselI[\[Nu], z]=Divide[1,2]*Exp[+ \[Nu]*Pi*I/ 2]*(HankelH1[\[Nu], z*Exp[- Pi*I/ 2]]+ HankelH2[\[Nu], z*Exp[- Pi*I/ 2]]) Failure Failure Error Error
10.27.E8 BesselK(nu, z)= BesselK[\[Nu], z]= Error Failure - -
10.27.E8 Error Failure - Error
10.27.E9 Pi*I*BesselJ(nu, z)= exp(- nu*Pi*I/ 2)*BesselK(nu, z*exp(- Pi*I/ 2))- exp(nu*Pi*I/ 2)*BesselK(nu, z*exp(Pi*I/ 2)) Pi*I*BesselJ[\[Nu], z]= Exp[- \[Nu]*Pi*I/ 2]*BesselK[\[Nu], z*Exp[- Pi*I/ 2]]- Exp[\[Nu]*Pi*I/ 2]*BesselK[\[Nu], z*Exp[Pi*I/ 2]] Failure Failure Skip Successful
10.27.E10 - Pi*BesselY(nu, z)= exp(- nu*Pi*I/ 2)*BesselK(nu, z*exp(- Pi*I/ 2))+ exp(nu*Pi*I/ 2)*BesselK(nu, z*exp(Pi*I/ 2)) - Pi*BesselY[\[Nu], z]= Exp[- \[Nu]*Pi*I/ 2]*BesselK[\[Nu], z*Exp[- Pi*I/ 2]]+ Exp[\[Nu]*Pi*I/ 2]*BesselK[\[Nu], z*Exp[Pi*I/ 2]] Failure Failure Skip Successful
10.27.E11 BesselY(nu, z)= exp(+(nu + 1)* Pi*I/ 2)*BesselI(nu, z*exp(- Pi*I/ 2))-(2/ Pi)* exp(- nu*Pi*I/ 2)*BesselK(nu, z*exp(- Pi*I/ 2)) BesselY[\[Nu], z]= Exp[+(\[Nu]+ 1)* Pi*I/ 2]*BesselI[\[Nu], z*Exp[- Pi*I/ 2]]-(2/ Pi)* Exp[- \[Nu]*Pi*I/ 2]*BesselK[\[Nu], z*Exp[- Pi*I/ 2]] Failure Failure Error Error
10.27.E11 BesselY(nu, z)= exp(-(nu + 1)* Pi*I/ 2)*BesselI(nu, z*exp(+ Pi*I/ 2))-(2/ Pi)* exp(+ nu*Pi*I/ 2)*BesselK(nu, z*exp(+ Pi*I/ 2)) BesselY[\[Nu], z]= Exp[-(\[Nu]+ 1)* Pi*I/ 2]*BesselI[\[Nu], z*Exp[+ Pi*I/ 2]]-(2/ Pi)* Exp[+ \[Nu]*Pi*I/ 2]*BesselK[\[Nu], z*Exp[+ Pi*I/ 2]] Failure Failure Error Error
10.28.E1 (BesselI(nu, z))*diff(BesselI(- nu, z), x)-diff(BesselI(nu, z), x)*(BesselI(- nu, z))= BesselI(nu, z)*BesselI(- nu - 1, z)- BesselI(nu + 1, z)*BesselI(- nu, z) Wronskian[{BesselI[\[Nu], z], BesselI[- \[Nu], z]}, x]= BesselI[\[Nu], z]*BesselI[- \[Nu]- 1, z]- BesselI[\[Nu]+ 1, z]*BesselI[- \[Nu], z] Failure Failure
Fail
-11.77116916+6.676770606*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-6.676770612-11.77116916*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
11.77116916-6.676770609*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
6.676770609+11.77116916*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-11.771169167436858, 6.676770630088813] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-6.676770630088807, 11.771169167436854] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[11.771169167436858, -6.676770630088829] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[6.67677063008883, -11.771169167436856] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.28.E1 BesselI(nu, z)*BesselI(- nu - 1, z)- BesselI(nu + 1, z)*BesselI(- nu, z)= - 2*sin(nu*Pi)/(Pi*z) BesselI[\[Nu], z]*BesselI[- \[Nu]- 1, z]- BesselI[\[Nu]+ 1, z]*BesselI[- \[Nu], z]= - 2*Sin[\[Nu]*Pi]/(Pi*z) Failure Successful Successful -
10.28.E2 (BesselK(nu, z))*diff(BesselI(nu, z), x)-diff(BesselK(nu, z), x)*(BesselI(nu, z))= BesselI(nu, z)*BesselK(nu + 1, z)+ BesselI(nu + 1, z)*BesselK(nu, z) Wronskian[{BesselK[\[Nu], z], BesselI[\[Nu], z]}, x]= BesselI[\[Nu], z]*BesselK[\[Nu]+ 1, z]+ BesselI[\[Nu]+ 1, z]*BesselK[\[Nu], z] Failure Failure
Fail
-.3535533907+.3535533906*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
-.3535533908-.3535533907*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
.353553388-.35355339*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
.3535533905+.3535533907*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.35355339059327384, 0.3535533905932732] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.35355339059327306, 0.3535533905932739] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.35355339059327395, 0.35355339059327373] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.3535533905932736, 0.3535533905932738] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.28.E2 BesselI(nu, z)*BesselK(nu + 1, z)+ BesselI(nu + 1, z)*BesselK(nu, z)= 1/ z BesselI[\[Nu], z]*BesselK[\[Nu]+ 1, z]+ BesselI[\[Nu]+ 1, z]*BesselK[\[Nu], z]= 1/ z Failure Successful Successful -
10.29#Ex5 subs( temp=z, diff( BesselI(0, temp), temp$(1) ) )= BesselI(1, z) (D[BesselI[0, temp], {temp, 1}]/.temp-> z)= BesselI[1, z] Successful Successful - -
10.29#Ex6 subs( temp=z, diff( BesselK(0, temp), temp$(1) ) )= - BesselK(1, z) (D[BesselK[0, temp], {temp, 1}]/.temp-> z)= - BesselK[1, z] Successful Successful - -
10.31.E1 BesselK(n, z)=(1)/(2)*((1)/(2)*z)^(- n)* sum((factorial(n - k - 1))/(factorial(k))*(-(1)/(4)*(z)^(2))^(k), k = 0..n - 1)+(- 1)^(n + 1)* ln((1)/(2)*z)*BesselI(n, z)+(- 1)^(n)*(1)/(2)*((1)/(2)*z)^(n)* sum((Psi(k + 1)+ Psi(n + k + 1))*(((1)/(4)*(z)^(2))^(k))/(factorial(k)*factorial(n + k)), k = 0..infinity) BesselK[n, z]=Divide[1,2]*(Divide[1,2]*z)^(- n)* Sum[Divide[(n - k - 1)!,(k)!]*(-Divide[1,4]*(z)^(2))^(k), {k, 0, n - 1}]+(- 1)^(n + 1)* Log[Divide[1,2]*z]*BesselI[n, z]+(- 1)^(n)*Divide[1,2]*(Divide[1,2]*z)^(n)* Sum[(PolyGamma[k + 1]+ PolyGamma[n + k + 1])*Divide[(Divide[1,4]*(z)^(2))^(k),(k)!*(n + k)!], {k, 0, Infinity}] Error Failure - Successful
10.32.E1 BesselI(0, z)=(1)/(Pi)*int(exp(+ z*cos(theta)), theta = 0..Pi) BesselI[0, z]=Divide[1,Pi]*Integrate[Exp[+ z*Cos[\[Theta]]], {\[Theta], 0, Pi}] Successful Successful - -
10.32.E1 BesselI(0, z)=(1)/(Pi)*int(exp(- z*cos(theta)), theta = 0..Pi) BesselI[0, z]=Divide[1,Pi]*Integrate[Exp[- z*Cos[\[Theta]]], {\[Theta], 0, Pi}] Successful Successful - -
10.32.E1 (1)/(Pi)*int(exp(+ z*cos(theta)), theta = 0..Pi)=(1)/(Pi)*int(cosh(z*cos(theta)), theta = 0..Pi) Divide[1,Pi]*Integrate[Exp[+ z*Cos[\[Theta]]], {\[Theta], 0, Pi}]=Divide[1,Pi]*Integrate[Cosh[z*Cos[\[Theta]]], {\[Theta], 0, Pi}] Failure Failure Skip Successful
10.32.E1 (1)/(Pi)*int(exp(- z*cos(theta)), theta = 0..Pi)=(1)/(Pi)*int(cosh(z*cos(theta)), theta = 0..Pi) Divide[1,Pi]*Integrate[Exp[- z*Cos[\[Theta]]], {\[Theta], 0, Pi}]=Divide[1,Pi]*Integrate[Cosh[z*Cos[\[Theta]]], {\[Theta], 0, Pi}] Failure Failure Skip Successful
10.32.E2 BesselI(nu, z)=(((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int(exp(+ z*cos(theta))*(sin(theta))^(2*nu), theta = 0..Pi) BesselI[\[Nu], z]=Divide[(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[+ z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*\[Nu]), {\[Theta], 0, Pi}] Failure Failure Skip Error
10.32.E2 BesselI(nu, z)=(((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int(exp(- z*cos(theta))*(sin(theta))^(2*nu), theta = 0..Pi) BesselI[\[Nu], z]=Divide[(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[- z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*\[Nu]), {\[Theta], 0, Pi}] Failure Failure Skip Error
10.32.E2 (((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int(exp(+ z*cos(theta))*(sin(theta))^(2*nu), theta = 0..Pi)=(((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int((1 - (t)^(2))^(nu -(1)/(2))* exp(+ z*t), t = - 1..1) Divide[(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[+ z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*\[Nu]), {\[Theta], 0, Pi}]=Divide[(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[(1 - (t)^(2))^(\[Nu]-Divide[1,2])* Exp[+ z*t], {t, - 1, 1}] Failure Failure Skip Error
10.32.E2 (((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int(exp(- z*cos(theta))*(sin(theta))^(2*nu), theta = 0..Pi)=(((1)/(2)*z)^(nu))/((Pi)^((1)/(2))* GAMMA(nu +(1)/(2)))*int((1 - (t)^(2))^(nu -(1)/(2))* exp(- z*t), t = - 1..1) Divide[(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[- z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*\[Nu]), {\[Theta], 0, Pi}]=Divide[(Divide[1,2]*z)^(\[Nu]),(Pi)^(Divide[1,2])* Gamma[\[Nu]+Divide[1,2]]]*Integrate[(1 - (t)^(2))^(\[Nu]-Divide[1,2])* Exp[- z*t], {t, - 1, 1}] Failure Failure Skip Error
10.32.E3 BesselI(n, z)=(1)/(Pi)*int(exp(z*cos(theta))*cos(n*theta), theta = 0..Pi) BesselI[n, z]=Divide[1,Pi]*Integrate[Exp[z*Cos[\[Theta]]]*Cos[n*\[Theta]], {\[Theta], 0, Pi}] Failure Failure Skip Error
10.32.E4 BesselI(nu, z)=(1)/(Pi)*int(exp(z*cos(theta))*cos(nu*theta), theta = 0..Pi)-(sin(nu*Pi))/(Pi)*int(exp(- z*cosh(t)- nu*t), t = 0..infinity) BesselI[\[Nu], z]=Divide[1,Pi]*Integrate[Exp[z*Cos[\[Theta]]]*Cos[\[Nu]*\[Theta]], {\[Theta], 0, Pi}]-Divide[Sin[\[Nu]*Pi],Pi]*Integrate[Exp[- z*Cosh[t]- \[Nu]*t], {t, 0, Infinity}] Failure Failure Skip Error
10.32.E5 BesselK(0, z)= -(1)/(Pi)*int(exp(+ z*cos(theta))*(gamma + ln(2*z*(sin(theta))^(2))), theta = 0..Pi) BesselK[0, z]= -Divide[1,Pi]*Integrate[Exp[+ z*Cos[\[Theta]]]*(EulerGamma + Log[2*z*(Sin[\[Theta]])^(2)]), {\[Theta], 0, Pi}] Failure Failure Skip Error
10.32.E5 BesselK(0, z)= -(1)/(Pi)*int(exp(- z*cos(theta))*(gamma + ln(2*z*(sin(theta))^(2))), theta = 0..Pi) BesselK[0, z]= -Divide[1,Pi]*Integrate[Exp[- z*Cos[\[Theta]]]*(EulerGamma + Log[2*z*(Sin[\[Theta]])^(2)]), {\[Theta], 0, Pi}] Failure Failure Skip Error
10.32.E6 BesselK(0, x)= int(cos(x*sinh(t)), t = 0..infinity) BesselK[0, x]= Integrate[Cos[x*Sinh[t]], {t, 0, Infinity}] Successful Failure - Error
10.32.E6 int(cos(x*sinh(t)), t = 0..infinity)= int((cos(x*t))/(sqrt((t)^(2)+ 1)), t = 0..infinity) Integrate[Cos[x*Sinh[t]], {t, 0, Infinity}]= Integrate[Divide[Cos[x*t],Sqrt[(t)^(2)+ 1]], {t, 0, Infinity}] Successful Failure - Error
10.32.E7 BesselK(nu, x)= sec((1)/(2)*nu*Pi)*int(cos(x*sinh(t))*cosh(nu*t), t = 0..infinity) BesselK[\[Nu], x]= Sec[Divide[1,2]*\[Nu]*Pi]*Integrate[Cos[x*Sinh[t]]*Cosh[\[Nu]*t], {t, 0, Infinity}] Failure Failure Skip Error
10.32.E7 sec((1)/(2)*nu*Pi)*int(cos(x*sinh(t))*cosh(nu*t), t = 0..infinity)= csc((1)/(2)*nu*Pi)*int(sin(x*sinh(t))*sinh(nu*t), t = 0..infinity) Sec[Divide[1,2]*\[Nu]*Pi]*Integrate[Cos[x*Sinh[t]]*Cosh[\[Nu]*t], {t, 0, Infinity}]= Csc[Divide[1,2]*\[Nu]*Pi]*Integrate[Sin[x*Sinh[t]]*Sinh[\[Nu]*t], {t, 0, Infinity}] Failure Failure Skip Error
10.32.E8 BesselK(nu, z)=((Pi)^((1)/(2))*((1)/(2)*z)^(nu))/(GAMMA(nu +(1)/(2)))*int(exp(- z*cosh(t))*(sinh(t))^(2*nu), t = 0..infinity) BesselK[\[Nu], z]=Divide[(Pi)^(Divide[1,2])*(Divide[1,2]*z)^(\[Nu]),Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[- z*Cosh[t]]*(Sinh[t])^(2*\[Nu]), {t, 0, Infinity}] Failure Failure Skip Error
10.32.E8 ((Pi)^((1)/(2))*((1)/(2)*z)^(nu))/(GAMMA(nu +(1)/(2)))*int(exp(- z*cosh(t))*(sinh(t))^(2*nu), t = 0..infinity)=((Pi)^((1)/(2))*((1)/(2)*z)^(nu))/(GAMMA(nu +(1)/(2)))*int(exp(- z*t)*((t)^(2)- 1)^(nu -(1)/(2)), t = 1..infinity) Divide[(Pi)^(Divide[1,2])*(Divide[1,2]*z)^(\[Nu]),Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[- z*Cosh[t]]*(Sinh[t])^(2*\[Nu]), {t, 0, Infinity}]=Divide[(Pi)^(Divide[1,2])*(Divide[1,2]*z)^(\[Nu]),Gamma[\[Nu]+Divide[1,2]]]*Integrate[Exp[- z*t]*((t)^(2)- 1)^(\[Nu]-Divide[1,2]), {t, 1, Infinity}] Failure Failure Skip Error
