Results of Error Functions, Dawson’s and Fresnel Integrals

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DLMF Formula Maple Mathematica Symbolic
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7.2.E1 erf ⁡ z = 2 π ⁢ ∫ 0 z e - t 2 ⁢ d t error-function 𝑧 2 𝜋 superscript subscript 0 𝑧 superscript 𝑒 superscript 𝑡 2 𝑡 {\displaystyle{\displaystyle\operatorname{erf}z=\frac{2}{\sqrt{\pi}}\int_{0}^{% z}e^{-t^{2}}\mathrm{d}t}} erf(z)=(2)/(sqrt(Pi))*int(exp(- (t)^(2)), t = 0..z) Erf[z]=Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, 0, z}] Successful Successful - -
7.2.E2 erfc ⁡ z = 2 π ⁢ ∫ z ∞ e - t 2 ⁢ d t complementary-error-function 𝑧 2 𝜋 superscript subscript 𝑧 superscript 𝑒 superscript 𝑡 2 𝑡 {\displaystyle{\displaystyle\operatorname{erfc}z=\frac{2}{\sqrt{\pi}}\int_{z}^% {\infty}e^{-t^{2}}\mathrm{d}t}} erfc(z)=(2)/(sqrt(Pi))*int(exp(- (t)^(2)), t = z..infinity) Erfc[z]=Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, z, Infinity}] Successful Successful - -
7.2.E2 2 π ⁢ ∫ z ∞ e - t 2 ⁢ d t = 1 - erf ⁡ z 2 𝜋 superscript subscript 𝑧 superscript 𝑒 superscript 𝑡 2 𝑡 1 error-function 𝑧 {\displaystyle{\displaystyle\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}% \mathrm{d}t=1-\operatorname{erf}z}} (2)/(sqrt(Pi))*int(exp(- (t)^(2)), t = z..infinity)= 1 - erf(z) Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, z, Infinity}]= 1 - Erf[z] Successful Successful - -
7.2.E3 e - z 2 ⁢ ( 1 + 2 ⁢ i π ⁢ ∫ 0 z e t 2 ⁢ d t ) = e - z 2 ⁢ erfc ⁡ ( - i ⁢ z ) superscript 𝑒 superscript 𝑧 2 1 2 𝑖 𝜋 superscript subscript 0 𝑧 superscript 𝑒 superscript 𝑡 2 𝑡 superscript 𝑒 superscript 𝑧 2 complementary-error-function 𝑖 𝑧 {\displaystyle{\displaystyle e^{-z^{2}}\left(1+\frac{2i}{\sqrt{\pi}}\int_{0}^{% z}e^{t^{2}}\mathrm{d}t\right)=e^{-z^{2}}\operatorname{erfc}\left(-iz\right)}} exp(- (z)^(2))*(1 +(2*I)/(sqrt(Pi))*int(exp((t)^(2)), t = 0..z))= exp(- (z)^(2))*erfc(- I*z) Exp[- (z)^(2)]*(1 +Divide[2*I,Sqrt[Pi]]*Integrate[Exp[(t)^(2)], {t, 0, z}])= Exp[- (z)^(2)]*Erfc[- I*z] Successful Successful - -
7.2#Ex1 lim z → ∞ ⁡ erf ⁡ z = 1 subscript → 𝑧 error-function 𝑧 1 {\displaystyle{\displaystyle\lim_{z\to\infty}\operatorname{erf}z=1}} limit(erf(z), z = infinity)= 1 Limit[Erf[z], z -> Infinity]= 1 Successful Successful - -
7.2#Ex2 lim z → ∞ ⁡ erfc ⁡ z = 0 subscript → 𝑧 complementary-error-function 𝑧 0 {\displaystyle{\displaystyle\lim_{z\to\infty}\operatorname{erfc}z=0}} limit(erfc(z), z = infinity)= 0 Limit[Erfc[z], z -> Infinity]= 0 Successful Successful - -
7.2.E5 F ⁡ ( z ) = e - z 2 ⁢ ∫ 0 z e t 2 ⁢ d t Dawsons-integral 𝑧 superscript 𝑒 superscript 𝑧 2 superscript subscript 0 𝑧 superscript 𝑒 superscript 𝑡 2 𝑡 {\displaystyle{\displaystyle F\left(z\right)=e^{-z^{2}}\int_{0}^{z}e^{t^{2}}% \mathrm{d}t}} dawson(z)= exp(- (z)^(2))*int(exp((t)^(2)), t = 0..z) DawsonF[z]= Exp[- (z)^(2)]*Integrate[Exp[(t)^(2)], {t, 0, z}] Successful Successful - -
7.2.E7 C ⁡ ( z ) = ∫ 0 z cos ⁡ ( 1 2 ⁢ π ⁢ t 2 ) ⁢ d t Fresnel-cosine-integral 𝑧 superscript subscript 0 𝑧 1 2 𝜋 superscript 𝑡 2 𝑡 {\displaystyle{\displaystyle C\left(z\right)=\int_{0}^{z}\cos\left(\tfrac{1}{2% }\pi t^{2}\right)\mathrm{d}t}} FresnelC(z)= int(cos((1)/(2)*Pi*(t)^(2)), t = 0..z) FresnelC[z]= Integrate[Cos[Divide[1,2]*Pi*(t)^(2)], {t, 0, z}] Successful Successful - -
7.2.E8 S ⁡ ( z ) = ∫ 0 z sin ⁡ ( 1 2 ⁢ π ⁢ t 2 ) ⁢ d t Fresnel-sine-integral 𝑧 superscript subscript 0 𝑧 1 2 𝜋 superscript 𝑡 2 𝑡 {\displaystyle{\displaystyle S\left(z\right)=\int_{0}^{z}\sin\left(\tfrac{1}{2% }\pi t^{2}\right)\mathrm{d}t}} FresnelS(z)= int(sin((1)/(2)*Pi*(t)^(2)), t = 0..z) FresnelS[z]= Integrate[Sin[Divide[1,2]*Pi*(t)^(2)], {t, 0, z}] Successful Successful - -
7.2#Ex3 lim x → ∞ ⁡ C ⁡ ( x ) = 1 2 subscript → 𝑥 Fresnel-cosine-integral 𝑥 1 2 {\displaystyle{\displaystyle\lim_{x\to\infty}C\left(x\right)=\tfrac{1}{2}}} limit(FresnelC(x), x = infinity)=(1)/(2) Limit[FresnelC[x], x -> Infinity]=Divide[1,2] Successful Successful - -
7.2#Ex4 lim x → ∞ ⁡ S ⁡ ( x ) = 1 2 subscript → 𝑥 Fresnel-sine-integral 𝑥 1 2 {\displaystyle{\displaystyle\lim_{x\to\infty}S\left(x\right)=\tfrac{1}{2}}} limit(FresnelS(x), x = infinity)=(1)/(2) Limit[FresnelS[x], x -> Infinity]=Divide[1,2] Successful Successful - -