10.32.E9 BesselK(nu, z)= int(exp(- z*cosh(t))*cosh(nu*t), t = 0..infinity) BesselK[\[Nu], z]= Integrate[Exp[- z*Cosh[t]]*Cosh[\[Nu]*t], {t, 0, Infinity}] Failure Failure Skip Error
10.32.E10 BesselK(nu, z)=(1)/(2)*((1)/(2)*z)^(nu)* int(exp(- t -((z)^(2))/(4*t))*(1)/((t)^(nu + 1)), t = 0..infinity) BesselK[\[Nu], z]=Divide[1,2]*(Divide[1,2]*z)^(\[Nu])* Integrate[Exp[- t -Divide[(z)^(2),4*t]]*Divide[1,(t)^(\[Nu]+ 1)], {t, 0, Infinity}] Successful Failure - Skip
10.32.E11 BesselK(nu, x*z)=(GAMMA(nu +(1)/(2))*(2*z)^(nu))/((Pi)^((1)/(2))* (x)^(nu))*int((cos(x*t))/(((t)^(2)+ (z)^(2))^(nu +(1)/(2))), t = 0..infinity) BesselK[\[Nu], x*z]=Divide[Gamma[\[Nu]+Divide[1,2]]*(2*z)^(\[Nu]),(Pi)^(Divide[1,2])* (x)^(\[Nu])]*Integrate[Divide[Cos[x*t],((t)^(2)+ (z)^(2))^(\[Nu]+Divide[1,2])], {t, 0, Infinity}] Failure Failure Skip Error
10.32.E12 BesselI(nu, z)=(1)/(2*Pi*I)*int(exp(z*cosh(t)- nu*t), t = infinity - I*Pi..infinity + I*Pi) BesselI[\[Nu], z]=Divide[1,2*Pi*I]*Integrate[Exp[z*Cosh[t]- \[Nu]*t], {t, Infinity - I*Pi, Infinity + I*Pi}] Failure Failure Skip
Fail
Complex[0.47377882604348887, 0.17987673448701852] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-2.5712028891376235, 2.011728577446344] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-5.630703522037926, 7.3730343474306625] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.4462596814449855, 3.2604536086998377] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.32.E13 BesselK(nu, z)=(((1)/(2)*z)^(nu))/(4*Pi*I)*int(GAMMA(t)*GAMMA(t - nu)*((1)/(2)*z)^(- 2*t), t = c - I*infinity..c + I*infinity) BesselK[\[Nu], z]=Divide[(Divide[1,2]*z)^(\[Nu]),4*Pi*I]*Integrate[Gamma[t]*Gamma[t - \[Nu]]*(Divide[1,2]*z)^(- 2*t), {t, c - I*Infinity, c + I*Infinity}] Failure Failure Skip Error
10.32.E14 BesselK(nu, z)=(1)/(2*(Pi)^(2)* I)*((Pi)/(2*z))^((1)/(2))* exp(- z)*cos(nu*Pi)* int(GAMMA(t)*GAMMA((1)/(2)- t - nu)*GAMMA((1)/(2)- t + nu)*(2*z)^(t), t = - I*infinity..I*infinity) BesselK[\[Nu], z]=Divide[1,2*(Pi)^(2)* I]*(Divide[Pi,2*z])^(Divide[1,2])* Exp[- z]*Cos[\[Nu]*Pi]* Integrate[Gamma[t]*Gamma[Divide[1,2]- t - \[Nu]]*Gamma[Divide[1,2]- t + \[Nu]]*(2*z)^(t), {t, - I*Infinity, I*Infinity}] Failure Failure Skip Error
10.32.E15 BesselI(mu, z)*BesselI(nu, z)=(2)/(Pi)*int(BesselI(mu + nu, 2*z*cos(theta))*cos((mu - nu)* theta), theta = 0..(1)/(2)*Pi) BesselI[\[Mu], z]*BesselI[\[Nu], z]=Divide[2,Pi]*Integrate[BesselI[\[Mu]+ \[Nu], 2*z*Cos[\[Theta]]]*Cos[(\[Mu]- \[Nu])* \[Theta]], {\[Theta], 0, Divide[1,2]*Pi}] Failure Failure Skip Skip
10.32.E16 BesselI(mu, x)*BesselK(nu, x)= int(BesselJ(mu + nu, 2*x*sinh(t))*exp((- mu + nu)* t), t = 0..infinity) BesselI[\[Mu], x]*BesselK[\[Nu], x]= Integrate[BesselJ[\[Mu]+ \[Nu], 2*x*Sinh[t]]*Exp[(- \[Mu]+ \[Nu])* t], {t, 0, Infinity}] Failure Failure Skip Error
10.32.E16 BesselI(mu, x)*BesselK(nu, x)= int(BesselJ(mu - nu, 2*x*sinh(t))*exp((- mu - nu)* t), t = 0..infinity) BesselI[\[Mu], x]*BesselK[\[Nu], x]= Integrate[BesselJ[\[Mu]- \[Nu], 2*x*Sinh[t]]*Exp[(- \[Mu]- \[Nu])* t], {t, 0, Infinity}] Failure Failure Skip Error
10.32.E17 BesselK(mu, z)*BesselK(nu, z)= 2*int(BesselK(mu + nu, 2*z*cosh(t))*cosh((mu - nu)* t), t = 0..infinity) BesselK[\[Mu], z]*BesselK[\[Nu], z]= 2*Integrate[BesselK[\[Mu]+ \[Nu], 2*z*Cosh[t]]*Cosh[(\[Mu]- \[Nu])* t], {t, 0, Infinity}] Failure Failure Skip Error
10.32.E17 BesselK(mu, z)*BesselK(nu, z)= 2*int(BesselK(mu - nu, 2*z*cosh(t))*cosh((mu + nu)* t), t = 0..infinity) BesselK[\[Mu], z]*BesselK[\[Nu], z]= 2*Integrate[BesselK[\[Mu]- \[Nu], 2*z*Cosh[t]]*Cosh[(\[Mu]+ \[Nu])* t], {t, 0, Infinity}] Failure Failure Skip Error
10.32.E18 BesselK(nu, z)*BesselK(nu, zeta)=(1)/(2)*int(exp(-(t)/(2)-((z)^(2)+ (zeta)^(2))/(2*t))*BesselK(nu, ((z*zeta)/(t))*)*(1)/(t), t = 0..infinity) BesselK[\[Nu], z]*BesselK[\[Nu], \[zeta]]=Divide[1,2]*Integrate[Exp[-Divide[t,2]-Divide[(z)^(2)+ (\[zeta])^(2),2*t]]*BesselK[\[Nu], (Divide[z*\[zeta],t])*]*Divide[1,t], {t, 0, Infinity}] Error Failure - Error
10.32.E19 BesselK(mu, z)*BesselK(nu, z)=(1)/(8*Pi*I)*int((GAMMA(t +(1)/(2)*mu +(1)/(2)*nu)*GAMMA(t +(1)/(2)*mu -(1)/(2)*nu)*GAMMA(t -(1)/(2)*mu +(1)/(2)*nu)*GAMMA(t -(1)/(2)*mu -(1)/(2)*nu))/(GAMMA(2*t))*((1)/(2)*z)^(- 2*t), t = c - I*infinity..c + I*infinity) BesselK[\[Mu], z]*BesselK[\[Nu], z]=Divide[1,8*Pi*I]*Integrate[Divide[Gamma[t +Divide[1,2]*\[Mu]+Divide[1,2]*\[Nu]]*Gamma[t +Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]]*Gamma[t -Divide[1,2]*\[Mu]+Divide[1,2]*\[Nu]]*Gamma[t -Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]],Gamma[2*t]]*(Divide[1,2]*z)^(- 2*t), {t, c - I*Infinity, c + I*Infinity}] Failure Failure Skip Error
10.34.E1 BesselI(nu, z*exp(m*Pi*I))= exp(m*nu*Pi*I)*BesselI(nu, z) BesselI[\[Nu], z*Exp[m*Pi*I]]= Exp[m*\[Nu]*Pi*I]*BesselI[\[Nu], z] Failure Failure
Fail
-25.46648651+34.76058054*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}
.4738478497+.1798644481*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}
-25.46593127+34.75464498*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}
-2.571651015-2.011784848*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2}
... skip entries to safe data
Fail
Complex[-25.466486497459893, 34.76058068352855] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.47384785056443346, 0.17986444785635414] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-25.465931246406512, 34.754645199849094] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-223.08150600961898, -165.2079311070147] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.34.E2 BesselK(nu, z*exp(m*Pi*I))= exp(- m*nu*Pi*I)*BesselK(nu, z)- Pi*I*sin(m*nu*Pi)*csc(nu*Pi)*BesselI(nu, z) BesselK[\[Nu], z*Exp[m*Pi*I]]= Exp[- m*\[Nu]*Pi*I]*BesselK[\[Nu], z]- Pi*I*Sin[m*\[Nu]*Pi]*Csc[\[Nu]*Pi]*BesselI[\[Nu], z] Failure Failure
Fail
-23.72816996-16.20095675*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}
1974.016674-1497.794581*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}
78036.43688+195659.8555*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}
-346.8741250+807.6398259*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2}
... skip entries to safe data
Fail
Complex[-23.728169968169517, -16.200956740051907] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1974.0166738862135, -1497.7945856695665] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[78036.4381344157, 195659.85598804062] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-16.563048824383813, -6.675705970582722] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.34.E3 BesselI(nu, z*exp(m*Pi*I))=(I/ Pi)*(+ exp(m*nu*Pi*I)*BesselK(nu, z*exp(+ Pi*I))- exp((m - 1)* nu*Pi*I)*BesselK(nu, z)) BesselI[\[Nu], z*Exp[m*Pi*I]]=(I/ Pi)*(+ Exp[m*\[Nu]*Pi*I]*BesselK[\[Nu], z*Exp[+ Pi*I]]- Exp[(m - 1)* \[Nu]*Pi*I]*BesselK[\[Nu], z]) Failure Failure
Fail
-25.36470622+34.79539330*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}
.4739237916+.1786015012*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}
-25.46594582+34.75464807*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}
-2.571651015-2.011784848*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2}
... skip entries to safe data
Fail
Complex[-25.36470621174673, 34.79539343891294] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.47392379247751504, 0.17860150099441258] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-25.465945802770374, 34.75464829402328] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[257.07974135711925, -110.41346285737623] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.34.E3 BesselI(nu, z*exp(m*Pi*I))=(I/ Pi)*(- exp(m*nu*Pi*I)*BesselK(nu, z*exp(- Pi*I))+ exp((m + 1)* nu*Pi*I)*BesselK(nu, z)) BesselI[\[Nu], z*Exp[m*Pi*I]]=(I/ Pi)*(- Exp[m*\[Nu]*Pi*I]*BesselK[\[Nu], z*Exp[- Pi*I]]+ Exp[(m + 1)* \[Nu]*Pi*I]*BesselK[\[Nu], z]) Failure Failure
Fail
-25.46648651+34.76058054*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}
.4738478497+.1798644481*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}
-25.46593127+34.75464498*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}
-.5311818950e-1+.4060217813e-1*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 1}
... skip entries to safe data
Fail
Complex[-25.466486497459893, 34.76058068352855] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.47384785056443346, 0.17986444785635414] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-25.465931246406512, 34.754645199849094] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-223.0815060096189, -165.20793110701467] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.34.E4 BesselK(nu, z*exp(m*Pi*I))= csc(nu*Pi)*(+ sin(m*nu*Pi)*BesselK(nu, z*exp(+ Pi*I))- sin((m - 1)* nu*Pi)*BesselK(nu, z)) BesselK[\[Nu], z*Exp[m*Pi*I]]= Csc[\[Nu]*Pi]*(+ Sin[m*\[Nu]*Pi]*BesselK[\[Nu], z*Exp[+ Pi*I]]- Sin[(m - 1)* \[Nu]*Pi]*BesselK[\[Nu], z]) Failure Failure
Fail
109.3129522+79.68557470*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}
-9027.967286+7136.744811*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}
-346.8741247+807.6398251*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 2}
-58340.79702-46700.99889*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 3}
... skip entries to safe data
Fail
Complex[109.31295240645538, 79.68557469528692] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-9027.967291383398, 7136.744876012727] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-346.8741237701426, -807.6398268342901] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-58340.79750295953, 46700.99881352048] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.34.E4 BesselK(nu, z*exp(m*Pi*I))= csc(nu*Pi)*(- sin(m*nu*Pi)*BesselK(nu, z*exp(- Pi*I))+ sin((m + 1)* nu*Pi)*BesselK(nu, z)) BesselK[\[Nu], z*Exp[m*Pi*I]]= Csc[\[Nu]*Pi]*(- Sin[m*\[Nu]*Pi]*BesselK[\[Nu], z*Exp[- Pi*I]]+ Sin[(m + 1)* \[Nu]*Pi]*BesselK[\[Nu], z]) Failure Failure
Fail
-23.72816993-16.20095676*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 1}
1974.016672-1497.794570*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 2}
78036.44012+195659.8571*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), m = 3}
-16.56304884+6.675705955*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), m = 1}
... skip entries to safe data
Fail
Complex[-23.728169968169517, -16.2009567400519] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1974.0166738862144, -1497.7945856695726] <- {Rule[m, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[78036.43813441524, 195659.85598804036] <- {Rule[m, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-16.563048824383813, -6.675705970582721] <- {Rule[m, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.34.E5 BesselK(n, z*exp(m*Pi*I))=(- 1)^(m*n)* BesselK(n, z)+(- 1)^(n*(m - 1)- 1)* m*Pi*I*BesselI(n, z) BesselK[n, z*Exp[m*Pi*I]]=(- 1)^(m*n)* BesselK[n, z]+(- 1)^(n*(m - 1)- 1)* m*Pi*I*BesselI[n, z] Failure Failure