7.2.E10 f ⁡ ( z ) = ( 1 2 - S ⁡ ( z ) ) ⁢ cos ⁡ ( 1 2 ⁢ π ⁢ z 2 ) - ( 1 2 - C ⁡ ( z ) ) ⁢ sin ⁡ ( 1 2 ⁢ π ⁢ z 2 ) Fresnel-auxilliary-function-f 𝑧 1 2 Fresnel-sine-integral 𝑧 1 2 𝜋 superscript 𝑧 2 1 2 Fresnel-cosine-integral 𝑧 1 2 𝜋 superscript 𝑧 2 {\displaystyle{\displaystyle\mathrm{f}\left(z\right)=\left(\tfrac{1}{2}-S\left% (z\right)\right)\cos\left(\tfrac{1}{2}\pi z^{2}\right)-\left(\tfrac{1}{2}-C% \left(z\right)\right)\sin\left(\tfrac{1}{2}\pi z^{2}\right)}} Fresnelf(z)=((1)/(2)- FresnelS(z))* cos((1)/(2)*Pi*(z)^(2))-((1)/(2)- FresnelC(z))* sin((1)/(2)*Pi*(z)^(2)) FresnelF[z]=(Divide[1,2]- FresnelS[z])* Cos[Divide[1,2]*Pi*(z)^(2)]-(Divide[1,2]- FresnelC[z])* Sin[Divide[1,2]*Pi*(z)^(2)] Successful Successful - -
7.2.E11 g ⁡ ( z ) = ( 1 2 - C ⁡ ( z ) ) ⁢ cos ⁡ ( 1 2 ⁢ π ⁢ z 2 ) + ( 1 2 - S ⁡ ( z ) ) ⁢ sin ⁡ ( 1 2 ⁢ π ⁢ z 2 ) Fresnel-auxilliary-function-g 𝑧 1 2 Fresnel-cosine-integral 𝑧 1 2 𝜋 superscript 𝑧 2 1 2 Fresnel-sine-integral 𝑧 1 2 𝜋 superscript 𝑧 2 {\displaystyle{\displaystyle\mathrm{g}\left(z\right)=\left(\tfrac{1}{2}-C\left% (z\right)\right)\cos\left(\tfrac{1}{2}\pi z^{2}\right)+\left(\tfrac{1}{2}-S% \left(z\right)\right)\sin\left(\tfrac{1}{2}\pi z^{2}\right)}} Fresnelg(z)=((1)/(2)- FresnelC(z))* cos((1)/(2)*Pi*(z)^(2))+((1)/(2)- FresnelS(z))* sin((1)/(2)*Pi*(z)^(2)) FresnelG[z]=(Divide[1,2]- FresnelC[z])* Cos[Divide[1,2]*Pi*(z)^(2)]+(Divide[1,2]- FresnelS[z])* Sin[Divide[1,2]*Pi*(z)^(2)] Successful Successful - -
7.4.E1 erf ⁡ ( - z ) = - erf ⁡ ( z ) error-function 𝑧 error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erf}\left(-z\right)=-\operatorname{% erf}\left(z\right)}} erf(- z)= - erf(z) Erf[- z]= - Erf[z] Successful Successful - -
7.4.E2 erfc ⁡ ( - z ) = 2 - erfc ⁡ ( z ) complementary-error-function 𝑧 2 complementary-error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erfc}\left(-z\right)=2-\operatorname% {erfc}\left(z\right)}} erfc(- z)= 2 - erfc(z) Erfc[- z]= 2 - Erfc[z] Successful Successful - -
7.4.E4 F ⁡ ( - z ) = - F ⁡ ( z ) Dawsons-integral 𝑧 Dawsons-integral 𝑧 {\displaystyle{\displaystyle F\left(-z\right)=-F\left(z\right)}} dawson(- z)= - dawson(z) DawsonF[- z]= - DawsonF[z] Successful Successful - -
7.4#Ex1 C ⁡ ( - z ) = - C ⁡ ( z ) Fresnel-cosine-integral 𝑧 Fresnel-cosine-integral 𝑧 {\displaystyle{\displaystyle C\left(-z\right)=-C\left(z\right)}} FresnelC(- z)= - FresnelC(z) FresnelC[- z]= - FresnelC[z] Successful Successful - -
7.4#Ex2 S ⁡ ( - z ) = - S ⁡ ( z ) Fresnel-sine-integral 𝑧 Fresnel-sine-integral 𝑧 {\displaystyle{\displaystyle S\left(-z\right)=-S\left(z\right)}} FresnelS(- z)= - FresnelS(z) FresnelS[- z]= - FresnelS[z] Successful Successful - -
7.4#Ex3 C ⁡ ( i ⁢ z ) = i ⁢ C ⁡ ( z ) Fresnel-cosine-integral 𝑖 𝑧 𝑖 Fresnel-cosine-integral 𝑧 {\displaystyle{\displaystyle C\left(iz\right)=iC\left(z\right)}} FresnelC(I*z)= I*FresnelC(z) FresnelC[I*z]= I*FresnelC[z] Successful Successful - -
7.4#Ex4 S ⁡ ( i ⁢ z ) = - i ⁢ S ⁡ ( z ) Fresnel-sine-integral 𝑖 𝑧 𝑖 Fresnel-sine-integral 𝑧 {\displaystyle{\displaystyle S\left(iz\right)=-iS\left(z\right)}} FresnelS(I*z)= - I*FresnelS(z) FresnelS[I*z]= - I*FresnelS[z] Successful Successful - -
7.4#Ex5 f ⁡ ( i ⁢ z ) = ( 1 / 2 ) ⁢ e 1 4 ⁢ π ⁢ i - 1 2 ⁢ π ⁢ i ⁢ z 2 - i ⁢ f ⁡ ( z ) Fresnel-auxilliary-function-f 𝑖 𝑧 1 2 superscript 𝑒 1 4 𝜋 𝑖 1 2 𝜋 𝑖 superscript 𝑧 2 𝑖 Fresnel-auxilliary-function-f 𝑧 {\displaystyle{\displaystyle\mathrm{f}\left(iz\right)=(1/\sqrt{2})e^{\frac{1}{% 4}\pi i-\frac{1}{2}\pi iz^{2}}-i\mathrm{f}\left(z\right)}} Fresnelf(I*z)=(1/sqrt(2))* exp((1)/(4)*Pi*I -(1)/(2)*Pi*I*(z)^(2))- I*Fresnelf(z) FresnelF[I*z]=(1/Sqrt[2])* Exp[Divide[1,4]*Pi*I -Divide[1,2]*Pi*I*(z)^(2)]- I*FresnelF[z] Failure Failure Successful Successful
7.4#Ex6 g ⁡ ( i ⁢ z ) = ( 1 / 2 ) ⁢ e - 1 4 ⁢ π ⁢ i - 1 2 ⁢ π ⁢ i ⁢ z 2 + i ⁢ g ⁡ ( z ) Fresnel-auxilliary-function-g 𝑖 𝑧 1 2 superscript 𝑒 1 4 𝜋 𝑖 1 2 𝜋 𝑖 superscript 𝑧 2 𝑖 Fresnel-auxilliary-function-g 𝑧 {\displaystyle{\displaystyle\mathrm{g}\left(iz\right)=(1/\sqrt{2})e^{-\frac{1}% {4}\pi i-\frac{1}{2}\pi iz^{2}}+i\mathrm{g}\left(z\right)}} Fresnelg(I*z)=(1/sqrt(2))* exp(-(1)/(4)*Pi*I -(1)/(2)*Pi*I*(z)^(2))+ I*Fresnelg(z) FresnelG[I*z]=(1/Sqrt[2])* Exp[-Divide[1,4]*Pi*I -Divide[1,2]*Pi*I*(z)^(2)]+ I*FresnelG[z] Failure Failure Successful Successful