Fail
-6.264823649+1.883544620*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 1}
-3.011056351-1.038481291*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 2}
-.5379125927-.9060977874*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 3}
6.264823651-1.883544623*I <- {z = 2^(1/2)+I*2^(1/2), m = 2, n = 1}
... skip entries to safe data
Fail
Complex[-6.264823652701258, 1.8835446212245166] <- {Rule[m, 1], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-3.0110563535593116, -1.0384812903661844] <- {Rule[m, 1], Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.537912595321433, -0.9060977859421694] <- {Rule[m, 1], Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[6.264823652701258, -1.8835446212245166] <- {Rule[m, 2], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.34.E6 BesselK(n, z*exp(m*Pi*I))= +(- 1)^(n*(m - 1))* m*BesselK(n, z*exp(+ Pi*I))-(- 1)^(n*m)*(m - 1)* BesselK(n, z) BesselK[n, z*Exp[m*Pi*I]]= +(- 1)^(n*(m - 1))* m*BesselK[n, z*Exp[+ Pi*I]]-(- 1)^(n*m)*(m - 1)* BesselK[n, z] Failure Failure
Fail
-6.264823650+1.883544622*I <- {z = 2^(1/2)+I*2^(1/2), m = 2, n = 1}
3.011056351+1.038481289*I <- {z = 2^(1/2)+I*2^(1/2), m = 2, n = 2}
-.5379125964-.9060977833*I <- {z = 2^(1/2)+I*2^(1/2), m = 2, n = 3}
6.264823652-1.883544629*I <- {z = 2^(1/2)+I*2^(1/2), m = 3, n = 1}
... skip entries to safe data
Fail
Complex[-6.264823652701258, 1.8835446212245168] <- {Rule[m, 2], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[3.0110563535593116, 1.0384812903661842] <- {Rule[m, 2], Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.537912595321433, -0.9060977859421694] <- {Rule[m, 2], Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[6.264823652701258, -1.8835446212245168] <- {Rule[m, 3], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.34.E6 BesselK(n, z*exp(m*Pi*I))= -(- 1)^(n*(m - 1))* m*BesselK(n, z*exp(- Pi*I))+(- 1)^(n*m)*(m + 1)* BesselK(n, z) BesselK[n, z*Exp[m*Pi*I]]= -(- 1)^(n*(m - 1))* m*BesselK[n, z*Exp[- Pi*I]]+(- 1)^(n*m)*(m + 1)* BesselK[n, z] Failure Failure
Fail
-6.264823648+1.883544620*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 1}
-3.011056351-1.038481291*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 2}
-.537912592-.9060977875*I <- {z = 2^(1/2)+I*2^(1/2), m = 1, n = 3}
6.264823649-1.883544623*I <- {z = 2^(1/2)+I*2^(1/2), m = 2, n = 1}
... skip entries to safe data
Fail
Complex[-6.264823652701258, 1.8835446212245168] <- {Rule[m, 1], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-3.0110563535593116, -1.0384812903661842] <- {Rule[m, 1], Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.537912595321433, -0.9060977859421694] <- {Rule[m, 1], Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[6.264823652701258, -1.8835446212245168] <- {Rule[m, 2], Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.34#Ex1 BesselI(nu, conjugate(z))= conjugate(BesselI(nu, z)) BesselI[\[Nu], Conjugate[z]]= Conjugate[BesselI[\[Nu], z]] Failure Failure
Fail
-3.044981713-1.831851844*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
3.044981713-1.831851844*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
25.45117532+34.79009675*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
-25.45117532+34.79009675*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-3.0449817151811125, -1.8318518429593253] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[3.0449817151811125, 1.8318518429593253] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[6.0769632034829115, 4.112580738730825] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-6.0769632034829115, -4.112580738730825] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.34#Ex2 BesselK(nu, conjugate(z))= conjugate(BesselK(nu, z)) BesselK[\[Nu], Conjugate[z]]= Conjugate[BesselK[\[Nu], z]] Failure Failure
Fail
-.2418444739-.2420650681*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
.2418444739-.2420650681*I <- {nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
-9.672734607+.86628375e-1*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
9.672734607+.86628375e-1*I <- {nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.2418444736872933, -0.24206506816430606] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.2418444736872933, 0.24206506816430606] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.2418444736872933, -0.24206506816430606] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.2418444736872933, 0.24206506816430606] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[ν, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.35.E1 exp((1)/(2)*z*(t + (t)^(- 1)))= sum((t)^(m)* BesselI(m, z), m = - infinity..infinity) Exp[Divide[1,2]*z*(t + (t)^(- 1))]= Sum[(t)^(m)* BesselI[m, z], {m, - Infinity, Infinity}] Failure Failure Skip Error
10.35.E2 exp(z*cos(theta))= BesselI(0, z)+ 2*sum(BesselI(k, z)*cos(k*theta), k = 1..infinity) Exp[z*Cos[\[Theta]]]= BesselI[0, z]+ 2*Sum[BesselI[k, z]*Cos[k*\[Theta]], {k, 1, Infinity}] Failure Successful Skip -
10.35.E3 exp(z*sin(theta))= BesselI(0, z)+ 2*sum((- 1)^(k)* BesselI(2*k + 1, z)*sin((2*k + 1)* theta), k = 0..infinity)+ 2*sum((- 1)^(k)* BesselI(2*k, z)*cos(2*k*theta), k = 1..infinity) Exp[z*Sin[\[Theta]]]= BesselI[0, z]+ 2*Sum[(- 1)^(k)* BesselI[2*k + 1, z]*Sin[(2*k + 1)* \[Theta]], {k, 0, Infinity}]+ 2*Sum[(- 1)^(k)* BesselI[2*k, z]*Cos[2*k*\[Theta]], {k, 1, Infinity}] Failure Failure Skip Skip
10.37.E1 abs(BesselK(nu, z))<abs(BesselK(mu, z)) Abs[BesselK[\[Nu], z]]<Abs[BesselK[\[Mu], z]] Failure Failure
Fail
.3485691514 < .3485691514 <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2)}
.1206554296 < .1206554296 <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2)}
1.592895962 < 1.592895962 <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)-I*2^(1/2)}
10.33727274 < 10.33727274 <- {mu = 2^(1/2)+I*2^(1/2), nu = 2^(1/2)+I*2^(1/2), z = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Successful
10.38.E1 diff(BesselI(+ nu, z), nu)= + BesselI(+ nu, z)*ln((1)/(2)*z)-((1)/(2)*z)^(+ nu)* sum((Psi(k + 1 + nu))/(GAMMA(k + 1 + nu))*(((1)/(4)*(z)^(2))^(k))/(factorial(k)), k = 0..infinity) D[BesselI[+ \[Nu], z], \[Nu]]= + BesselI[+ \[Nu], z]*Log[Divide[1,2]*z]-(Divide[1,2]*z)^(+ \[Nu])* Sum[Divide[PolyGamma[k + 1 + \[Nu]],Gamma[k + 1 + \[Nu]]]*Divide[(Divide[1,4]*(z)^(2))^(k),(k)!], {k, 0, Infinity}] Failure Failure Skip Successful
10.38.E1 diff(BesselI(- nu, z), nu)= - BesselI(- nu, z)*ln((1)/(2)*z)+((1)/(2)*z)^(- nu)* sum((Psi(k + 1 - nu))/(GAMMA(k + 1 - nu))*(((1)/(4)*(z)^(2))^(k))/(factorial(k)), k = 0..infinity) D[BesselI[- \[Nu], z], \[Nu]]= - BesselI[- \[Nu], z]*Log[Divide[1,2]*z]+(Divide[1,2]*z)^(- \[Nu])* Sum[Divide[PolyGamma[k + 1 - \[Nu]],Gamma[k + 1 - \[Nu]]]*Divide[(Divide[1,4]*(z)^(2))^(k),(k)!], {k, 0, Infinity}] Failure Failure Skip Successful
10.38.E2 diff(BesselK(nu, z), nu)=(1)/(2)*Pi*csc(nu*Pi)*(diff(BesselI(- nu, z), nu)- diff(BesselI(nu, z), nu))- Pi*cot(nu*Pi)*BesselK(nu, z) D[BesselK[\[Nu], z], \[Nu]]=Divide[1,2]*Pi*Csc[\[Nu]*Pi]*(D[BesselI[- \[Nu], z], \[Nu]]- D[BesselI[\[Nu], z], \[Nu]])- Pi*Cot[\[Nu]*Pi]*BesselK[\[Nu], z] Successful Failure - Successful
10.39#Ex1 BesselI((1)/(2), z)=((2)/(Pi*z))^((1)/(2))* sinh(z) BesselI[Divide[1,2], z]=(Divide[2,Pi*z])^(Divide[1,2])* Sinh[z] Failure Failure Successful Successful
10.39#Ex2 BesselI(-(1)/(2), z)=((2)/(Pi*z))^((1)/(2))* cosh(z) BesselI[-Divide[1,2], z]=(Divide[2,Pi*z])^(Divide[1,2])* Cosh[z] Failure Failure Successful Successful
10.39.E2 BesselK((1)/(2), z)= BesselK(-(1)/(2), z) BesselK[Divide[1,2], z]= BesselK[-Divide[1,2], z] Successful Successful - -
10.39.E2 BesselK(-(1)/(2), z)=((Pi)/(2*z))^((1)/(2))* exp(- z) BesselK[-Divide[1,2], z]=(Divide[Pi,2*z])^(Divide[1,2])* Exp[- z] Failure Failure Successful Successful
10.39.E3 BesselK((1)/(4), z)= (Pi)^((1)/(2))* (z)^(-(1)/(4))* CylinderU(0, 2*(z)^((1)/(2))) BesselK[Divide[1,4], z]= (Pi)^(Divide[1,2])* (z)^(-Divide[1,4])* ParabolicCylinderD[-0 - 1/2, 2*(z)^(Divide[1,2])] Successful Failure - Successful
10.39.E4 BesselK((3)/(4), z)=(1)/(2)*(Pi)^((1)/(2))* (z)^(-(3)/(4))*((1)/(2)*CylinderU(1, 2*(z)^((1)/(2)))+ CylinderU(- 1, 2*(z)^((1)/(2)))) BesselK[Divide[3,4], z]=Divide[1,2]*(Pi)^(Divide[1,2])* (z)^(-Divide[3,4])*(Divide[1,2]*ParabolicCylinderD[-1 - 1/2, 2*(z)^(Divide[1,2])]+ ParabolicCylinderD[-- 1 - 1/2, 2*(z)^(Divide[1,2])]) Failure Failure Successful
Fail
Complex[-0.24654129480515125, -0.14875200582767972] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[PreDecrement[1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.5994174244432623, -0.2209182787401364] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[PreDecrement[1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.045375099662282176, -0.23186773164961136] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[PreDecrement[1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.05931481647092712, -0.20328323070426188] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[PreDecrement[1], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.39.E5 BesselI(nu, z)=(((1)/(2)*z)^(nu)* exp(+ z))/(GAMMA(nu + 1))*KummerM(nu +(1)/(2), 2*nu + 1, - 2*z) BesselI[\[Nu], z]=Divide[(Divide[1,2]*z)^(\[Nu])* Exp[+ z],Gamma[\[Nu]+ 1]]*Hypergeometric1F1[\[Nu]+Divide[1,2], 2*\[Nu]+ 1, - 2*z] Failure Successful Successful -
10.39.E5 BesselI(nu, z)=(((1)/(2)*z)^(nu)* exp(- z))/(GAMMA(nu + 1))*KummerM(nu +(1)/(2), 2*nu + 1, + 2*z) BesselI[\[Nu], z]=Divide[(Divide[1,2]*z)^(\[Nu])* Exp[- z],Gamma[\[Nu]+ 1]]*Hypergeometric1F1[\[Nu]+Divide[1,2], 2*\[Nu]+ 1, + 2*z] Successful Successful - -
10.39.E6 BesselK(nu, z)= (Pi)^((1)/(2))*(2*z)^(nu)* exp(- z)*KummerU(nu +(1)/(2), 2*nu + 1, 2*z) BesselK[\[Nu], z]= (Pi)^(Divide[1,2])*(2*z)^(\[Nu])* Exp[- z]*HypergeometricU[\[Nu]+Divide[1,2], 2*\[Nu]+ 1, 2*z] Successful Successful - -
10.39.E7 BesselI(nu, z)=((2*z)^(-(1)/(2))* WhittakerM(0, nu, 2*z))/((2)^(2*nu)* GAMMA(nu + 1)) BesselI[\[Nu], z]=Divide[(2*z)^(-Divide[1,2])* WhittakerM[0, \[Nu], 2*z],(2)^(2*\[Nu])* Gamma[\[Nu]+ 1]] Successful Successful - -
10.39.E8 BesselK(nu, z)=((Pi)/(2*z))^((1)/(2))* WhittakerW(0, nu, 2*z) BesselK[\[Nu], z]=(Divide[Pi,2*z])^(Divide[1,2])* WhittakerW[0, \[Nu], 2*z] Failure Failure Successful Successful