7.4#Ex7 f ⁡ ( - z ) = 2 ⁢ cos ⁡ ( 1 4 ⁢ π + 1 2 ⁢ π ⁢ z 2 ) - f ⁡ ( z ) Fresnel-auxilliary-function-f 𝑧 2 1 4 𝜋 1 2 𝜋 superscript 𝑧 2 Fresnel-auxilliary-function-f 𝑧 {\displaystyle{\displaystyle\mathrm{f}\left(-z\right)=\sqrt{2}\cos\left(\tfrac% {1}{4}\pi+\tfrac{1}{2}\pi z^{2}\right)-\mathrm{f}\left(z\right)}} Fresnelf(- z)=sqrt(2)*cos((1)/(4)*Pi +(1)/(2)*Pi*(z)^(2))- Fresnelf(z) FresnelF[- z]=Sqrt[2]*Cos[Divide[1,4]*Pi +Divide[1,2]*Pi*(z)^(2)]- FresnelF[z] Failure Successful Successful -
7.4#Ex8 g ⁡ ( - z ) = 2 ⁢ sin ⁡ ( 1 4 ⁢ π + 1 2 ⁢ π ⁢ z 2 ) - g ⁡ ( z ) Fresnel-auxilliary-function-g 𝑧 2 1 4 𝜋 1 2 𝜋 superscript 𝑧 2 Fresnel-auxilliary-function-g 𝑧 {\displaystyle{\displaystyle\mathrm{g}\left(-z\right)=\sqrt{2}\sin\left(\tfrac% {1}{4}\pi+\tfrac{1}{2}\pi z^{2}\right)-\mathrm{g}\left(z\right)}} Fresnelg(- z)=sqrt(2)*sin((1)/(4)*Pi +(1)/(2)*Pi*(z)^(2))- Fresnelg(z) FresnelG[- z]=Sqrt[2]*Sin[Divide[1,4]*Pi +Divide[1,2]*Pi*(z)^(2)]- FresnelG[z] Failure Failure Successful Successful
7.5.E3 C ⁡ ( z ) = 1 2 + f ⁡ ( z ) ⁢ sin ⁡ ( 1 2 ⁢ π ⁢ z 2 ) - g ⁡ ( z ) ⁢ cos ⁡ ( 1 2 ⁢ π ⁢ z 2 ) Fresnel-cosine-integral 𝑧 1 2 Fresnel-auxilliary-function-f 𝑧 1 2 𝜋 superscript 𝑧 2 Fresnel-auxilliary-function-g 𝑧 1 2 𝜋 superscript 𝑧 2 {\displaystyle{\displaystyle C\left(z\right)=\tfrac{1}{2}+\mathrm{f}\left(z% \right)\sin\left(\tfrac{1}{2}\pi z^{2}\right)-\mathrm{g}\left(z\right)\cos% \left(\tfrac{1}{2}\pi z^{2}\right)}} FresnelC(z)=(1)/(2)+ Fresnelf(z)*sin((1)/(2)*Pi*(z)^(2))- Fresnelg(z)*cos((1)/(2)*Pi*(z)^(2)) FresnelC[z]=Divide[1,2]+ FresnelF[z]*Sin[Divide[1,2]*Pi*(z)^(2)]- FresnelG[z]*Cos[Divide[1,2]*Pi*(z)^(2)] Successful Failure - Successful
7.5.E4 S ⁡ ( z ) = 1 2 - f ⁡ ( z ) ⁢ cos ⁡ ( 1 2 ⁢ π ⁢ z 2 ) - g ⁡ ( z ) ⁢ sin ⁡ ( 1 2 ⁢ π ⁢ z 2 ) Fresnel-sine-integral 𝑧 1 2 Fresnel-auxilliary-function-f 𝑧 1 2 𝜋 superscript 𝑧 2 Fresnel-auxilliary-function-g 𝑧 1 2 𝜋 superscript 𝑧 2 {\displaystyle{\displaystyle S\left(z\right)=\tfrac{1}{2}-\mathrm{f}\left(z% \right)\cos\left(\tfrac{1}{2}\pi z^{2}\right)-\mathrm{g}\left(z\right)\sin% \left(\tfrac{1}{2}\pi z^{2}\right)}} FresnelS(z)=(1)/(2)- Fresnelf(z)*cos((1)/(2)*Pi*(z)^(2))- Fresnelg(z)*sin((1)/(2)*Pi*(z)^(2)) FresnelS[z]=Divide[1,2]- FresnelF[z]*Cos[Divide[1,2]*Pi*(z)^(2)]- FresnelG[z]*Sin[Divide[1,2]*Pi*(z)^(2)] Successful Failure - Successful
7.5.E6 e + 1 2 ⁢ π ⁢ i ⁢ z 2 ⁢ ( g ⁡ ( z ) + i ⁢ f ⁡ ( z ) ) = 1 2 ⁢ ( 1 + i ) - ( C ⁡ ( z ) + i ⁢ S ⁡ ( z ) ) superscript 𝑒 1 2 𝜋 𝑖 superscript 𝑧 2 Fresnel-auxilliary-function-g 𝑧 𝑖 Fresnel-auxilliary-function-f 𝑧 1 2 1 𝑖 Fresnel-cosine-integral 𝑧 𝑖 Fresnel-sine-integral 𝑧 {\displaystyle{\displaystyle e^{+\frac{1}{2}\pi iz^{2}}(\mathrm{g}\left(z% \right)+i\mathrm{f}\left(z\right))=\tfrac{1}{2}(1+i)-(C\left(z\right)+iS\left(% z\right))}} exp(+(1)/(2)*Pi*I*(z)^(2))*(Fresnelg(z)+ I*Fresnelf(z))=(1)/(2)*(1 + I)-(FresnelC(z)+ I*FresnelS(z)) Exp[+Divide[1,2]*Pi*I*(z)^(2)]*(FresnelG[z]+ I*FresnelF[z])=Divide[1,2]*(1 + I)-(FresnelC[z]+ I*FresnelS[z]) Failure Failure
Fail
.149314e-2-.173022e-2*I <- {z = 2^(1/2)-I*2^(1/2)}
-.119473e-2+.149314e-2*I <- {z = -2^(1/2)+I*2^(1/2)}
 Successful
7.5.E6 e - 1 2 ⁢ π ⁢ i ⁢ z 2 ⁢ ( g ⁡ ( z ) - i ⁢ f ⁡ ( z ) ) = 1 2 ⁢ ( 1 - i ) - ( C ⁡ ( z ) - i ⁢ S ⁡ ( z ) ) superscript 𝑒 1 2 𝜋 𝑖 superscript 𝑧 2 Fresnel-auxilliary-function-g 𝑧 𝑖 Fresnel-auxilliary-function-f 𝑧 1 2 1 𝑖 Fresnel-cosine-integral 𝑧 𝑖 Fresnel-sine-integral 𝑧 {\displaystyle{\displaystyle e^{-\frac{1}{2}\pi iz^{2}}(\mathrm{g}\left(z% \right)-i\mathrm{f}\left(z\right))=\tfrac{1}{2}(1-i)-(C\left(z\right)-iS\left(% z\right))}} exp(-(1)/(2)*Pi*I*(z)^(2))*(Fresnelg(z)- I*Fresnelf(z))=(1)/(2)*(1 - I)-(FresnelC(z)- I*FresnelS(z)) Exp[-Divide[1,2]*Pi*I*(z)^(2)]*(FresnelG[z]- I*FresnelF[z])=Divide[1,2]*(1 - I)-(FresnelC[z]- I*FresnelS[z]) Failure Failure
Fail
.149314e-2+.173022e-2*I <- {z = 2^(1/2)+I*2^(1/2)}
-.119473e-2-.149314e-2*I <- {z = -2^(1/2)-I*2^(1/2)}
 Successful
7.5.E8 C ⁡ ( z ) + i ⁢ S ⁡ ( z ) = 1 2 ⁢ ( 1 + i ) ⁢ erf ⁡ ζ Fresnel-cosine-integral 𝑧 𝑖 Fresnel-sine-integral 𝑧 1 2 1 𝑖 error-function 𝜁 {\displaystyle{\displaystyle C\left(z\right)+iS\left(z\right)=\tfrac{1}{2}(1+i% )\operatorname{erf}\zeta}} FresnelC(z)+ I*FresnelS(z)=(1)/(2)*(1 + I)* erf(zeta) FresnelC[z]+ I*FresnelS[z]=Divide[1,2]*(1 + I)* Erf[\[zeta]] Failure Failure