10.39.E9 BesselI(nu, z)=(((1)/(2)*z)^(nu))/(GAMMA(nu + 1))*hypergeom([-], [nu + 1], (1)/(4)*(z)^(2)) BesselI[\[Nu], z]=Divide[(Divide[1,2]*z)^(\[Nu]),Gamma[\[Nu]+ 1]]*HypergeometricPFQ[{-}, {\[Nu]+ 1}, Divide[1,4]*(z)^(2)] Error Failure - Error
10.40.E10 BesselK(nu, z)=((Pi)/(2*z))^((1)/(2))* exp(- z)*(sum((a[k]*(nu))/((z)^(k)), k = 0..ell - 1)+ R[ell]*(nu , z)) BesselK[\[Nu], z]=(Divide[Pi,2*z])^(Divide[1,2])* Exp[- z]*(Sum[Divide[Subscript[a, k]*(\[Nu]),(z)^(k)], {k, 0, \[ScriptL]- 1}]+ Subscript[R, \[ScriptL]]*(\[Nu], z)) Failure Failure Skip Error
10.40.E13 R[ell]*(nu , z)=(- 1)^(ell)* 2*cos(nu*Pi)*(sum((a[k]*(nu))/((z)^(k))*(exp(2*z)/(2*Pi))*GAMMA(ell - k)*GAMMA(1-ell - k,2*z), k = 0..m - 1)+ R[m , ell]*(nu , z)) Error Failure Error Skip -
10.43.E4 int((BesselI(0, t)- 1)/(t), t = 0..x)=(1)/(2)*sum((- 1)^(k - 1)*(Psi(k + 1)- Psi(1))/(factorial(k))*((1)/(2)*x)^(k)* BesselI(k, x), k = 1..infinity) Integrate[Divide[BesselI[0, t]- 1,t], {t, 0, x}]=Divide[1,2]*Sum[(- 1)^(k - 1)*Divide[PolyGamma[k + 1]- PolyGamma[1],(k)!]*(Divide[1,2]*x)^(k)* BesselI[k, x], {k, 1, Infinity}] Failure Failure Skip Skip
10.43.E4 (1)/(2)*sum((- 1)^(k - 1)*(Psi(k + 1)- Psi(1))/(factorial(k))*((1)/(2)*x)^(k)* BesselI(k, x), k = 1..infinity)=(2)/(x)*sum((- 1)^(k)*(2*k + 3)*(Psi(k + 2)- Psi(1))* BesselI(2*k + 3, x), k = 0..infinity) Divide[1,2]*Sum[(- 1)^(k - 1)*Divide[PolyGamma[k + 1]- PolyGamma[1],(k)!]*(Divide[1,2]*x)^(k)* BesselI[k, x], {k, 1, Infinity}]=Divide[2,x]*Sum[(- 1)^(k)*(2*k + 3)*(PolyGamma[k + 2]- PolyGamma[1])* BesselI[2*k + 3, x], {k, 0, Infinity}] Failure Failure Skip Skip
10.43.E5 int((BesselK(0, t))/(t), t = x..infinity)=(1)/(2)*(ln((1)/(2)*x)+ gamma)^(2)+((Pi)^(2))/(24)- sum((Psi(k + 1)+(1)/(2*k)- ln((1)/(2)*x))*(((1)/(2)*x)^(2*k))/(2*k*(factorial(k))^(2)), k = 1..infinity) Integrate[Divide[BesselK[0, t],t], {t, x, Infinity}]=Divide[1,2]*(Log[Divide[1,2]*x]+ EulerGamma)^(2)+Divide[(Pi)^(2),24]- Sum[(PolyGamma[k + 1]+Divide[1,2*k]- Log[Divide[1,2]*x])*Divide[(Divide[1,2]*x)^(2*k),2*k*((k)!)^(2)], {k, 1, Infinity}] Failure Failure Skip Skip
10.43.E6 int(exp(- t)*BesselI(n, t), t = 0..x)= x*exp(- x)*(BesselI(0, x)+ BesselI(1, x))+ n*(exp(- x)*BesselI(0, x)- 1)+ 2*exp(- x)*sum((n - k)* BesselI(k, x), k = 1..n - 1) Integrate[Exp[- t]*BesselI[n, t], {t, 0, x}]= x*Exp[- x]*(BesselI[0, x]+ BesselI[1, x])+ n*(Exp[- x]*BesselI[0, x]- 1)+ 2*Exp[- x]*Sum[(n - k)* BesselI[k, x], {k, 1, n - 1}] Failure Failure Skip Successful
10.43.E7 int(exp(+ t)*(t)^(nu)* BesselI(nu, t), t = 0..x)=(exp(+ x)*(x)^(nu + 1))/(2*nu + 1)*(BesselI(nu, x)- BesselI(nu + 1, x)) Integrate[Exp[+ t]*(t)^(\[Nu])* BesselI[\[Nu], t], {t, 0, x}]=Divide[Exp[+ x]*(x)^(\[Nu]+ 1),2*\[Nu]+ 1]*(BesselI[\[Nu], x]- BesselI[\[Nu]+ 1, x]) Failure Failure Skip Successful
10.43.E7 int(exp(- t)*(t)^(nu)* BesselI(nu, t), t = 0..x)=(exp(- x)*(x)^(nu + 1))/(2*nu + 1)*(BesselI(nu, x)+ BesselI(nu + 1, x)) Integrate[Exp[- t]*(t)^(\[Nu])* BesselI[\[Nu], t], {t, 0, x}]=Divide[Exp[- x]*(x)^(\[Nu]+ 1),2*\[Nu]+ 1]*(BesselI[\[Nu], x]+ BesselI[\[Nu]+ 1, x]) Failure Failure Skip Successful
10.43.E8 int(exp(+ t)*(t)^(- nu)* BesselI(nu, t), t = 0..x)= -(exp(+ x)*(x)^(- nu + 1))/(2*nu - 1)*(BesselI(nu, x)- BesselI(nu - 1, x))-((2)^(- nu + 1))/((2*nu - 1)* GAMMA(nu)) Integrate[Exp[+ t]*(t)^(- \[Nu])* BesselI[\[Nu], t], {t, 0, x}]= -Divide[Exp[+ x]*(x)^(- \[Nu]+ 1),2*\[Nu]- 1]*(BesselI[\[Nu], x]- BesselI[\[Nu]- 1, x])-Divide[(2)^(- \[Nu]+ 1),(2*\[Nu]- 1)* Gamma[\[Nu]]] Failure Failure Skip Successful
10.43.E8 int(exp(- t)*(t)^(- nu)* BesselI(nu, t), t = 0..x)= -(exp(- x)*(x)^(- nu + 1))/(2*nu - 1)*(BesselI(nu, x)+ BesselI(nu - 1, x))+((2)^(- nu + 1))/((2*nu - 1)* GAMMA(nu)) Integrate[Exp[- t]*(t)^(- \[Nu])* BesselI[\[Nu], t], {t, 0, x}]= -Divide[Exp[- x]*(x)^(- \[Nu]+ 1),2*\[Nu]- 1]*(BesselI[\[Nu], x]+ BesselI[\[Nu]- 1, x])+Divide[(2)^(- \[Nu]+ 1),(2*\[Nu]- 1)* Gamma[\[Nu]]] Successful Failure - Successful
10.43.E9 int(exp(+ t)*(t)^(nu)* BesselK(nu, t), t = 0..x)=(exp(+ x)*(x)^(nu + 1))/(2*nu + 1)*(BesselK(nu, x)+ BesselK(nu + 1, x))-((2)^(nu)* GAMMA(nu + 1))/(2*nu + 1) Integrate[Exp[+ t]*(t)^(\[Nu])* BesselK[\[Nu], t], {t, 0, x}]=Divide[Exp[+ x]*(x)^(\[Nu]+ 1),2*\[Nu]+ 1]*(BesselK[\[Nu], x]+ BesselK[\[Nu]+ 1, x])-Divide[(2)^(\[Nu])* Gamma[\[Nu]+ 1],2*\[Nu]+ 1] Failure Failure Skip Error
10.43.E9 int(exp(- t)*(t)^(nu)* BesselK(nu, t), t = 0..x)=(exp(- x)*(x)^(nu + 1))/(2*nu + 1)*(BesselK(nu, x)- BesselK(nu + 1, x))+((2)^(nu)* GAMMA(nu + 1))/(2*nu + 1) Integrate[Exp[- t]*(t)^(\[Nu])* BesselK[\[Nu], t], {t, 0, x}]=Divide[Exp[- x]*(x)^(\[Nu]+ 1),2*\[Nu]+ 1]*(BesselK[\[Nu], x]- BesselK[\[Nu]+ 1, x])+Divide[(2)^(\[Nu])* Gamma[\[Nu]+ 1],2*\[Nu]+ 1] Failure Failure Skip Error
10.43.E10 int(exp(t)*(t)^(- nu)* BesselK(nu, t), t = x..infinity)=(exp(x)*(x)^(- nu + 1))/(2*nu - 1)*(BesselK(nu, x)+ BesselK(nu - 1, x)) Integrate[Exp[t]*(t)^(- \[Nu])* BesselK[\[Nu], t], {t, x, Infinity}]=Divide[Exp[x]*(x)^(- \[Nu]+ 1),2*\[Nu]- 1]*(BesselK[\[Nu], x]+ BesselK[\[Nu]- 1, x]) Failure Failure Skip Skip
10.43.E18 int(BesselK(nu, t), t = 0..infinity)=(1)/(2)*Pi*sec((1)/(2)*Pi*nu) Integrate[BesselK[\[Nu], t], {t, 0, Infinity}]=Divide[1,2]*Pi*Sec[Divide[1,2]*Pi*\[Nu]] Successful Failure - Successful
10.43.E19 int((t)^(mu - 1)* BesselK(nu, t), t = 0..infinity)= (2)^(mu - 2)* GAMMA((1)/(2)*mu -(1)/(2)*nu)*GAMMA((1)/(2)*mu +(1)/(2)*nu) Integrate[(t)^(\[Mu]- 1)* BesselK[\[Nu], t], {t, 0, Infinity}]= (2)^(\[Mu]- 2)* Gamma[Divide[1,2]*\[Mu]-Divide[1,2]*\[Nu]]*Gamma[Divide[1,2]*\[Mu]+Divide[1,2]*\[Nu]] Successful Failure - Successful
10.43.E20 int(cos(a*t)*BesselK(0, t), t = 0..infinity)=(Pi)/(2*(1 + (a)^(2))^((1)/(2))) Integrate[Cos[a*t]*BesselK[0, t], {t, 0, Infinity}]=Divide[Pi,2*(1 + (a)^(2))^(Divide[1,2])] Successful Failure - Successful
10.43.E21 int(sin(a*t)*BesselK(0, t), t = 0..infinity)=(arcsinh(a))/((1 + (a)^(2))^((1)/(2))) Integrate[Sin[a*t]*BesselK[0, t], {t, 0, Infinity}]=Divide[ArcSinh[a],(1 + (a)^(2))^(Divide[1,2])] Failure Failure - Successful
10.43.E22 int((t)^(mu - 1)* exp(- a*t)*BesselK(nu, t), t = 0..infinity)= Integrate[(t)^(\[Mu]- 1)* Exp[- a*t]*BesselK[\[Nu], t], {t, 0, Infinity}]= Error Failure - -
10.43.E22 Error Failure - Error
10.43.E23 int((t)^(nu + 1)* BesselI(nu, b*t)*exp(- (p)^(2)* (t)^(2)), t = 0..infinity)=((b)^(nu))/((2*(p)^(2))^(nu + 1))*exp(((b)^(2))/(4*(p)^(2))) Integrate[(t)^(\[Nu]+ 1)* BesselI[\[Nu], b*t]*Exp[- (p)^(2)* (t)^(2)], {t, 0, Infinity}]=Divide[(b)^(\[Nu]),(2*(p)^(2))^(\[Nu]+ 1)]*Exp[Divide[(b)^(2),4*(p)^(2)]] Failure Failure Skip Error
10.43.E24 int(BesselI(nu, b*t)*exp(- (p)^(2)* (t)^(2)), t = 0..infinity)=(sqrt(Pi))/(2*p)*exp(((b)^(2))/(8*(p)^(2)))*BesselI((1)/(2)*nu, ((b)^(2))/(8*(p)^(2))) Integrate[BesselI[\[Nu], b*t]*Exp[- (p)^(2)* (t)^(2)], {t, 0, Infinity}]=Divide[Sqrt[Pi],2*p]*Exp[Divide[(b)^(2),8*(p)^(2)]]*BesselI[Divide[1,2]*\[Nu], Divide[(b)^(2),8*(p)^(2)]] Failure Failure Skip Error
10.43.E25 int(BesselK(nu, b*t)*exp(- (p)^(2)* (t)^(2)), t = 0..infinity)=(sqrt(Pi))/(4*p)*sec((1)/(2)*Pi*nu)*exp(((b)^(2))/(8*(p)^(2)))*BesselK((1)/(2)*nu, ((b)^(2))/(8*(p)^(2))) Integrate[BesselK[\[Nu], b*t]*Exp[- (p)^(2)* (t)^(2)], {t, 0, Infinity}]=Divide[Sqrt[Pi],4*p]*Sec[Divide[1,2]*Pi*\[Nu]]*Exp[Divide[(b)^(2),8*(p)^(2)]]*BesselK[Divide[1,2]*\[Nu], Divide[(b)^(2),8*(p)^(2)]] Failure Failure Skip Error
10.43.E26 int((BesselK(mu, a*t)*BesselJ(nu, b*t))/((t)^(lambda)), t = 0..infinity)=((b)^(nu)* GAMMA((1)/(2)*nu -(1)/(2)*lambda +(1)/(2)*mu +(1)/(2))*GAMMA((1)/(2)*nu -(1)/(2)*lambda -(1)/(2)*mu +(1)/(2)))/((2)^(lambda + 1)* (a)^(nu - lambda + 1))* hypergeom([(nu - lambda + mu + 1)/(2), (nu - lambda - mu + 1)/(2)], [nu + 1], -((b)^(2))/((a)^(2)))/GAMMA(nu + 1) Integrate[Divide[BesselK[\[Mu], a*t]*BesselJ[\[Nu], b*t],(t)^(\[Lambda])], {t, 0, Infinity}]=Divide[(b)^(\[Nu])* Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Lambda]+Divide[1,2]*\[Mu]+Divide[1,2]]*Gamma[Divide[1,2]*\[Nu]-Divide[1,2]*\[Lambda]-Divide[1,2]*\[Mu]+Divide[1,2]],(2)^(\[Lambda]+ 1)* (a)^(\[Nu]- \[Lambda]+ 1)]* Hypergeometric2F1Regularized[Divide[\[Nu]- \[Lambda]+ \[Mu]+ 1,2], Divide[\[Nu]- \[Lambda]- \[Mu]+ 1,2], \[Nu]+ 1, -Divide[(b)^(2),(a)^(2)]] Error Failure - Error
10.43.E27 int((t)^(mu + nu + 1)* BesselK(mu, a*t)*BesselJ(nu, b*t), t = 0..infinity)=((2*a)^(mu)*(2*b)^(nu)* GAMMA(mu + nu + 1))/(((a)^(2)+ (b)^(2))^(mu + nu + 1)) Integrate[(t)^(\[Mu]+ \[Nu]+ 1)* BesselK[\[Mu], a*t]*BesselJ[\[Nu], b*t], {t, 0, Infinity}]=Divide[(2*a)^(\[Mu])*(2*b)^(\[Nu])* Gamma[\[Mu]+ \[Nu]+ 1],((a)^(2)+ (b)^(2))^(\[Mu]+ \[Nu]+ 1)] Error Failure - Error
10.43.E28 int(t*exp(- (p)^(2)* (t)^(2))*BesselI(nu, a*t)*BesselI(nu, b*t), t = 0..infinity)=(1)/(2*(p)^(2))*exp(((a)^(2)+ (b)^(2))/(4*(p)^(2)))*BesselI(nu, (a*b)/(2*(p)^(2))) Integrate[t*Exp[- (p)^(2)* (t)^(2)]*BesselI[\[Nu], a*t]*BesselI[\[Nu], b*t], {t, 0, Infinity}]=Divide[1,2*(p)^(2)]*Exp[Divide[(a)^(2)+ (b)^(2),4*(p)^(2)]]*BesselI[\[Nu], Divide[a*b,2*(p)^(2)]] Failure Failure Skip Error
10.43.E29 int(t*exp(- (p)^(2)* (t)^(2))*BesselI(0, a*t)*BesselK(0, a*t), t = 0..infinity)=(1)/(4*(p)^(2))*exp(((a)^(2))/(2*(p)^(2)))*BesselK(0, ((a)^(2))/(2*(p)^(2))) Integrate[t*Exp[- (p)^(2)* (t)^(2)]*BesselI[0, a*t]*BesselK[0, a*t], {t, 0, Infinity}]=Divide[1,4*(p)^(2)]*Exp[Divide[(a)^(2),2*(p)^(2)]]*BesselK[0, Divide[(a)^(2),2*(p)^(2)]] Failure Failure Skip Error
10.44#Ex1 BesselI(nu, z)= sum(((z)^(k))/(factorial(k))*BesselJ(nu + k, z), k = 0..infinity) BesselI[\[Nu], z]= Sum[Divide[(z)^(k),(k)!]*BesselJ[\[Nu]+ k, z], {k, 0, Infinity}] Failure Successful Skip -
10.44#Ex2 BesselJ(nu, z)= sum((- 1)^(k)*((z)^(k))/(factorial(k))*BesselI(nu + k, z), k = 0..infinity) BesselJ[\[Nu], z]= Sum[(- 1)^(k)*Divide[(z)^(k),(k)!]*BesselI[\[Nu]+ k, z], {k, 0, Infinity}] Failure Failure Skip Skip
10.44.E4 ((1)/(2)*z)^(nu)= sum((- 1)^(k)*((nu + 2*k)* GAMMA(nu + k))/(factorial(k))*BesselI(nu + 2*k, z), k = 0..infinity) (Divide[1,2]*z)^(\[Nu])= Sum[(- 1)^(k)*Divide[(\[Nu]+ 2*k)* Gamma[\[Nu]+ k],(k)!]*BesselI[\[Nu]+ 2*k, z], {k, 0, Infinity}] Failure Failure Skip Skip
10.44.E5 BesselK(0, z)= -(ln((1)/(2)*z)+ gamma)* BesselI(0, z)+ 2*sum((BesselI(2*k, z))/(k), k = 1..infinity) BesselK[0, z]= -(Log[Divide[1,2]*z]+ EulerGamma)* BesselI[0, z]+ 2*Sum[Divide[BesselI[2*k, z],k], {k, 1, Infinity}] Failure Successful Skip -
10.44.E6 BesselK(n, z)=(factorial(n)*((1)/(2)*z)^(- n))/(2)*sum((- 1)^(k)*(((1)/(2)*z)^(k)* BesselI(k, z))/(factorial(k)*(n - k)), k = 0..n - 1)+(- 1)^(n - 1)*(ln((1)/(2)*z)- Psi(n + 1))* BesselI(n, z)+(- 1)^(n)* sum(((n + 2*k)* BesselI(n + 2*k, z))/(k*(n + k)), k = 1..infinity) BesselK[n, z]=Divide[(n)!*(Divide[1,2]*z)^(- n),2]*Sum[(- 1)^(k)*Divide[(Divide[1,2]*z)^(k)* BesselI[k, z],(k)!*(n - k)], {k, 0, n - 1}]+(- 1)^(n - 1)*(Log[Divide[1,2]*z]- PolyGamma[n + 1])* BesselI[n, z]+(- 1)^(n)* Sum[Divide[(n + 2*k)* BesselI[n + 2*k, z],k*(n + k)], {k, 1, Infinity}] Failure Failure Skip Skip