Fail
-.1423151062+.1316106532*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)}
.1316106532-.1423151062*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)}
1.141922366+.8679966068*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)}
.8679966068+1.141922366*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
7.5.E8 C ⁡ ( z ) - i ⁢ S ⁡ ( z ) = 1 2 ⁢ ( 1 - i ) ⁢ erf ⁡ ζ Fresnel-cosine-integral 𝑧 𝑖 Fresnel-sine-integral 𝑧 1 2 1 𝑖 error-function 𝜁 {\displaystyle{\displaystyle C\left(z\right)-iS\left(z\right)=\tfrac{1}{2}(1-i% )\operatorname{erf}\zeta}} FresnelC(z)- I*FresnelS(z)=(1)/(2)*(1 - I)* erf(zeta) FresnelC[z]- I*FresnelS[z]=Divide[1,2]*(1 - I)* Erf[\[zeta]] Failure Failure
Fail
66.79933367+67.80964539*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)}
66.52540791+67.53571963*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)}
67.53571963+66.52540791*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)}
67.80964539+66.79933367*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
7.5.E10 g ⁡ ( z ) + i ⁢ f ⁡ ( z ) = 1 2 ⁢ ( 1 + i ) ⁢ e ζ 2 ⁢ erfc ⁡ ζ Fresnel-auxilliary-function-g 𝑧 𝑖 Fresnel-auxilliary-function-f 𝑧 1 2 1 𝑖 superscript 𝑒 superscript 𝜁 2 complementary-error-function 𝜁 {\displaystyle{\displaystyle\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)% =\tfrac{1}{2}(1+i)e^{\zeta^{2}}\operatorname{erfc}\zeta}} Fresnelg(z)+ I*Fresnelf(z)=(1)/(2)*(1 + I)* exp((zeta)^(2))*erfc(zeta) FresnelG[z]+ I*FresnelF[z]=Divide[1,2]*(1 + I)* Exp[(\[zeta])^(2)]*Erfc[\[zeta]] Failure Failure
Fail
-.874918896e-1+.8375300635e-1*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)}
.8375400635e-1-.874928896e-1*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)}
.1946430180+1.537001110*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)}
1.537002110+.1946420180*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
7.5.E10 g ⁡ ( z ) - i ⁢ f ⁡ ( z ) = 1 2 ⁢ ( 1 - i ) ⁢ e ζ 2 ⁢ erfc ⁡ ζ Fresnel-auxilliary-function-g 𝑧 𝑖 Fresnel-auxilliary-function-f 𝑧 1 2 1 𝑖 superscript 𝑒 superscript 𝜁 2 complementary-error-function 𝜁 {\displaystyle{\displaystyle\mathrm{g}\left(z\right)-i\mathrm{f}\left(z\right)% =\tfrac{1}{2}(1-i)e^{\zeta^{2}}\operatorname{erfc}\zeta}} Fresnelg(z)- I*Fresnelf(z)=(1)/(2)*(1 - I)* exp((zeta)^(2))*erfc(zeta) FresnelG[z]- I*FresnelF[z]=Divide[1,2]*(1 - I)* Exp[(\[zeta])^(2)]*Erfc[\[zeta]] Failure Failure
Fail
-.1458959936+.662848896e-1*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)}
-.3171418896-.1049610064*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)}
1.307352110-.2158500180*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)}
-.3500698200e-1-1.558209110*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
 Error
7.6.E1 erf ⁡ z = 2 π ⁢ ∑ n = 0 ∞ ( - 1 ) n ⁢ z 2 ⁢ n + 1 n ! ⁢ ( 2 ⁢ n + 1 ) error-function 𝑧 2 𝜋 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑧 2 𝑛 1 𝑛 2 𝑛 1 {\displaystyle{\displaystyle\operatorname{erf}z=\frac{2}{\sqrt{\pi}}\sum_{n=0}% ^{\infty}\frac{(-1)^{n}z^{2n+1}}{n!(2n+1)}}} erf(z)=(2)/(sqrt(Pi))*sum(((- 1)^(n)* (z)^(2*n + 1))/(factorial(n)*(2*n + 1)), n = 0..infinity) Erf[z]=Divide[2,Sqrt[Pi]]*Sum[Divide[(- 1)^(n)* (z)^(2*n + 1),(n)!*(2*n + 1)], {n, 0, Infinity}] Successful Successful - -
7.6.E4 C ⁡ ( z ) = ∑ n = 0 ∞ ( - 1 ) n ⁢ ( 1 2 ⁢ π ) 2 ⁢ n ( 2 ⁢ n ) ! ⁢ ( 4 ⁢ n + 1 ) ⁢ z 4 ⁢ n + 1 Fresnel-cosine-integral 𝑧 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 1 2 𝜋 2 𝑛 2 𝑛 4 𝑛 1 superscript 𝑧 4 𝑛 1 {\displaystyle{\displaystyle C\left(z\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}% (\frac{1}{2}\pi)^{2n}}{(2n)!(4n+1)}z^{4n+1}}} FresnelC(z)= sum(((- 1)^(n)*((1)/(2)*Pi)^(2*n))/(factorial(2*n)*(4*n + 1))*(z)^(4*n + 1), n = 0..infinity) FresnelC[z]= Sum[Divide[(- 1)^(n)*(Divide[1,2]*Pi)^(2*n),(2*n)!*(4*n + 1)]*(z)^(4*n + 1), {n, 0, Infinity}] Successful Successful - -
7.6.E6 S ⁡ ( z ) = ∑ n = 0 ∞ ( - 1 ) n ⁢ ( 1 2 ⁢ π ) 2 ⁢ n + 1 ( 2 ⁢ n + 1 ) ! ⁢ ( 4 ⁢ n + 3 ) ⁢ z 4 ⁢ n + 3 Fresnel-sine-integral 𝑧 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 1 2 𝜋 2 𝑛 1 2 𝑛 1 4 𝑛 3 superscript 𝑧 4 𝑛 3 {\displaystyle{\displaystyle S\left(z\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}% (\frac{1}{2}\pi)^{2n+1}}{(2n+1)!(4n+3)}z^{4n+3}}} FresnelS(z)= sum(((- 1)^(n)*((1)/(2)*Pi)^(2*n + 1))/(factorial(2*n + 1)*(4*n + 3))*(z)^(4*n + 3), n = 0..infinity) FresnelS[z]= Sum[Divide[(- 1)^(n)*(Divide[1,2]*Pi)^(2*n + 1),(2*n + 1)!*(4*n + 3)]*(z)^(4*n + 3), {n, 0, Infinity}] Successful Successful - -