10.45.E1 (x)^(2)* diff(w, [x$(2)])+ x*diff(w, x)+((nu)^(2)- (x)^(2))* w = 0 (x)^(2)* D[w, {x, 2}]+ x*D[w, x]+((\[Nu])^(2)- (x)^(2))* w = 0 Failure Failure
Fail
-7.071067807+4.242640683*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 1}
-11.31370849-.2828427124e-8*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 2}
-18.38477630-7.071067813*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 3}
4.242640683+7.071067807*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Fail
Complex[-7.0710678118654755, 4.242640687119286] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
-11.313708498984761 <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-18.38477631085024, -7.0710678118654755] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[4.242640687119286, -7.0710678118654755] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.45.E2 Re(BesselI(I*(nu), x))= Re(BesselI(I*nu, x)) Re[BesselI[I*\[Nu], x]]= Re[BesselI[I*\[Nu], x]] Successful Successful - -
10.45.E2 BesselK(I*(nu), x)= BesselK(I*nu, x) BesselK[I*\[Nu], x]= BesselK[I*\[Nu], x] Successful Successful - -
10.45.E4 (BesselK(I*(nu), x))*diff(Re(BesselI(I*(nu), x)), x)-diff(BesselK(I*(nu), x), x)*(Re(BesselI(I*(nu), x)))= 1/ x Wronskian[{BesselK[I*\[Nu], x], Re[BesselI[I*\[Nu], x]]}, x]= 1/ x Failure Failure Error Successful
10.45.E8 BesselK(I*(0), x)= BesselK(0, x) BesselK[I*0, x]= BesselK[0, x] Successful Successful - -
10.47.E1 (z)^(2)* diff(w, [z$(2)])+ 2*z*diff(w, z)+((z)^(2)- n*(n + 1))* w = 0 (z)^(2)* D[w, {z, 2}]+ 2*z*D[w, z]+((z)^(2)- n*(n + 1))* w = 0 Failure Failure
Fail
-8.485281369+2.828427121*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 1}
-14.14213562-2.828427127*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 2}
-22.62741699-11.31370850*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 3}
2.828427121-8.485281369*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
Fail
Complex[-8.485281374238571, 2.8284271247461903] <- {Rule[n, 1], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-14.142135623730951, -2.8284271247461903] <- {Rule[n, 2], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-22.627416997969522, -11.313708498984761] <- {Rule[n, 3], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.8284271247461903, -8.485281374238571] <- {Rule[n, 1], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.47.E2 (z)^(2)* diff(w, [z$(2)])+ 2*z*diff(w, z)-((z)^(2)+ n*(n + 1))* w = 0 (z)^(2)* D[w, {z, 2}]+ 2*z*D[w, z]-((z)^(2)+ n*(n + 1))* w = 0 Failure Failure
Fail
2.828427121-8.485281369*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 1}
-2.828427127-14.14213562*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 2}
-11.31370850-22.62741699*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)+I*2^(1/2), n = 3}
-8.485281369+2.828427121*I <- {w = 2^(1/2)+I*2^(1/2), z = 2^(1/2)-I*2^(1/2), n = 1}
... skip entries to safe data
Fail
Complex[2.8284271247461903, -8.485281374238571] <- {Rule[n, 1], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-2.8284271247461903, -14.142135623730951] <- {Rule[n, 2], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-11.313708498984761, -22.627416997969522] <- {Rule[n, 3], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-8.485281374238571, 2.8284271247461903] <- {Rule[n, 1], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.47.E3 Error SphericalBesselJ[n, z]=Sqrt[Divide[1,2]*Pi/ z]*BesselJ[n +Divide[1,2], z] Error Failure - Successful
10.47.E3 sqrt((1)/(2)*Pi/ z)*BesselJ(n +(1)/(2), z)=(- 1)^(n)*sqrt((1)/(2)*Pi/ z)*BesselY(- n -(1)/(2), z) Sqrt[Divide[1,2]*Pi/ z]*BesselJ[n +Divide[1,2], z]=(- 1)^(n)*Sqrt[Divide[1,2]*Pi/ z]*BesselY[- n -Divide[1,2], z] Failure Failure Successful Successful
10.47.E4 Error SphericalBesselY[n, z]=Sqrt[Divide[1,2]*Pi/ z]*BesselY[n +Divide[1,2], z] Error Failure - Successful
10.47.E4 sqrt((1)/(2)*Pi/ z)*BesselY(n +(1)/(2), z)=(- 1)^(n + 1)*sqrt((1)/(2)*Pi/ z)*BesselJ(- n -(1)/(2), z) Sqrt[Divide[1,2]*Pi/ z]*BesselY[n +Divide[1,2], z]=(- 1)^(n + 1)*Sqrt[Divide[1,2]*Pi/ z]*BesselJ[- n -Divide[1,2], z] Failure Failure Successful Successful
10.47.E5 Error \|SphericalHankelH2[1, n]* z =Sqrt[Divide[1,2]*Pi/ z]*HankelH1[n +Divide[1,2], z] Error Failure - Error
10.47.E5 sqrt((1)/(2)*Pi/ z)*HankelH1(n +(1)/(2), z)=(- 1)^(n + 1)* I*sqrt((1)/(2)*Pi/ z)*HankelH1(- n -(1)/(2), z) Sqrt[Divide[1,2]*Pi/ z]*HankelH1[n +Divide[1,2], z]=(- 1)^(n + 1)* I*Sqrt[Divide[1,2]*Pi/ z]*HankelH1[- n -Divide[1,2], z] Failure Failure Successful Successful
10.47.E6 Error \|SphericalHankelH2[2, n]* z =Sqrt[Divide[1,2]*Pi/ z]*HankelH2[n +Divide[1,2], z] Error Failure - Error
10.47.E6 sqrt((1)/(2)*Pi/ z)*HankelH2(n +(1)/(2), z)=(- 1)^(n)* I*sqrt((1)/(2)*Pi/ z)*HankelH2(- n -(1)/(2), z) Sqrt[Divide[1,2]*Pi/ z]*HankelH2[n +Divide[1,2], z]=(- 1)^(n)* I*Sqrt[Divide[1,2]*Pi/ z]*HankelH2[- n -Divide[1,2], z] Failure Failure Successful Successful
10.47.E7 Error \|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z =Sqrt[Divide[1,2]*Pi/ z]*BesselI[n +Divide[1,2], z] Error Error - -
10.47.E8 Error \|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z =Sqrt[Divide[1,2]*Pi/ z]*BesselI[- n -Divide[1,2], z] Error Error - -
10.47.E9 Error Sqrt[1/2 Pi /z] BesselK[n + 1/2, z]=Sqrt[Divide[1,2]*Pi/ z]*BesselK[n +Divide[1,2], z] Error Successful - -
10.47.E9 sqrt((1)/(2)*Pi/ z)*BesselK(n +(1)/(2), z)=sqrt((1)/(2)*Pi/ z)*BesselK(- n -(1)/(2), z) Sqrt[Divide[1,2]*Pi/ z]*BesselK[n +Divide[1,2], z]=Sqrt[Divide[1,2]*Pi/ z]*BesselK[- n -Divide[1,2], z] Successful Successful - -
10.47#Ex1 Error \|SphericalHankelH2[1, n]* z = SphericalBesselJ[n, z]+ I*SphericalBesselY[n, z] Error Failure - Error
10.47#Ex2 Error \|SphericalHankelH2[2, n]* z = SphericalBesselJ[n, z]- I*SphericalBesselY[n, z] Error Failure - Error
10.47.E11 Error \|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z - Sqrt[1/2 Pi /$2] BesselI[(-1)^(2-1)*n + 1/2, n]\|\|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z) Error Error - -
10.47#Ex3 Error \|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z = (I)^(- n)* SphericalBesselJ[n, I*z] Error Error - -
10.47#Ex4 Error \|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z = (I)^(- n - 1)* SphericalBesselY[n, I*z] Error Error - -
10.47.E13 Error \|SphericalHankelH2[1, n]*I*z Error Failure - Error
10.47.E13 Error \|SphericalHankelH2[1, n]*I*z*= -Divide[1,2]*Pi*(I)^(- n)* SphericalHankelH1[2, n]\|\|SphericalHankelH2[2, n]*- I*z Error Failure - Error
10.47.E17 Error \|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z + Sqrt[1/2 Pi /$2] BesselI[(-1)^(2-1)*n + 1/2, n]\|\|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z) Error Error - -
10.49.E2 Error SphericalBesselJ[n, z]= Sin[z -Divide[1,2]*n*Pi]*Sum[(- 1)^(k)*Divide[Subscript[a, 2*k]*(n +Divide[1,2]),(z)^(2*k + 1)], {k, 0, Floor[n/ 2]}]+ Cos[z -Divide[1,2]*n*Pi]*Sum[(- 1)^(k)*Divide[Subscript[a, 2*k + 1]*(n +Divide[1,2]),(z)^(2*k + 2)], {k, 0, Floor[(n - 1)/ 2]}] Error Failure - Skip
10.49#Ex1 Error SphericalBesselJ[0, z]=Divide[Sin[z],z] Error Successful - -
10.49#Ex2 Error SphericalBesselJ[1, z]=Divide[Sin[z],(z)^(2)]-Divide[Cos[z],z] Error Successful - -
10.49#Ex3 Error SphericalBesselJ[2, z]=(-Divide[1,z]+Divide[3,(z)^(3)])* Sin[z]-Divide[3,(z)^(2)]*Cos[z] Error Successful - -
10.49.E4 Error SphericalBesselY[n, z]= - Cos[z -Divide[1,2]*n*Pi]*Sum[(- 1)^(k)*Divide[Subscript[a, 2*k]*(n +Divide[1,2]),(z)^(2*k + 1)], {k, 0, Floor[n/ 2]}]+ Sin[z -Divide[1,2]*n*Pi]*Sum[(- 1)^(k)*Divide[Subscript[a, 2*k + 1]*(n +Divide[1,2]),(z)^(2*k + 2)], {k, 0, Floor[(n - 1)/ 2]}] Error Failure - Skip
10.49#Ex4 Error SphericalBesselY[0, z]= -Divide[Cos[z],z] Error Successful - -
10.49#Ex5 Error SphericalBesselY[1, z]= -Divide[Cos[z],(z)^(2)]-Divide[Sin[z],z] Error Successful - -
10.49#Ex6 Error SphericalBesselY[2, z]=(Divide[1,z]-Divide[3,(z)^(3)])* Cos[z]-Divide[3,(z)^(2)]*Sin[z] Error Successful - -
10.49.E6 Error \|SphericalHankelH2[1, n]* z = Exp[I*z]*Sum[(I)^(k - n - 1)*Divide[Subscript[a, k]*(n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}] Error Failure - Error
10.49.E7 Error \|SphericalHankelH2[2, n]* z = Exp[- I*z]*Sum[(- I)^(k - n - 1)*Divide[Subscript[a, k]*(n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}] Error Failure - Error
10.49.E8 Error \|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z =Divide[1,2]*Exp[z]*Sum[(- 1)^(k)*Divide[Subscript[a, k]*(n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}]+(- 1)^(n + 1)*Divide[1,2]*(E)^(- z)* Sum[Divide[Subscript[a, k]*(n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}] Error Error - -
10.49#Ex7 Error \|Sqrt[1/2 Pi /$2] BesselI[-0 - 1/2, 0]* z =Divide[Sinh[z],z] Error Error - -
10.49#Ex8 Error \|Sqrt[1/2 Pi /$2] BesselI[-1 - 1/2, 1]* z = -Divide[Sinh[z],(z)^(2)]+Divide[Cosh[z],z] Error Error - -
10.49#Ex9 Error \|Sqrt[1/2 Pi /$2] BesselI[-2 - 1/2, 2]* z =(Divide[1,z]+Divide[3,(z)^(3)])* Sinh[z]-Divide[3,(z)^(2)]*Cosh[z] Error Error - -
10.49.E10 Error \|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z =Divide[1,2]*Exp[z]*Sum[(- 1)^(k)*Divide[Subscript[a, k]*(n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}]+(- 1)^(n)*Divide[1,2]*(E)^(- z)* Sum[Divide[Subscript[a, k]*(n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}] Error Error - -
10.49#Ex10 Error \|Sqrt[1/2 Pi /$2] BesselI[-0 - 1/2, 0]* z =Divide[Cosh[z],z] Error Error - -
10.49#Ex11 Error \|Sqrt[1/2 Pi /$2] BesselI[-1 - 1/2, 1]* z = -Divide[Cosh[z],(z)^(2)]+Divide[Sinh[z],z] Error Error - -
10.49#Ex12 Error \|Sqrt[1/2 Pi /$2] BesselI[-2 - 1/2, 2]* z =(Divide[1,z]+Divide[3,(z)^(3)])* Cosh[z]-Divide[3,(z)^(2)]*Sinh[z] Error Error - -
10.49.E12 Error Sqrt[1/2 Pi /z] BesselK[n + 1/2, z]=Divide[1,2]*Pi*Exp[- z]*Sum[Divide[Subscript[a, k]*(n +Divide[1,2]),(z)^(k + 1)], {k, 0, n}] Error Failure - Skip
10.49#Ex13 Error Sqrt[1/2 Pi /z] BesselK[0 + 1/2, z]=Divide[1,2]*Pi*Divide[Exp[- z],z] Error Failure - Successful
10.49#Ex14 Error Sqrt[1/2 Pi /z] BesselK[1 + 1/2, z]=Divide[1,2]*Pi*Exp[- z]*(Divide[1,z]+Divide[1,(z)^(2)]) Error Failure - Successful
10.49#Ex15 Error Sqrt[1/2 Pi /z] BesselK[2 + 1/2, z]=Divide[1,2]*Pi*Exp[- z]*(Divide[1,z]+Divide[3,(z)^(2)]+Divide[3,(z)^(3)]) Error Failure - Successful
10.49.E18 Error (SphericalBesselJ[n, z])^(2)+ (SphericalBesselY[n, z])^(2)= Sum[Divide[Subscript[s, k]*(n +Divide[1,2]),(z)^(2*k + 2)], {k, 0, n}] Error Failure - Successful
10.49#Ex20 Error (SphericalBesselJ[0, z])^(2)+ (SphericalBesselY[0, z])^(2)= (z)^(- 2) Error Successful - -
10.49#Ex21 Error (SphericalBesselJ[1, z])^(2)+ (SphericalBesselY[1, z])^(2)= (z)^(- 2)+ (z)^(- 4) Error Successful - -
10.49#Ex22 Error (SphericalBesselJ[2, z])^(2)+ (SphericalBesselY[2, z])^(2)= (z)^(- 2)+ 3*(z)^(- 4)+ 9*(z)^(- 6) Error Successful - -
10.49.E20 Error \|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z))^(2)-((Sqrt[1/2 Pi /$2] BesselI[(-1)^(2-1)*n + 1/2, n]\|\|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z))^(2)=(- 1)^(n + 1)* Sum[(- 1)^(k)*Divide[Subscript[s, k]*(n +Divide[1,2]),(z)^(2*k + 2)], {k, 0, n}] Error Error - -
10.50#Ex1 Error Wronskian[{SphericalBesselJ[n, z], SphericalBesselY[n, z]}, z]= (z)^(- 2) Error Successful - -
10.50#Ex2 Unstrip size limit exceeded (5,000,000) Error \|SphericalHankelH2[1, n]* z, SphericalHankelH1[2, n]\|\|SphericalHankelH2[2, n]* z}, z]= - 2*I*(z)^(- 2) Error Failure - Error
10.50#Ex3 Unstrip size limit exceeded (5,000,000) Error \|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z] BesselI[(-1)^(1-1)*n + 1/2, n]\|\|Sqrt[1/2 Pi /Sqrt[1/2 Pi /$2] BesselI[(-1)^(2-1)*n + 1/2, n]\|\|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z] BesselI[-n - 1/2, n]* z, Sqrt[1/2 Pi /$2] BesselI[(-1)^(2-1)*n + 1/2, n]\|\|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z}, z]=(- 1)^(n + 1)* (z)^(- 2) Error Error - -
10.50#Ex4 Unstrip size limit exceeded (5,000,000) Error \|Sqrt[1/2 Pi /Sqrt[1/2 Pi /z] BesselK[n + 1/2, z]] BesselI[-n - 1/2, n]* z, Sqrt[1/2 Pi /z] BesselK[n + 1/2, z]}, z]= -Divide[1,2]*Pi*(z)^(- 2) Error Error - -
10.50#Ex5 Unstrip size limit exceeded (5,000,000) Error SphericalBesselJ[n + 1, z]*SphericalBesselY[n, z]- SphericalBesselJ[n, z]*SphericalBesselY[n + 1, z]= (z)^(- 2) Error Successful - -
10.50#Ex6 Unstrip size limit exceeded (5,000,000) Error SphericalBesselJ[n + 2, z]*SphericalBesselY[n, z]- SphericalBesselJ[n, z]*SphericalBesselY[n + 2, z]=(2*n + 3)* (z)^(- 3) Error Failure - Successful
10.50.E4 Unstrip size limit exceeded (5,000,000) Error SphericalBesselJ[0, z]*SphericalBesselJ[n, z]+ SphericalBesselY[0, z]*SphericalBesselY[n, z]= Cos[Divide[1,2]*n*Pi]*Sum[(- 1)^(k)*Divide[Subscript[a, 2*k]*(n +Divide[1,2]),(z)^(2*k + 2)], {k, 0, Floor[n/ 2]}]+ Sin[Divide[1,2]*n*Pi]*Sum[(- 1)^(k)*Divide[Subscript[a, 2*k + 1]*(n +Divide[1,2]),(z)^(2*k + 3)], {k, 0, Floor[(n - 1)/ 2]}] Error Failure - Skip
10.53.E1 Unstrip size limit exceeded (5,000,000) Error SphericalBesselJ[n, z]= (z)^(n)* Sum[Divide[(-Divide[1,2]*(z)^(2))^(k),(k)!*(2*n + 2*k + 1)!!], {k, 0, Infinity}] Error Failure - Successful
10.53.E2 Unstrip size limit exceeded (5,000,000) Error SphericalBesselY[n, z]= -Divide[1,(z)^(n + 1)]*Sum[Divide[(2*n - 2*k - 1)!!*(Divide[1,2]*(z)^(2))^(k),(k)!], {k, 0, n}]+Divide[(- 1)^(n + 1),(z)^(n + 1)]*Sum[Divide[(-Divide[1,2]*(z)^(2))^(k),(k)!*(2*k - 2*n - 1)!!], {k, n + 1, Infinity}] Error Failure - Successful
10.53.E3 Unstrip size limit exceeded (5,000,000) Error \|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z = (z)^(n)* Sum[Divide[(Divide[1,2]*(z)^(2))^(k),(k)!*(2*n + 2*k + 1)!!], {k, 0, Infinity}] Error Error - -
10.53.E4 Unstrip size limit exceeded (5,000,000) Error \|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z =Divide[(- 1)^(n),(z)^(n + 1)]*Sum[Divide[(2*n - 2*k - 1)!!*(-Divide[1,2]*(z)^(2))^(k),(k)!], {k, 0, n}]+Divide[1,(z)^(n + 1)]*Sum[Divide[(Divide[1,2]*(z)^(2))^(k),(k)!*(2*k - 2*n - 1)!!], {k, n + 1, Infinity}] Error Error - -
10.54.E1 Unstrip size limit exceeded (5,000,000) Error SphericalBesselJ[n, z]=Divide[(z)^(n),(2)^(n + 1)* (n)!]*Integrate[Cos[z*Cos[\[Theta]]]*(Sin[\[Theta]])^(2*n + 1), {\[Theta], 0, Pi}] Error Failure - Successful
10.54.E2 Unstrip size limit exceeded (5,000,000) Error SphericalBesselJ[n, z]=Divide[(- I)^(n),2]*Integrate[Exp[I*z*Cos[\[Theta]]]*LegendreP[n, 0, 3, Cos[\[Theta]]]*Sin[\[Theta]], {\[Theta], 0, Pi}] Error Failure - Error
10.54.E3 Unstrip size limit exceeded (5,000,000) Error Sqrt[1/2 Pi /z] BesselK[n + 1/2, z]=Divide[Pi,2]*Integrate[Exp[- z*t]*LegendreP[n, 0, 3, t], {t, 1, Infinity}] Error Failure - Error
10.54.E4 Unstrip size limit exceeded (5,000,000) Error SphericalBesselJ[n, z]=Divide[(- I)^(n + 1),2*Pi]*Integrate[Exp[I*z*t]*LegendreQ[n, 0, 3, t], {t, I*Infinity, (- 1 + , 1 +)}] Error Failure - Error
10.54#Ex1 Unstrip size limit exceeded (5,000,000) Error \|SphericalHankelH2[1, n]* z =Divide[(- I)^(n + 1),Pi]*Integrate[Exp[I*z*t]*LegendreQ[n, 0, 3, t], {t, I*Infinity, (1 +)}] Error Failure - Error
10.54#Ex2 Unstrip size limit exceeded (5,000,000) Error \|SphericalHankelH2[2, n]* z =Divide[(- I)^(n + 1),Pi]*Integrate[Exp[I*z*t]*LegendreQ[n, 0, 3, t], {t, I*Infinity, (- 1 +)}] Error Failure - Error
10.56.E5 Unstrip size limit exceeded (5,000,000) Error Divide[Exp[-Sqrt[(z)^(2)+ 2*I*z*t]],z]=Divide[Exp[- z],z]+Divide[2,Pi]*Sum[Divide[(- I*t)^(n),(n)!]*Sqrt[1/2 Pi /z] BesselK[n - 1 + 1/2, z], {n, 1, Infinity}] Error Failure - Error
10.57.E1 Unstrip size limit exceeded (5,000,000) Error (D[SphericalBesselJ[n, temp], {temp, 1}]/.temp-> (n +Divide[1,2])* z)=Divide[(Pi)^(Divide[1,2]),((2*n + 1)*z)^(Divide[1,2])]*(D[BesselJ[n +Divide[1,2], temp], {temp, 1}]/.temp-> (n +Divide[1,2])* z)-Divide[(Pi)^(Divide[1,2]),((2*n + 1)*z)^(Divide[3,2])]*BesselJ[n +Divide[1,2], (n +Divide[1,2])* z] Error Failure - Successful
10.59.E1 Unstrip size limit exceeded (5,000,000) Error Integrate[Exp[I*b*t]*SphericalBesselJ[n, t], {t, - Infinity, Infinity}]= Error Failure - -
10.59.E1 Unstrip size limit exceeded (5,000,000) Error Failure - -
10.59.E1 Unstrip size limit exceeded (5,000,000) b < 1 ,(1)/(2)*Pi*(+ I)^(n), b < 1 ,Divide[1,2]*Pi*(+ I)^(n), Error Failure - Error
10.59.E1 Unstrip size limit exceeded (5,000,000) b < 1 ,(1)/(2)*Pi*(- I)^(n), b < 1 ,Divide[1,2]*Pi*(- I)^(n), Error Failure - Error
10.60.E1 Unstrip size limit exceeded (5,000,000) Error Divide[Cos[w],w]= - Sum[(2*n + 1)* SphericalBesselJ[n, v]*SphericalBesselY[n, u]*LegendreP[n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}] Error Failure - Error
10.60.E2 Unstrip size limit exceeded (5,000,000) Error Divide[Sin[w],w]= Sum[(2*n + 1)* SphericalBesselJ[n, v]*SphericalBesselJ[n, u]*LegendreP[n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}] Error Failure -
Fail
Complex[-0.5468420859284989, -2.0682074571733775] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Plus[1, Times[2, n]], LegendreP[n, Cos[α]], SphericalBesselJ[n, u], SphericalBesselJ[n, v]], {n, 0, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.5468420859284989, 0.7602196675728127] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Plus[1, Times[2, n]], LegendreP[n, Cos[α]], SphericalBesselJ[n, u], SphericalBesselJ[n, v]], {n, 0, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.2815850388176915, 0.7602196675728127] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Plus[1, Times[2, n]], LegendreP[n, Cos[α]], SphericalBesselJ[n, u], SphericalBesselJ[n, v]], {n, 0, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.2815850388176915, -2.0682074571733775] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Plus[1, Times[2, n]], LegendreP[n, Cos[α]], SphericalBesselJ[n, u], SphericalBesselJ[n, v]], {n, 0, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.60.E3 Unstrip size limit exceeded (5,000,000) Error \|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* v*Sqrt[1/2 Pi /u] BesselK[n + 1/2, u]*LegendreP[n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}] Error Error - -
10.60.E4 Unstrip size limit exceeded (5,000,000) Error SphericalBesselJ[n, 2*z]= - (n)!*(z)^(n + 1)* Sum[Divide[2*n - 2*k + 1,(k)!*(2*n - k + 1)!]*SphericalBesselJ[n - k, z]*SphericalBesselY[n - k, z], {k, 0, n}] Error Failure - Skip
10.60.E5 Unstrip size limit exceeded (5,000,000) Error SphericalBesselY[n, 2*z]= (n)!*(z)^(n + 1)* Sum[Divide[n - k +Divide[1,2],(k)!*(2*n - k + 1)!]*((SphericalBesselJ[n - k, z])^(2)- (SphericalBesselY[n - k, z])^(2)), {k, 0, n}] Error Failure - Error
10.60.E6 Unstrip size limit exceeded (5,000,000) Error Sqrt[1/2 Pi /2*z] BesselK[n + 1/2, 2*z]=Divide[1,Pi]*(n)!*(z)^(n + 1)* Sum[(- 1)^(k)*Divide[2*n - 2*k + 1,(k)!*(2*n - k + 1)!]*(Sqrt[1/2 Pi /z] BesselK[n - k + 1/2, z])^(2), {k, 0, n}] Error Failure - Error
10.60.E7 Unstrip size limit exceeded (5,000,000) Error Exp[I*z*Cos[\[Alpha]]]= Sum[(2*n + 1)* (I)^(n)* SphericalBesselJ[n, z]*LegendreP[n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}] Error Failure - Skip
10.60.E8 Unstrip size limit exceeded (5,000,000) Error \|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z*LegendreP[n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}] Error Error - -
10.60.E9 Unstrip size limit exceeded (5,000,000) Error \|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]* z*LegendreP[n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}] Error Error - -
10.60.E10 Unstrip size limit exceeded (5,000,000) Error BesselJ[0, z*Sin[\[Alpha]]]= Sum[(4*n + 1)*Divide[(2*n)!,(2)^(2*n)*((n)!)^(2)]*SphericalBesselJ[2*n, z]*LegendreP[2*n, 0, 3, Cos[\[Alpha]]], {n, 0, Infinity}] Error Failure - Skip
10.60.E11 Unstrip size limit exceeded (5,000,000) Error Sum[(SphericalBesselJ[n, z])^(2), {n, 0, Infinity}]=Divide[SinIntegral[2*z],2*z] Error Failure - Successful
10.60.E12 Unstrip size limit exceeded (5,000,000) Error Sum[(2*n + 1)* (SphericalBesselJ[n, z])^(2), {n, 0, Infinity}]= 1 Error Failure -
Fail
Complex[0.41421356237309515, 1.4142135623730951] <- {Rule[Sum[Times[Plus[1, Times[2, n]], Power[SphericalBesselJ[n, z], 2]], {n, 0, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.41421356237309515, -1.4142135623730951] <- {Rule[Sum[Times[Plus[1, Times[2, n]], Power[SphericalBesselJ[n, z], 2]], {n, 0, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.414213562373095, -1.4142135623730951] <- {Rule[Sum[Times[Plus[1, Times[2, n]], Power[SphericalBesselJ[n, z], 2]], {n, 0, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.414213562373095, 1.4142135623730951] <- {Rule[Sum[Times[Plus[1, Times[2, n]], Power[SphericalBesselJ[n, z], 2]], {n, 0, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
10.60.E13 Unstrip size limit exceeded (5,000,000) Error Sum[(- 1)^(n)*(2*n + 1)* (SphericalBesselJ[n, z])^(2), {n, 0, Infinity}]=Divide[Sin[2*z],2*z] Error Failure - Skip
10.60.E14 Unstrip size limit exceeded (5,000,000) Error Sum[(2*n + 1)*(((D[SphericalBesselJ[n, temp], {temp, 1}]/.temp-> z)))^(2), {n, 0, Infinity}]=Divide[1,3] Error Failure - Error