7.6.E8 erf ⁡ z = 2 ⁢ z π ⁢ ∑ n = 0 ∞ ( - 1 ) n ⁢ ( 𝗂 2 ⁢ n ( 1 ) ⁡ ( z 2 ) - 𝗂 2 ⁢ n + 1 ( 1 ) ⁡ ( z 2 ) ) error-function 𝑧 2 𝑧 𝜋 superscript subscript 𝑛 0 superscript 1 𝑛 spherical-Bessel-I-1 2 𝑛 superscript 𝑧 2 spherical-Bessel-I-1 2 𝑛 1 superscript 𝑧 2 {\displaystyle{\displaystyle\operatorname{erf}z=\frac{2z}{\sqrt{\pi}}\sum_{n=0% }^{\infty}(-1)^{n}\left({\mathsf{i}^{(1)}_{2n}}\left(z^{2}\right)-{\mathsf{i}^% {(1)}_{2n+1}}\left(z^{2}\right)\right)}} Error \|Sqrt[1/2 Pi /$2] BesselI[-2*n - 1/2, 2*n]*(z)^(2)- Sqrt[1/2 Pi /$2] BesselI[(-1)^(1-1)*2*n + 1 + 1/2, 2*n + 1]\|\|Sqrt[1/2 Pi /$2] BesselI[-2*n + 1 - 1/2, 2*n + 1]*(z)^(2)), {n, 0, Infinity}] Error Error - -
7.6.E9 erf ⁡ ( a ⁢ z ) = 2 ⁢ z π ⁢ e ( 1 2 - a 2 ) ⁢ z 2 ⁢ ∑ n = 0 ∞ T 2 ⁢ n + 1 ⁡ ( a ) ⁢ 𝗂 n ( 1 ) ⁡ ( 1 2 ⁢ z 2 ) error-function 𝑎 𝑧 2 𝑧 𝜋 superscript 𝑒 1 2 superscript 𝑎 2 superscript 𝑧 2 superscript subscript 𝑛 0 Chebyshev-polynomial-first-kind-T 2 𝑛 1 𝑎 spherical-Bessel-I-1 𝑛 1 2 superscript 𝑧 2 {\displaystyle{\displaystyle\operatorname{erf}\left(az\right)=\frac{2z}{\sqrt{% \pi}}e^{(\frac{1}{2}-a^{2})z^{2}}\sum_{n=0}^{\infty}T_{2n+1}\left(a\right){% \mathsf{i}^{(1)}_{n}}\left(\tfrac{1}{2}z^{2}\right)}} Error \|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]*Divide[1,2]*(z)^(2), {n, 0, Infinity}] Error Error - -
7.6.E10 C ⁡ ( z ) = z ⁢ ∑ n = 0 ∞ 𝗃 2 ⁢ n ⁡ ( 1 2 ⁢ π ⁢ z 2 ) Fresnel-cosine-integral 𝑧 𝑧 superscript subscript 𝑛 0 spherical-Bessel-J 2 𝑛 1 2 𝜋 superscript 𝑧 2 {\displaystyle{\displaystyle C\left(z\right)=z\sum_{n=0}^{\infty}\mathsf{j}_{2% n}\left(\tfrac{1}{2}\pi z^{2}\right)}} Error FresnelC[z]= z*Sum[SphericalBesselJ[2*n, Divide[1,2]*Pi*(z)^(2)], {n, 0, Infinity}] Error Failure - Skip
7.6.E11 S ⁡ ( z ) = z ⁢ ∑ n = 0 ∞ 𝗃 2 ⁢ n + 1 ⁡ ( 1 2 ⁢ π ⁢ z 2 ) Fresnel-sine-integral 𝑧 𝑧 superscript subscript 𝑛 0 spherical-Bessel-J 2 𝑛 1 1 2 𝜋 superscript 𝑧 2 {\displaystyle{\displaystyle S\left(z\right)=z\sum_{n=0}^{\infty}\mathsf{j}_{2% n+1}\left(\tfrac{1}{2}\pi z^{2}\right)}} Error FresnelS[z]= z*Sum[SphericalBesselJ[2*n + 1, Divide[1,2]*Pi*(z)^(2)], {n, 0, Infinity}] Error Failure - Skip
7.7.E1 erfc ⁡ z = 2 π ⁢ e - z 2 ⁢ ∫ 0 ∞ e - z 2 ⁢ t 2 t 2 + 1 ⁢ d t complementary-error-function 𝑧 2 𝜋 superscript 𝑒 superscript 𝑧 2 superscript subscript 0 superscript 𝑒 superscript 𝑧 2 superscript 𝑡 2 superscript 𝑡 2 1 𝑡 {\displaystyle{\displaystyle\operatorname{erfc}z=\frac{2}{\pi}e^{-z^{2}}\int_{% 0}^{\infty}\frac{e^{-z^{2}t^{2}}}{t^{2}+1}\mathrm{d}t}} erfc(z)=(2)/(Pi)*exp(- (z)^(2))*int((exp(- (z)^(2)* (t)^(2)))/((t)^(2)+ 1), t = 0..infinity) Erfc[z]=Divide[2,Pi]*Exp[- (z)^(2)]*Integrate[Divide[Exp[- (z)^(2)* (t)^(2)],(t)^(2)+ 1], {t, 0, Infinity}] Successful Failure - Error
7.7.E2 1 π ⁢ i ⁢ ∫ - ∞ ∞ e - t 2 ⁢ d t t - z = 2 ⁢ z π ⁢ i ⁢ ∫ 0 ∞ e - t 2 ⁢ d t t 2 - z 2 1 𝜋 𝑖 superscript subscript superscript 𝑒 superscript 𝑡 2 𝑡 𝑡 𝑧 2 𝑧 𝜋 𝑖 superscript subscript 0 superscript 𝑒 superscript 𝑡 2 𝑡 superscript 𝑡 2 superscript 𝑧 2 {\displaystyle{\displaystyle\frac{1}{\pi i}\int_{-\infty}^{\infty}\frac{e^{-t^% {2}}\mathrm{d}t}{t-z}=\frac{2z}{\pi i}\int_{0}^{\infty}\frac{e^{-t^{2}}\mathrm% {d}t}{t^{2}-z^{2}}}} (1)/(Pi*I)*int((exp(- (t)^(2)))/(t - z), t = - infinity..infinity)=(2*z)/(Pi*I)*int((exp(- (t)^(2)))/((t)^(2)- (z)^(2)), t = 0..infinity) Divide[1,Pi*I]*Integrate[Divide[Exp[- (t)^(2)],t - z], {t, - Infinity, Infinity}]=Divide[2*z,Pi*I]*Integrate[Divide[Exp[- (t)^(2)],(t)^(2)- (z)^(2)], {t, 0, Infinity}] Failure Failure Skip Successful
7.7.E3 ∫ 0 ∞ e - a ⁢ t 2 + 2 ⁢ i ⁢ z ⁢ t ⁢ d t = 1 2 ⁢ π a ⁢ e - z 2 / a + i a ⁢ F ⁡ ( z a ) superscript subscript 0 superscript 𝑒 𝑎 superscript 𝑡 2 2 𝑖 𝑧 𝑡 𝑡 1 2 𝜋 𝑎 superscript 𝑒 superscript 𝑧 2 𝑎 𝑖 𝑎 Dawsons-integral 𝑧 𝑎 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at^{2}+2izt}\mathrm{d}t=\frac% {1}{2}\sqrt{\frac{\pi}{a}}e^{-z^{2}/a}+\frac{i}{\sqrt{a}}F\left(\frac{z}{\sqrt% {a}}\right)}} int(exp(- a*(t)^(2)+ 2*I*z*t), t = 0..infinity)=(1)/(2)*sqrt((Pi)/(a))*exp(- (z)^(2)/ a)+(I)/(sqrt(a))*dawson((z)/(sqrt(a))) Integrate[Exp[- a*(t)^(2)+ 2*I*z*t], {t, 0, Infinity}]=Divide[1,2]*Sqrt[Divide[Pi,a]]*Exp[- (z)^(2)/ a]+Divide[I,Sqrt[a]]*DawsonF[Divide[z,Sqrt[a]]] Failure Successful Skip -
7.7.E4 ∫ 0 ∞ e - a ⁢ t t + z 2 ⁢ d t = π a ⁢ e a ⁢ z 2 ⁢ erfc ⁡ ( a ⁢ z ) superscript subscript 0 superscript 𝑒 𝑎 𝑡 𝑡 superscript 𝑧 2 𝑡 𝜋 𝑎 superscript 𝑒 𝑎 superscript 𝑧 2 complementary-error-function 𝑎 𝑧 {\displaystyle{\displaystyle\int_{0}^{\infty}\frac{e^{-at}}{\sqrt{t+z^{2}}}% \mathrm{d}t=\sqrt{\frac{\pi}{a}}e^{az^{2}}\operatorname{erfc}\left(\sqrt{a}z% \right)}} int((exp(- a*t))/(sqrt(t + (z)^(2))), t = 0..infinity)=sqrt((Pi)/(a))*exp(a*(z)^(2))*erfc(sqrt(a)*z) Integrate[Divide[Exp[- a*t],Sqrt[t + (z)^(2)]], {t, 0, Infinity}]=Sqrt[Divide[Pi,a]]*Exp[a*(z)^(2)]*Erfc[Sqrt[a]*z] Successful Failure - Successful
7.7.E6 ∫ x ∞ e - ( a ⁢ t 2 + 2 ⁢ b ⁢ t + c ) ⁢ d t = 1 2 ⁢ π a ⁢ e ( b 2 - a ⁢ c ) / a ⁢ erfc ⁡ ( a ⁢ x + b a ) superscript subscript 𝑥 superscript 𝑒 𝑎 superscript 𝑡 2 2 𝑏 𝑡 𝑐 𝑡 1 2 𝜋 𝑎 superscript 𝑒 superscript 𝑏 2 𝑎 𝑐 𝑎 complementary-error-function 𝑎 𝑥 𝑏 𝑎 {\displaystyle{\displaystyle\int_{x}^{\infty}e^{-(at^{2}+2bt+c)}\mathrm{d}t=% \frac{1}{2}\sqrt{\frac{\pi}{a}}e^{(b^{2}-ac)/a}\operatorname{erfc}\left(\sqrt{% a}x+\frac{b}{\sqrt{a}}\right)}} int(exp(-(a*(t)^(2)+ 2*b*t + c)), t = x..infinity)=(1)/(2)*sqrt((Pi)/(a))*exp(((b)^(2)- a*c)/ a)*erfc(sqrt(a)*x +(b)/(sqrt(a))) Integrate[Exp[-(a*(t)^(2)+ 2*b*t + c)], {t, x, Infinity}]=Divide[1,2]*Sqrt[Divide[Pi,a]]*Exp[((b)^(2)- a*c)/ a]*Erfc[Sqrt[a]*x +Divide[b,Sqrt[a]]] Failure Failure Skip Successful
7.7.E7 ∫ x ∞ e - a 2 ⁢ t 2 - ( b 2 / t 2 ) ⁢ d t = π 4 ⁢ a ⁢ ( e 2 ⁢ a ⁢ b ⁢ erfc ⁡ ( a ⁢ x + ( b / x ) ) + e - 2 ⁢ a ⁢ b ⁢ erfc ⁡ ( a ⁢ x - ( b / x ) ) ) superscript subscript 𝑥 superscript 𝑒 superscript 𝑎 2 superscript 𝑡 2 superscript 𝑏 2 superscript 𝑡 2 𝑡 𝜋 4 𝑎 superscript 𝑒 2 𝑎 𝑏 complementary-error-function 𝑎 𝑥 𝑏 𝑥 superscript 𝑒 2 𝑎 𝑏 complementary-error-function 𝑎 𝑥 𝑏 𝑥 {\displaystyle{\displaystyle\int_{x}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}% \mathrm{d}t=\frac{\sqrt{\pi}}{4a}\left(e^{2ab}\operatorname{erfc}\left(ax+(b/x% )\right)+e^{-2ab}\operatorname{erfc}\left(ax-(b/x)\right)\right)}} int(exp(- (a)^(2)* (t)^(2)-((b)^(2)/ (t)^(2))), t = x..infinity)=(sqrt(Pi))/(4*a)*(exp(2*a*b)*erfc(a*x +(b/ x))+ exp(- 2*a*b)*erfc(a*x -(b/ x))) Integrate[Exp[- (a)^(2)* (t)^(2)-((b)^(2)/ (t)^(2))], {t, x, Infinity}]=Divide[Sqrt[Pi],4*a]*(Exp[2*a*b]*Erfc[a*x +(b/ x)]+ Exp[- 2*a*b]*Erfc[a*x -(b/ x)]) Failure Failure Skip Error
7.7.E8 ∫ 0 ∞ e - a 2 ⁢ t 2 - ( b 2 / t 2 ) ⁢ d t = π 2 ⁢ a ⁢ e - 2 ⁢ a ⁢ b superscript subscript 0 superscript 𝑒 superscript 𝑎 2 superscript 𝑡 2 superscript 𝑏 2 superscript 𝑡 2 𝑡 𝜋 2 𝑎 superscript 𝑒 2 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}% \mathrm{d}t=\frac{\sqrt{\pi}}{2a}e^{-2ab}}} int(exp(- (a)^(2)* (t)^(2)-((b)^(2)/ (t)^(2))), t = 0..infinity)=(sqrt(Pi))/(2*a)*exp(- 2*a*b) Integrate[Exp[- (a)^(2)* (t)^(2)-((b)^(2)/ (t)^(2))], {t, 0, Infinity}]=Divide[Sqrt[Pi],2*a]*Exp[- 2*a*b] Successful Failure - Successful
7.7.E9 ∫ 0 x erf ⁡ t ⁢ d t = x ⁢ erf ⁡ x + 1 π ⁢ ( e - x 2 - 1 ) superscript subscript 0 𝑥 error-function 𝑡 𝑡 𝑥 error-function 𝑥 1 𝜋 superscript 𝑒 superscript 𝑥 2 1 {\displaystyle{\displaystyle\int_{0}^{x}\operatorname{erf}t\mathrm{d}t=x% \operatorname{erf}x+\frac{1}{\sqrt{\pi}}\left(e^{-x^{2}}-1\right)}} int(erf(t), t = 0..x)= x*erf(x)+(1)/(sqrt(Pi))*(exp(- (x)^(2))- 1) Integrate[Erf[t], {t, 0, x}]= x*Erf[x]+Divide[1,Sqrt[Pi]]*(Exp[- (x)^(2)]- 1) Successful Successful - -
7.7.E10 f ⁡ ( z ) = 1 π ⁢ 2 ⁢ ∫ 0 ∞ e - π ⁢ z 2 ⁢ t / 2 t ⁢ ( t 2 + 1 ) ⁢ d t Fresnel-auxilliary-function-f 𝑧 1 𝜋 2 superscript subscript 0 superscript 𝑒 𝜋 superscript 𝑧 2 𝑡 2 𝑡 superscript 𝑡 2 1 𝑡 {\displaystyle{\displaystyle\mathrm{f}\left(z\right)=\frac{1}{\pi\sqrt{2}}\int% _{0}^{\infty}\frac{e^{-\pi z^{2}t/2}}{\sqrt{t}(t^{2}+1)}\mathrm{d}t}} Fresnelf(z)=(1)/(Pi*sqrt(2))*int((exp(- Pi*(z)^(2)* t/ 2))/(sqrt(t)*((t)^(2)+ 1)), t = 0..infinity) FresnelF[z]=Divide[1,Pi*Sqrt[2]]*Integrate[Divide[Exp[- Pi*(z)^(2)* t/ 2],Sqrt[t]*((t)^(2)+ 1)], {t, 0, Infinity}] Failure Failure Skip Error