10.61.E1 Unstrip size limit exceeded (5,000,000) KelvinBer(nu, x)+ I*KelvinBei(nu, x)= BesselJ(nu, x*exp(3*Pi*I/ 4)) KelvinBer[\[Nu], x]+ I*KelvinBei[\[Nu], x]= BesselJ[\[Nu], x*Exp[3*Pi*I/ 4]] Successful Failure - Successful
10.61.E1 Unstrip size limit exceeded (5,000,000) BesselJ(nu, x*exp(3*Pi*I/ 4))= exp(nu*Pi*I)*BesselJ(nu, x*exp(- Pi*I/ 4)) BesselJ[\[Nu], x*Exp[3*Pi*I/ 4]]= Exp[\[Nu]*Pi*I]*BesselJ[\[Nu], x*Exp[- Pi*I/ 4]] Failure Successful Successful -
10.61.E1 Unstrip size limit exceeded (5,000,000) exp(nu*Pi*I)*BesselJ(nu, x*exp(- Pi*I/ 4))= exp(nu*Pi*I/ 2)*BesselI(nu, x*exp(Pi*I/ 4)) Exp[\[Nu]*Pi*I]*BesselJ[\[Nu], x*Exp[- Pi*I/ 4]]= Exp[\[Nu]*Pi*I/ 2]*BesselI[\[Nu], x*Exp[Pi*I/ 4]] Failure Failure Skip Successful
10.61.E1 Unstrip size limit exceeded (5,000,000) exp(nu*Pi*I/ 2)*BesselI(nu, x*exp(Pi*I/ 4))= exp(3*nu*Pi*I/ 2)*BesselI(nu, x*exp(- 3*Pi*I/ 4)) Exp[\[Nu]*Pi*I/ 2]*BesselI[\[Nu], x*Exp[Pi*I/ 4]]= Exp[3*\[Nu]*Pi*I/ 2]*BesselI[\[Nu], x*Exp[- 3*Pi*I/ 4]] Failure Failure Skip Successful
10.61.E2 Unstrip size limit exceeded (5,000,000) KelvinKer(nu, x)+ I*KelvinKei(nu, x)= exp(- nu*Pi*I/ 2)*BesselK(nu, x*exp(Pi*I/ 4)) KelvinKer[\[Nu], x]+ I*KelvinKei[\[Nu], x]= Exp[- \[Nu]*Pi*I/ 2]*BesselK[\[Nu], x*Exp[Pi*I/ 4]] Failure Failure Successful Successful
10.61.E2 Unstrip size limit exceeded (5,000,000) exp(- nu*Pi*I/ 2)*BesselK(nu, x*exp(Pi*I/ 4))=(1)/(2)*Pi*I*HankelH1(nu, x*exp(3*Pi*I/ 4)) Exp[- \[Nu]*Pi*I/ 2]*BesselK[\[Nu], x*Exp[Pi*I/ 4]]=Divide[1,2]*Pi*I*HankelH1[\[Nu], x*Exp[3*Pi*I/ 4]] Failure Failure Successful Successful
10.61.E2 Unstrip size limit exceeded (5,000,000) (1)/(2)*Pi*I*HankelH1(nu, x*exp(3*Pi*I/ 4))= -(1)/(2)*Pi*I*exp(- nu*Pi*I)*HankelH2(nu, x*exp(- Pi*I/ 4)) Divide[1,2]*Pi*I*HankelH1[\[Nu], x*Exp[3*Pi*I/ 4]]= -Divide[1,2]*Pi*I*Exp[- \[Nu]*Pi*I]*HankelH2[\[Nu], x*Exp[- Pi*I/ 4]] Failure Failure Skip Successful
10.61.E3 Unstrip size limit exceeded (5,000,000) (x)^(2)* diff(w, [x$(2)])+ x*diff(w, x)-(I*(x)^(2)+ (nu)^(2))* w = 0 (x)^(2)* D[w, {x, 2}]+ x*D[w, x]-(I*(x)^(2)+ (\[Nu])^(2))* w = 0 Failure Failure
Fail
7.071067807-7.071067807*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 1}
11.31370849-11.31370849*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 2}
18.38477631-18.38477631*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 3}
-7.071067807-7.071067807*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Fail
Complex[7.0710678118654755, -7.0710678118654755] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[11.313708498984761, -11.313708498984761] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[18.38477631085024, -18.38477631085024] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-4.242640687119286, 4.242640687119286] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.61.E4 Unstrip size limit exceeded (5,000,000) (x)^(4)* diff(w, [x$(4)])+ 2*(x)^(3)* diff(w, [x$(3)])-(1 + 2*(nu)^(2))*((x)^(2)* diff(w, [x$(2)])- x*diff(w, x))+((nu)^(4)- 4*(nu)^(2)+ (x)^(4))* w = 0 (x)^(4)* D[w, {x, 4}]+ 2*(x)^(3)* D[w, {x, 3}]-(1 + 2*(\[Nu])^(2))*((x)^(2)* D[w, {x, 2}]- x*D[w, x])+((\[Nu])^(4)- 4*(\[Nu])^(2)+ (x)^(4))* w = 0 Failure Failure
Fail
1.414213576-43.84062038*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 1}
22.62741701-22.62741695*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 2}
114.5512985+69.29646458*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)+I*2^(1/2), x = 3}
-43.84062038-1.414213576*I <- {nu = 2^(1/2)+I*2^(1/2), w = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Fail
Complex[1.4142135623730951, -43.84062043356595] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[22.627416997969522, -22.627416997969522] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[114.5512985522207, 69.29646455628166] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3], Rule[ν, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-43.84062043356595, 1.4142135623730951] <- {Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[ν, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
10.61#Ex1 Unstrip size limit exceeded (5,000,000) KelvinBer(n, - x)=(- 1)^(n)* KelvinBer(n, x) KelvinBer[n, - x]=(- 1)^(n)* KelvinBer[n, x] Failure Failure Successful Successful
10.61#Ex2 Unstrip size limit exceeded (5,000,000) KelvinBei(n, - x)=(- 1)^(n)* KelvinBei(n, x) KelvinBei[n, - x]=(- 1)^(n)* KelvinBei[n, x] Failure Failure Successful Successful
10.61#Ex3 Unstrip size limit exceeded (5,000,000) KelvinBer(- nu, x)= cos(nu*Pi)*KelvinBer(nu, x)+ sin(nu*Pi)*KelvinBei(nu, x)+(2/ Pi)* sin(nu*Pi)*KelvinKer(nu, x) KelvinBer[- \[Nu], x]= Cos[\[Nu]*Pi]*KelvinBer[\[Nu], x]+ Sin[\[Nu]*Pi]*KelvinBei[\[Nu], x]+(2/ Pi)* Sin[\[Nu]*Pi]*KelvinKer[\[Nu], x] Failure Failure Successful Successful
10.61#Ex4 Unstrip size limit exceeded (5,000,000) KelvinBei(- nu, x)= - sin(nu*Pi)*KelvinBer(nu, x)+ cos(nu*Pi)*KelvinBei(nu, x)+(2/ Pi)* sin(nu*Pi)*KelvinKei(nu, x) KelvinBei[- \[Nu], x]= - Sin[\[Nu]*Pi]*KelvinBer[\[Nu], x]+ Cos[\[Nu]*Pi]*KelvinBei[\[Nu], x]+(2/ Pi)* Sin[\[Nu]*Pi]*KelvinKei[\[Nu], x] Failure Failure Successful Successful
10.61#Ex5 Unstrip size limit exceeded (5,000,000) KelvinKer(- nu, x)= cos(nu*Pi)*KelvinKer(nu, x)- sin(nu*Pi)*KelvinKei(nu, x) KelvinKer[- \[Nu], x]= Cos[\[Nu]*Pi]*KelvinKer[\[Nu], x]- Sin[\[Nu]*Pi]*KelvinKei[\[Nu], x] Failure Failure Successful Successful
10.61#Ex6 Unstrip size limit exceeded (5,000,000) KelvinKei(- nu, x)= sin(nu*Pi)*KelvinKer(nu, x)+ cos(nu*Pi)*KelvinKei(nu, x) KelvinKei[- \[Nu], x]= Sin[\[Nu]*Pi]*KelvinKer[\[Nu], x]+ Cos[\[Nu]*Pi]*KelvinKei[\[Nu], x] Failure Failure Successful Successful
10.61#Ex7 Unstrip size limit exceeded (5,000,000) KelvinBer(- n, x)=(- 1)^(n)* KelvinBer(n, x),*KelvinBei(- n, x) KelvinBer[- n, x]=(- 1)^(n)* KelvinBer[n, x],*KelvinBei[- n, x] Error Failure - Error
10.61#Ex7 Unstrip size limit exceeded (5,000,000) (- 1)^(n)* KelvinBer(n, x),*KelvinBei(- n, x)=(- 1)^(n)* KelvinBei(n, x) (- 1)^(n)* KelvinBer[n, x],*KelvinBei[- n, x]=(- 1)^(n)* KelvinBei[n, x] Error Failure - Error
10.61#Ex8 Unstrip size limit exceeded (5,000,000) KelvinKer(- n, x)=(- 1)^(n)* KelvinKer(n, x),*KelvinKei(- n, x) KelvinKer[- n, x]=(- 1)^(n)* KelvinKer[n, x],*KelvinKei[- n, x] Error Failure - Error
10.61#Ex8 Unstrip size limit exceeded (5,000,000) (- 1)^(n)* KelvinKer(n, x),*KelvinKei(- n, x)=(- 1)^(n)* KelvinKei(n, x) (- 1)^(n)* KelvinKer[n, x],*KelvinKei[- n, x]=(- 1)^(n)* KelvinKei[n, x] Error Failure - Error
10.61#Ex9 Unstrip size limit exceeded (5,000,000) KelvinBer((1)/(2), x*sqrt(2))=((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*cos(x +(Pi)/(8))- exp(- x)*cos(x -(Pi)/(8))) KelvinBer[Divide[1,2], x*Sqrt[2]]=Divide[(2)^(-Divide[3,4]),Sqrt[Pi*x]]*(Exp[x]*Cos[x +Divide[Pi,8]]- Exp[- x]*Cos[x -Divide[Pi,8]]) Failure Failure Successful Successful
10.61#Ex10 Unstrip size limit exceeded (5,000,000) KelvinBei((1)/(2), x*sqrt(2))=((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*sin(x +(Pi)/(8))+ exp(- x)*sin(x -(Pi)/(8))) KelvinBei[Divide[1,2], x*Sqrt[2]]=Divide[(2)^(-Divide[3,4]),Sqrt[Pi*x]]*(Exp[x]*Sin[x +Divide[Pi,8]]+ Exp[- x]*Sin[x -Divide[Pi,8]]) Failure Failure Successful Successful
10.61#Ex11 Unstrip size limit exceeded (5,000,000) KelvinBer(-(1)/(2), x*sqrt(2))=((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*sin(x +(Pi)/(8))- exp(- x)*sin(x -(Pi)/(8))) KelvinBer[-Divide[1,2], x*Sqrt[2]]=Divide[(2)^(-Divide[3,4]),Sqrt[Pi*x]]*(Exp[x]*Sin[x +Divide[Pi,8]]- Exp[- x]*Sin[x -Divide[Pi,8]]) Failure Failure Successful Successful
10.61#Ex12 Unstrip size limit exceeded (5,000,000) KelvinBei(-(1)/(2), x*sqrt(2))= -((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*cos(x +(Pi)/(8))+ exp(- x)*cos(x -(Pi)/(8))) KelvinBei[-Divide[1,2], x*Sqrt[2]]= -Divide[(2)^(-Divide[3,4]),Sqrt[Pi*x]]*(Exp[x]*Cos[x +Divide[Pi,8]]+ Exp[- x]*Cos[x -Divide[Pi,8]]) Failure Failure Successful Successful
10.61.E11 Unstrip size limit exceeded (5,000,000) KelvinKer((1)/(2), x*sqrt(2))= KelvinKei(-(1)/(2), x*sqrt(2)) KelvinKer[Divide[1,2], x*Sqrt[2]]= KelvinKei[-Divide[1,2], x*Sqrt[2]] Successful Failure - Successful
10.61.E11 Unstrip size limit exceeded (5,000,000) KelvinKei(-(1)/(2), x*sqrt(2))= - (2)^(-(3)/(4))*sqrt((Pi)/(x))*exp(- x)*sin(x -(Pi)/(8)) KelvinKei[-Divide[1,2], x*Sqrt[2]]= - (2)^(-Divide[3,4])*Sqrt[Divide[Pi,x]]*Exp[- x]*Sin[x -Divide[Pi,8]] Failure Failure Successful Successful
10.61.E12 Unstrip size limit exceeded (5,000,000) KelvinKei((1)/(2), x*sqrt(2))= - KelvinKer(-(1)/(2), x*sqrt(2)) KelvinKei[Divide[1,2], x*Sqrt[2]]= - KelvinKer[-Divide[1,2], x*Sqrt[2]] Successful Successful - -
10.61.E12 Unstrip size limit exceeded (5,000,000) - KelvinKer(-(1)/(2), x*sqrt(2))= - (2)^(-(3)/(4))*sqrt((Pi)/(x))*exp(- x)*cos(x -(Pi)/(8)) - KelvinKer[-Divide[1,2], x*Sqrt[2]]= - (2)^(-Divide[3,4])*Sqrt[Divide[Pi,x]]*Exp[- x]*Cos[x -Divide[Pi,8]] Failure Successful Successful -
10.63#Ex9 Unstrip size limit exceeded (5,000,000) subs( temp=x, diff( KelvinBer(, temp), temp$(1) ) )= KelvinBer(1, x)+ KelvinBei(1, x) (D[KelvinBer[, temp], {temp, 1}]/.temp-> x)= KelvinBer[1, x]+ KelvinBei[1, x] Error Successful - -
10.63#Ex10 Unstrip size limit exceeded (5,000,000) subs( temp=x, diff( KelvinBei(, temp), temp$(1) ) )= - KelvinBer(1, x)+ KelvinBei(1, x) (D[KelvinBei[, temp], {temp, 1}]/.temp-> x)= - KelvinBer[1, x]+ KelvinBei[1, x] Error Successful - -
10.63#Ex11 Unstrip size limit exceeded (5,000,000) subs( temp=x, diff( KelvinKer(, temp), temp$(1) ) )= KelvinKer(1, x)+ KelvinKei(1, x) (D[KelvinKer[, temp], {temp, 1}]/.temp-> x)= KelvinKer[1, x]+ KelvinKei[1, x] Error Failure -
Fail
Complex[-1.1863575732592084, 14.181995430502623] <- {Rule[Null, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}
Complex[2.2791430867712648, 1.1716762871697879] <- {Rule[Null, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}
Complex[0.6426179371583077, -0.28537763977623365] <- {Rule[Null, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}
Complex[-1.186357573259207, -14.181995430502624] <- {Rule[Null, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}
... skip entries to safe data
10.63#Ex12 Unstrip size limit exceeded (5,000,000) subs( temp=x, diff( KelvinKei(, temp), temp$(1) ) )= - KelvinKer(1, x)+ KelvinKei(1, x) (D[KelvinKei[, temp], {temp, 1}]/.temp-> x)= - KelvinKer[1, x]+ KelvinKei[1, x] Error Successful - -
10.63#Ex13 Unstrip size limit exceeded (5,000,000) p[nu]= (KelvinBer(nu, x))^(2)+ (KelvinBei(nu, x))^(2) Subscript[p, \[Nu]]= (KelvinBer[\[Nu], x])^(2)+ (KelvinBei[\[Nu], x])^(2) Failure Failure
Fail
1.558095916+1.23230553*I <- {nu = 2^(1/2)+I*2^(1/2), p[nu] = 2^(1/2)+I*2^(1/2), x = 1}
2.2705317+2.829827546*I <- {nu = 2^(1/2)+I*2^(1/2), p[nu] = 2^(1/2)+I*2^(1/2), x = 2}
-.9860947+6.1991516*I <- {nu = 2^(1/2)+I*2^(1/2), p[nu] = 2^(1/2)+I*2^(1/2), x = 3}
1.558095916-1.59612159*I <- {nu = 2^(1/2)+I*2^(1/2), p[nu] = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Successful
10.63#Ex14 Unstrip size limit exceeded (5,000,000) q[nu]= KelvinBer(nu, x)*subs( temp=x, diff( KelvinBei(nu, temp), temp$(1) ) )- subs( temp=x, diff( KelvinBer(nu, temp), temp$(1) ) )*KelvinBei(nu, x) Subscript[q, \[Nu]]= KelvinBer[\[Nu], x]*(D[KelvinBei[\[Nu], temp], {temp, 1}]/.temp-> x)- (D[KelvinBer[\[Nu], temp], {temp, 1}]/.temp-> x)*KelvinBei[\[Nu], x] Failure Failure
Fail
1.42000721+1.37316601*I <- {nu = 2^(1/2)+I*2^(1/2), q[nu] = 2^(1/2)+I*2^(1/2), x = 1}
1.93113589+1.7044153*I <- {nu = 2^(1/2)+I*2^(1/2), q[nu] = 2^(1/2)+I*2^(1/2), x = 2}
1.38678669+4.3022397*I <- {nu = 2^(1/2)+I*2^(1/2), q[nu] = 2^(1/2)+I*2^(1/2), x = 3}
1.42000721-1.45526111*I <- {nu = 2^(1/2)+I*2^(1/2), q[nu] = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Successful
10.63#Ex15 Unstrip size limit exceeded (5,000,000) r[nu]= KelvinBer(nu, x)*subs( temp=x, diff( KelvinBer(nu, temp), temp$(1) ) )+ KelvinBei(nu, x)*subs( temp=x, diff( KelvinBei(nu, temp), temp$(1) ) ) Subscript[r, \[Nu]]= KelvinBer[\[Nu], x]*(D[KelvinBer[\[Nu], temp], {temp, 1}]/.temp-> x)+ KelvinBei[\[Nu], x]*(D[KelvinBei[\[Nu], temp], {temp, 1}]/.temp-> x) Failure Failure
Fail
1.87424466+1.35972583*I <- {nu = 2^(1/2)+I*2^(1/2), r[nu] = 2^(1/2)+I*2^(1/2), x = 1}
1.0714855+2.99841206*I <- {nu = 2^(1/2)+I*2^(1/2), r[nu] = 2^(1/2)+I*2^(1/2), x = 2}
-1.5010775+2.94632015*I <- {nu = 2^(1/2)+I*2^(1/2), r[nu] = 2^(1/2)+I*2^(1/2), x = 3}
1.87424466-1.46870129*I <- {nu = 2^(1/2)+I*2^(1/2), r[nu] = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Successful
10.63#Ex16 Unstrip size limit exceeded (5,000,000) s[nu]=(subs( temp=x, diff( KelvinBer(nu, temp), temp$(1) ) ))^(2)+(subs( temp=x, diff( KelvinBei(nu, temp), temp$(1) ) ))^(2) Subscript[s, \[Nu]]=(((D[KelvinBer[\[Nu], temp], {temp, 1}]/.temp-> x)))^(2)+(((D[KelvinBei[\[Nu], temp], {temp, 1}]/.temp-> x)))^(2) Failure Failure
Fail
2.1390486+1.97888025*I <- {nu = 2^(1/2)+I*2^(1/2), s[nu] = 2^(1/2)+I*2^(1/2), x = 1}
.31664947+2.31090336*I <- {nu = 2^(1/2)+I*2^(1/2), s[nu] = 2^(1/2)+I*2^(1/2), x = 2}
.7948503e-1+2.54111021*I <- {nu = 2^(1/2)+I*2^(1/2), s[nu] = 2^(1/2)+I*2^(1/2), x = 3}
2.1390486-.849546872*I <- {nu = 2^(1/2)+I*2^(1/2), s[nu] = 2^(1/2)-I*2^(1/2), x = 1}
... skip entries to safe data
Successful
10.64.E1 Unstrip size limit exceeded (5,000,000) KelvinBer(n, x*sqrt(2))=((- 1)^(n))/(Pi)*int(cos(x*sin(t)- n*t)*cosh(x*sin(t)), t = 0..Pi) KelvinBer[n, x*Sqrt[2]]=Divide[(- 1)^(n),Pi]*Integrate[Cos[x*Sin[t]- n*t]*Cosh[x*Sin[t]], {t, 0, Pi}] Failure Failure Skip Error
10.64.E2 Unstrip size limit exceeded (5,000,000) KelvinBei(n, x*sqrt(2))=((- 1)^(n))/(Pi)*int(sin(x*sin(t)- n*t)*sinh(x*sin(t)), t = 0..Pi) KelvinBei[n, x*Sqrt[2]]=Divide[(- 1)^(n),Pi]*Integrate[Sin[x*Sin[t]- n*t]*Sinh[x*Sin[t]], {t, 0, Pi}] Failure Failure Skip Error
10.65#Ex1 Unstrip size limit exceeded (5,000,000) KelvinBer(nu, x)=((1)/(2)*x)^(nu)* sum((cos((3)/(4)*nu*Pi +(1)/(2)*k*Pi))/(factorial(k)*GAMMA(nu + k + 1))*((1)/(4)*(x)^(2))^(k), k = 0..infinity) KelvinBer[\[Nu], x]=(Divide[1,2]*x)^(\[Nu])* Sum[Divide[Cos[Divide[3,4]*\[Nu]*Pi +Divide[1,2]*k*Pi],(k)!*Gamma[\[Nu]+ k + 1]]*(Divide[1,4]*(x)^(2))^(k), {k, 0, Infinity}] Failure Failure Skip Successful
10.65#Ex2 Unstrip size limit exceeded (5,000,000) KelvinBei(nu, x)=((1)/(2)*x)^(nu)* sum((sin((3)/(4)*nu*Pi +(1)/(2)*k*Pi))/(factorial(k)*GAMMA(nu + k + 1))*((1)/(4)*(x)^(2))^(k), k = 0..infinity) KelvinBei[\[Nu], x]=(Divide[1,2]*x)^(\[Nu])* Sum[Divide[Sin[Divide[3,4]*\[Nu]*Pi +Divide[1,2]*k*Pi],(k)!*Gamma[\[Nu]+ k + 1]]*(Divide[1,4]*(x)^(2))^(k), {k, 0, Infinity}] Failure Failure Skip Successful
10.65#Ex5 Unstrip size limit exceeded (5,000,000) KelvinBei(, x)+ sum((- 1)^(k)*(Psi(2*k + 1))/((factorial(2*k))^(2))*((1)/(4)*(x)^(2))^(2*k), k = 0..infinity) KelvinBei[, x]+ Sum[(- 1)^(k)*Divide[PolyGamma[2*k + 1],((2*k)!)^(2)]*(Divide[1,4]*(x)^(2))^(2*k), {k, 0, Infinity}] Error Failure - Error
10.65#Ex6 Unstrip size limit exceeded (5,000,000) KelvinBer(, x)+ sum((- 1)^(k)*(Psi(2*k + 2))/((factorial(2*k + 1))^(2))*((1)/(4)*(x)^(2))^(2*k + 1), k = 0..infinity) KelvinBer[, x]+ Sum[(- 1)^(k)*Divide[PolyGamma[2*k + 2],((2*k + 1)!)^(2)]*(Divide[1,4]*(x)^(2))^(2*k + 1), {k, 0, Infinity}] Error Failure -
Fail
Complex[-9.498271428543017, -1.4209618670710054] <- {Rule[Null, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}
Complex[0.18667337329748546, -13.49491576100636] <- {Rule[Null, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}
Complex[2.6690254116240313, -21.496718507485472] <- {Rule[Null, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}
Complex[-9.498271428543028, 1.4209618670710045] <- {Rule[Null, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}
... skip entries to safe data
10.65.E6 Unstrip size limit exceeded (5,000,000) (KelvinBer(nu, x))^(2)+ (KelvinBei(nu, x))^(2)=((1)/(2)*x)^(2*nu)* sum((1)/(GAMMA(nu + k + 1)*GAMMA(nu + 2*k + 1))*(((1)/(4)*(x)^(2))^(2*k))/(factorial(k)), k = 0..infinity) (KelvinBer[\[Nu], x])^(2)+ (KelvinBei[\[Nu], x])^(2)=(Divide[1,2]*x)^(2*\[Nu])* Sum[Divide[1,Gamma[\[Nu]+ k + 1]*Gamma[\[Nu]+ 2*k + 1]]*Divide[(Divide[1,4]*(x)^(2))^(2*k),(k)!], {k, 0, Infinity}] Successful Failure - Successful
10.65.E7 Unstrip size limit exceeded (5,000,000) KelvinBer(nu, x)*subs( temp=x, diff( KelvinBei(nu, temp), temp$(1) ) )- subs( temp=x, diff( KelvinBer(nu, temp), temp$(1) ) )*KelvinBei(nu, x)=((1)/(2)*x)^(2*nu + 1)* sum((1)/(GAMMA(nu + k + 1)*GAMMA(nu + 2*k + 2))*(((1)/(4)*(x)^(2))^(2*k))/(factorial(k)), k = 0..infinity) KelvinBer[\[Nu], x]*(D[KelvinBei[\[Nu], temp], {temp, 1}]/.temp-> x)- (D[KelvinBer[\[Nu], temp], {temp, 1}]/.temp-> x)*KelvinBei[\[Nu], x]=(Divide[1,2]*x)^(2*\[Nu]+ 1)* Sum[Divide[1,Gamma[\[Nu]+ k + 1]*Gamma[\[Nu]+ 2*k + 2]]*Divide[(Divide[1,4]*(x)^(2))^(2*k),(k)!], {k, 0, Infinity}] Failure Failure Skip Successful
10.65.E8 Unstrip size limit exceeded (5,000,000) KelvinBer(nu, x)*subs( temp=x, diff( KelvinBer(nu, temp), temp$(1) ) )+ KelvinBei(nu, x)*subs( temp=x, diff( KelvinBei(nu, temp), temp$(1) ) )=(1)/(2)*((1)/(2)*x)^(2*nu - 1)* sum((1)/(GAMMA(nu + k + 1)*GAMMA(nu + 2*k))*(((1)/(4)*(x)^(2))^(2*k))/(factorial(k)), k = 0..infinity) KelvinBer[\[Nu], x]*(D[KelvinBer[\[Nu], temp], {temp, 1}]/.temp-> x)+ KelvinBei[\[Nu], x]*(D[KelvinBei[\[Nu], temp], {temp, 1}]/.temp-> x)=Divide[1,2]*(Divide[1,2]*x)^(2*\[Nu]- 1)* Sum[Divide[1,Gamma[\[Nu]+ k + 1]*Gamma[\[Nu]+ 2*k]]*Divide[(Divide[1,4]*(x)^(2))^(2*k),(k)!], {k, 0, Infinity}] Failure Failure Skip Successful
10.65.E9 Unstrip size limit exceeded (5,000,000) (subs( temp=x, diff( KelvinBer(nu, temp), temp$(1) ) ))^(2)+(subs( temp=x, diff( KelvinBei(nu, temp), temp$(1) ) ))^(2)=((1)/(2)*x)^(2*nu - 2)* sum((2*(k)^(2)+ 2*nu*k +(1)/(4)*(nu)^(2))/(GAMMA(nu + k + 1)*GAMMA(nu + 2*k + 1))*(((1)/(4)*(x)^(2))^(2*k))/(factorial(k)), k = 0..infinity) (((D[KelvinBer[\[Nu], temp], {temp, 1}]/.temp-> x)))^(2)+(((D[KelvinBei[\[Nu], temp], {temp, 1}]/.temp-> x)))^(2)=(Divide[1,2]*x)^(2*\[Nu]- 2)* Sum[Divide[2*(k)^(2)+ 2*\[Nu]*k +Divide[1,4]*(\[Nu])^(2),Gamma[\[Nu]+ k + 1]*Gamma[\[Nu]+ 2*k + 1]]*Divide[(Divide[1,4]*(x)^(2))^(2*k),(k)!], {k, 0, Infinity}] Failure Failure Skip Successful
10.66.E1 Unstrip size limit exceeded (5,000,000) KelvinBer(nu, x)+ I*KelvinBei(nu, x)= sum((exp((3*nu + k)* Pi*I/ 4)*(x)^(k)* BesselJ(nu + k, x))/((2)^(k/ 2)* factorial(k)), k = 0..infinity) KelvinBer[\[Nu], x]+ I*KelvinBei[\[Nu], x]= Sum[Divide[Exp[(3*\[Nu]+ k)* Pi*I/ 4]*(x)^(k)* BesselJ[\[Nu]+ k, x],(2)^(k/ 2)* (k)!], {k, 0, Infinity}] Failure Failure Skip Skip
10.66.E1 Unstrip size limit exceeded (5,000,000) sum((exp((3*nu + k)* Pi*I/ 4)*(x)^(k)* BesselJ(nu + k, x))/((2)^(k/ 2)* factorial(k)), k = 0..infinity)= sum((exp((3*nu + 3*k)* Pi*I/ 4)*(x)^(k)* BesselI(nu + k, x))/((2)^(k/ 2)* factorial(k)), k = 0..infinity) Sum[Divide[Exp[(3*\[Nu]+ k)* Pi*I/ 4]*(x)^(k)* BesselJ[\[Nu]+ k, x],(2)^(k/ 2)* (k)!], {k, 0, Infinity}]= Sum[Divide[Exp[(3*\[Nu]+ 3*k)* Pi*I/ 4]*(x)^(k)* BesselI[\[Nu]+ k, x],(2)^(k/ 2)* (k)!], {k, 0, Infinity}] Failure Successful Skip -
10.66#Ex1 Unstrip size limit exceeded (5,000,000) KelvinBer(n, x*sqrt(2))= sum((- 1)^(n + k)* BesselJ(n + 2*k, x)*BesselI(2*k, x), k = - infinity..infinity) KelvinBer[n, x*Sqrt[2]]= Sum[(- 1)^(n + k)* BesselJ[n + 2*k, x]*BesselI[2*k, x], {k, - Infinity, Infinity}] Failure Successful Skip -
10.66#Ex2 Unstrip size limit exceeded (5,000,000) KelvinBei(n, x*sqrt(2))= sum((- 1)^(n + k)* BesselJ(n + 2*k + 1, x)*BesselI(2*k + 1, x), k = - infinity..infinity) KelvinBei[n, x*Sqrt[2]]= Sum[(- 1)^(n + k)* BesselJ[n + 2*k + 1, x]*BesselI[2*k + 1, x], {k, - Infinity, Infinity}] Failure Successful Skip -
10.71.E1 Unstrip size limit exceeded (5,000,000) int((x)^(1 + nu)* f[nu], x)= -((x)^(1 + nu))/(sqrt(2))*(f[nu + 1]- g[nu + 1]) Integrate[(x)^(1 + \[Nu])* Subscript[f, \[Nu]], x]= -Divide[(x)^(1 + \[Nu]),Sqrt[2]]*(Subscript[f, \[Nu]+ 1]- Subscript[g, \[Nu]+ 1]) Failure Failure Skip Successful
10.71.E2 Unstrip size limit exceeded (5,000,000) int((x)^(1 - nu)* f[nu], x)=((x)^(1 - nu))/(sqrt(2))*(f[nu - 1]- g[nu - 1]) Integrate[(x)^(1 - \[Nu])* Subscript[f, \[Nu]], x]=Divide[(x)^(1 - \[Nu]),Sqrt[2]]*(Subscript[f, \[Nu]- 1]- Subscript[g, \[Nu]- 1]) Failure Failure Skip Successful
10.71.E6 Unstrip size limit exceeded (5,000,000) int(x*f[nu]*g[nu], x)=(1)/(4)*(x)^(2)*(2*f[nu]*g[nu]- f[nu - 1]*g[nu + 1]- f[nu + 1]*g[nu - 1]) Integrate[x*Subscript[f, \[Nu]]*Subscript[g, \[Nu]], x]=Divide[1,4]*(x)^(2)*(2*Subscript[f, \[Nu]]*Subscript[g, \[Nu]]- Subscript[f, \[Nu]- 1]*Subscript[g, \[Nu]+ 1]- Subscript[f, \[Nu]+ 1]*Subscript[g, \[Nu]- 1]) Failure Failure Skip Skip
10.71.E7 Unstrip size limit exceeded (5,000,000) int(x*(f(f[nu])^(2)- g(g[nu])^(2)), x)=(f(f[nu])^(2)- f[nu - 1]*f[nu + 1]- g(g[nu])^(2)+ g[nu - 1]*g[nu + 1]) Integrate[x*(f(Subscript[f, \[Nu]])^(2)- g(Subscript[g, \[Nu]])^(2)), x]=(f(Subscript[f, \[Nu]])^(2)- Subscript[f, \[Nu]- 1]*Subscript[f, \[Nu]+ 1]- g(Subscript[g, \[Nu]])^(2)+ Subscript[g, \[Nu]- 1]*Subscript[g, \[Nu]+ 1]) Failure Failure Skip Successful
10.73.E1 Unstrip size limit exceeded (5,000,000) (1)/(r)*diff((r*diff(V, r))+(1)/((r)^(2))*diff(V, [phi$(2)]), r)+ diff(V, [z$(2)])= 0 Divide[1,r]*D[(r*D[V, r])+Divide[1,(r)^(2)]*D[V, {\[Phi], 2}], r]+ D[V, {z, 2}]= 0 Successful Failure - Successful