7.7.E11 g ⁡ ( z ) = 1 π ⁢ 2 ⁢ ∫ 0 ∞ t ⁢ e - π ⁢ z 2 ⁢ t / 2 t 2 + 1 ⁢ d t Fresnel-auxilliary-function-g 𝑧 1 𝜋 2 superscript subscript 0 𝑡 superscript 𝑒 𝜋 superscript 𝑧 2 𝑡 2 superscript 𝑡 2 1 𝑡 {\displaystyle{\displaystyle\mathrm{g}\left(z\right)=\frac{1}{\pi\sqrt{2}}\int% _{0}^{\infty}\frac{\sqrt{t}e^{-\pi z^{2}t/2}}{t^{2}+1}\mathrm{d}t}} Fresnelg(z)=(1)/(Pi*sqrt(2))*int((sqrt(t)*exp(- Pi*(z)^(2)* t/ 2))/((t)^(2)+ 1), t = 0..infinity) FresnelG[z]=Divide[1,Pi*Sqrt[2]]*Integrate[Divide[Sqrt[t]*Exp[- Pi*(z)^(2)* t/ 2],(t)^(2)+ 1], {t, 0, Infinity}] Failure Failure Skip Error
7.7.E12 g ⁡ ( z ) + i ⁢ f ⁡ ( z ) = e - π ⁢ i ⁢ z 2 / 2 ⁢ ∫ z ∞ e π ⁢ i ⁢ t 2 / 2 ⁢ d t Fresnel-auxilliary-function-g 𝑧 𝑖 Fresnel-auxilliary-function-f 𝑧 superscript 𝑒 𝜋 𝑖 superscript 𝑧 2 2 superscript subscript 𝑧 superscript 𝑒 𝜋 𝑖 superscript 𝑡 2 2 𝑡 {\displaystyle{\displaystyle\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)% =e^{-\pi iz^{2}/2}\int_{z}^{\infty}e^{\pi it^{2}/2}\mathrm{d}t}} Fresnelg(z)+ I*Fresnelf(z)= exp(- Pi*I*(z)^(2)/ 2)*int(exp(Pi*I*(t)^(2)/ 2), t = z..infinity) FresnelG[z]+ I*FresnelF[z]= Exp[- Pi*I*(z)^(2)/ 2]*Integrate[Exp[Pi*I*(t)^(2)/ 2], {t, z, Infinity}] Failure Failure Skip
Fail
Complex[-0.12449815517713354, 0.12449815517716199] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.12449815517710515, -0.12449815517713354] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
7.7.E13 f ⁡ ( z ) = ( 2 ⁢ π ) - 3 / 2 2 ⁢ π ⁢ i ⁢ ∫ c - i ⁢ ∞ c + i ⁢ ∞ ζ - s ⁢ Γ ⁡ ( s ) ⁢ Γ ⁡ ( s + 1 2 ) ⁢ Γ ⁡ ( s + 3 4 ) ⁢ Γ ⁡ ( 1 4 - s ) ⁢ d s Fresnel-auxilliary-function-f 𝑧 superscript 2 𝜋 3 2 2 𝜋 𝑖 superscript subscript 𝑐 𝑖 𝑐 𝑖 superscript 𝜁 𝑠 Euler-Gamma 𝑠 Euler-Gamma 𝑠 1 2 Euler-Gamma 𝑠 3 4 Euler-Gamma 1 4 𝑠 𝑠 {\displaystyle{\displaystyle\mathrm{f}\left(z\right)=\frac{(2\pi)^{-3/2}}{2\pi i% }\int_{c-i\infty}^{c+i\infty}\zeta^{-s}\Gamma\left(s\right)\Gamma\left(s+% \tfrac{1}{2}\right)\*\Gamma\left(s+\tfrac{3}{4}\right)\Gamma\left(\tfrac{1}{4}% -s\right)\mathrm{d}s}} Fresnelf(z)=((2*Pi)^(- 3/ 2))/(2*Pi*I)*int((zeta)^(- s)* GAMMA(s)*GAMMA(s +(1)/(2))* GAMMA(s +(3)/(4))*GAMMA((1)/(4)- s), s = c - I*infinity..c + I*infinity) FresnelF[z]=Divide[(2*Pi)^(- 3/ 2),2*Pi*I]*Integrate[(\[zeta])^(- s)* Gamma[s]*Gamma[s +Divide[1,2]]* Gamma[s +Divide[3,4]]*Gamma[Divide[1,4]- s], {s, c - I*Infinity, c + I*Infinity}] Failure Failure Skip Error
7.7.E14 g ⁡ ( z ) = ( 2 ⁢ π ) - 3 / 2 2 ⁢ π ⁢ i ⁢ ∫ c - i ⁢ ∞ c + i ⁢ ∞ ζ - s ⁢ Γ ⁡ ( s ) ⁢ Γ ⁡ ( s + 1 2 ) ⁢ Γ ⁡ ( s + 1 4 ) ⁢ Γ ⁡ ( 3 4 - s ) ⁢ d s Fresnel-auxilliary-function-g 𝑧 superscript 2 𝜋 3 2 2 𝜋 𝑖 superscript subscript 𝑐 𝑖 𝑐 𝑖 superscript 𝜁 𝑠 Euler-Gamma 𝑠 Euler-Gamma 𝑠 1 2 Euler-Gamma 𝑠 1 4 Euler-Gamma 3 4 𝑠 𝑠 {\displaystyle{\displaystyle\mathrm{g}\left(z\right)=\frac{(2\pi)^{-3/2}}{2\pi i% }\int_{c-i\infty}^{c+i\infty}\zeta^{-s}\Gamma\left(s\right)\Gamma\left(s+% \tfrac{1}{2}\right)\*\Gamma\left(s+\tfrac{1}{4}\right)\Gamma\left(\tfrac{3}{4}% -s\right)\mathrm{d}s}} Fresnelg(z)=((2*Pi)^(- 3/ 2))/(2*Pi*I)*int((zeta)^(- s)* GAMMA(s)*GAMMA(s +(1)/(2))* GAMMA(s +(1)/(4))*GAMMA((3)/(4)- s), s = c - I*infinity..c + I*infinity) FresnelG[z]=Divide[(2*Pi)^(- 3/ 2),2*Pi*I]*Integrate[(\[zeta])^(- s)* Gamma[s]*Gamma[s +Divide[1,2]]* Gamma[s +Divide[1,4]]*Gamma[Divide[3,4]- s], {s, c - I*Infinity, c + I*Infinity}] Failure Failure Skip Error
7.7.E15 ∫ 0 ∞ e - a ⁢ t ⁢ cos ⁡ ( t 2 ) ⁢ d t = π 2 ⁢ f ⁡ ( a 2 ⁢ π ) superscript subscript 0 superscript 𝑒 𝑎 𝑡 superscript 𝑡 2 𝑡 𝜋 2 Fresnel-auxilliary-function-f 𝑎 2 𝜋 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}\cos\left(t^{2}\right)% \mathrm{d}t=\sqrt{\frac{\pi}{2}}\mathrm{f}\left(\frac{a}{\sqrt{2\pi}}\right)}} int(exp(- a*t)*cos((t)^(2)), t = 0..infinity)=sqrt((Pi)/(2))*Fresnelf((a)/(sqrt(2*Pi))) Integrate[Exp[- a*t]*Cos[(t)^(2)], {t, 0, Infinity}]=Sqrt[Divide[Pi,2]]*FresnelF[Divide[a,Sqrt[2*Pi]]] Successful Failure - Error
7.7.E16 ∫ 0 ∞ e - a ⁢ t ⁢ sin ⁡ ( t 2 ) ⁢ d t = π 2 ⁢ g ⁡ ( a 2 ⁢ π ) superscript subscript 0 superscript 𝑒 𝑎 𝑡 superscript 𝑡 2 𝑡 𝜋 2 Fresnel-auxilliary-function-g 𝑎 2 𝜋 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}\sin\left(t^{2}\right)% \mathrm{d}t=\sqrt{\frac{\pi}{2}}\mathrm{g}\left(\frac{a}{\sqrt{2\pi}}\right)}} int(exp(- a*t)*sin((t)^(2)), t = 0..infinity)=sqrt((Pi)/(2))*Fresnelg((a)/(sqrt(2*Pi))) Integrate[Exp[- a*t]*Sin[(t)^(2)], {t, 0, Infinity}]=Sqrt[Divide[Pi,2]]*FresnelG[Divide[a,Sqrt[2*Pi]]] Successful Failure - Error
7.8.E1 ∫ x ∞ e - t 2 ⁢ d t e - x 2 = e x 2 ⁢ ∫ x ∞ e - t 2 ⁢ d t superscript subscript 𝑥 superscript 𝑒 superscript 𝑡 2 𝑡 superscript 𝑒 superscript 𝑥 2 superscript 𝑒 superscript 𝑥 2 superscript subscript 𝑥 superscript 𝑒 superscript 𝑡 2 𝑡 {\displaystyle{\displaystyle\frac{\int_{x}^{\infty}e^{-t^{2}}\mathrm{d}t}{e^{-% x^{2}}}=e^{x^{2}}\int_{x}^{\infty}e^{-t^{2}}\mathrm{d}t}} (int(exp(- (t)^(2)), t = x..infinity))/(exp(- (x)^(2)))= exp((x)^(2))*int(exp(- (t)^(2)), t = x..infinity) Divide[Integrate[Exp[- (t)^(2)], {t, x, Infinity}],Exp[- (x)^(2)]]= Exp[(x)^(2)]*Integrate[Exp[- (t)^(2)], {t, x, Infinity}] Successful Successful - -
7.8.E5 x 2 2 ⁢ x 2 + 1 ≤ x 2 ⁢ ( 2 ⁢ x 2 + 5 ) 4 ⁢ x 4 + 12 ⁢ x 2 + 3 superscript 𝑥 2 2 superscript 𝑥 2 1 superscript 𝑥 2 2 superscript 𝑥 2 5 4 superscript 𝑥 4 12 superscript 𝑥 2 3 {\displaystyle{\displaystyle\frac{x^{2}}{2x^{2}+1}<=\frac{x^{2}(2x^{2}+5)}{4x^% {4}+12x^{2}+3}}} ((x)^(2))/(2*(x)^(2)+ 1)< =((x)^(2)*(2*(x)^(2)+ 5))/(4*(x)^(4)+ 12*(x)^(2)+ 3) Divide[(x)^(2),2*(x)^(2)+ 1]< =Divide[(x)^(2)*(2*(x)^(2)+ 5),4*(x)^(4)+ 12*(x)^(2)+ 3] Failure Failure Successful Successful
7.8.E5 2 ⁢ x 4 + 9 ⁢ x 2 + 4 4 ⁢ x 4 + 20 ⁢ x 2 + 15 < x 2 + 1 2 ⁢ x 2 + 3 2 superscript 𝑥 4 9 superscript 𝑥 2 4 4 superscript 𝑥 4 20 superscript 𝑥 2 15 superscript 𝑥 2 1 2 superscript 𝑥 2 3 {\displaystyle{\displaystyle\frac{2x^{4}+9x^{2}+4}{4x^{4}+20x^{2}+15}<\frac{x^% {2}+1}{2x^{2}+3}}} (2*(x)^(4)+ 9*(x)^(2)+ 4)/(4*(x)^(4)+ 20*(x)^(2)+ 15)<((x)^(2)+ 1)/(2*(x)^(2)+ 3) Divide[2*(x)^(4)+ 9*(x)^(2)+ 4,4*(x)^(4)+ 20*(x)^(2)+ 15]<Divide[(x)^(2)+ 1,2*(x)^(2)+ 3] Failure Failure Skip Successful
7.8.E6 ∫ 0 x e a ⁢ t 2 ⁢ d t < 1 3 ⁢ a ⁢ x ⁢ ( 2 ⁢ e a ⁢ x 2 + a ⁢ x 2 - 2 ) superscript subscript 0 𝑥 superscript 𝑒 𝑎 superscript 𝑡 2 𝑡 1 3 𝑎 𝑥 2 superscript 𝑒 𝑎 superscript 𝑥 2 𝑎 superscript 𝑥 2 2 {\displaystyle{\displaystyle\int_{0}^{x}e^{at^{2}}\mathrm{d}t<\frac{1}{3ax}% \left(2e^{ax^{2}}+ax^{2}-2\right)}} int(exp(a*(t)^(2)), t = 0..x)<(1)/(3*a*x)*(2*exp(a*(x)^(2))+ a*(x)^(2)- 2) Integrate[Exp[a*(t)^(2)], {t, 0, x}]<Divide[1,3*a*x]*(2*Exp[a*(x)^(2)]+ a*(x)^(2)- 2) Error Failure - Successful
7.8.E7 ∫ 0 x e t 2 ⁢ d t < e x 2 - 1 x superscript subscript 0 𝑥 superscript 𝑒 superscript 𝑡 2 𝑡 superscript 𝑒 superscript 𝑥 2 1 𝑥 {\displaystyle{\displaystyle\int_{0}^{x}e^{t^{2}}\mathrm{d}t<\frac{e^{x^{2}}-1% }{x}}} int(exp((t)^(2)), t = 0..x)<(exp((x)^(2))- 1)/(x) Integrate[Exp[(t)^(2)], {t, 0, x}]<Divide[Exp[(x)^(2)]- 1,x] Failure Failure Skip Successful
7.8.E8 erf ⁡ x < 1 - e - 4 ⁢ x 2 / π error-function 𝑥 1 4 superscript 𝑥 2 {\displaystyle{\displaystyle\operatorname{erf}x<\sqrt{1-{\mathrm{e}^{-4x^{2}/% \pi}}}}} erf(x)<sqrt(1 - exp(- 4*(x)^(2)/ Pi)) Erf[x]<Sqrt[1 - Exp[- 4*(x)^(2)/ Pi]] Failure Failure Skip Successful
7.10.E1 d n + 1 erf ⁡ z d z n + 1 = ( - 1 ) n ⁢ 2 π ⁢ H n ⁡ ( z ) ⁢ e - z 2 derivative error-function 𝑧 𝑧 𝑛 1 superscript 1 𝑛 2 𝜋 Hermite-polynomial-H 𝑛 𝑧 superscript 𝑒 superscript 𝑧 2 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n+1}\operatorname{erf}z}{{% \mathrm{d}z}^{n+1}}=(-1)^{n}\frac{2}{\sqrt{\pi}}H_{n}\left(z\right)e^{-z^{2}}}} diff(erf(z), [z$(n + 1)])=(- 1)^(n)*(2)/(sqrt(Pi))*HermiteH(n, z)*exp(- (z)^(2)) D[Erf[z], {z, n + 1}]=(- 1)^(n)*Divide[2,Sqrt[Pi]]*HermiteH[n, z]*Exp[- (z)^(2)] Failure Failure Skip Successful
7.10#Ex1 d f ⁡ ( z ) d z = - π ⁢ z ⁢ g ⁡ ( z ) derivative Fresnel-auxilliary-function-f 𝑧 𝑧 𝜋 𝑧 Fresnel-auxilliary-function-g 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}\mathrm{f}\left(z\right)}{\mathrm{% d}z}=-\pi z\mathrm{g}\left(z\right)}} diff(Fresnelf(z), z)= - Pi*z*Fresnelg(z) D[FresnelF[z], z]= - Pi*z*FresnelG[z] Successful Successful - -
7.10#Ex2 d g ⁡ ( z ) d z = π ⁢ z ⁢ f ⁡ ( z ) - 1 derivative Fresnel-auxilliary-function-g 𝑧 𝑧 𝜋 𝑧 Fresnel-auxilliary-function-f 𝑧 1 {\displaystyle{\displaystyle\frac{\mathrm{d}\mathrm{g}\left(z\right)}{\mathrm{% d}z}=\pi z\mathrm{f}\left(z\right)-1}} diff(Fresnelg(z), z)= Pi*z*Fresnelf(z)- 1 D[FresnelG[z], z]= Pi*z*FresnelF[z]- 1 Successful Successful - -
7.11.E1 erf ⁡ z = 1 π ⁢ γ ⁡ ( 1 2 , z 2 ) error-function 𝑧 1 𝜋 incomplete-gamma 1 2 superscript 𝑧 2 {\displaystyle{\displaystyle\operatorname{erf}z=\frac{1}{\sqrt{\pi}}\gamma% \left(\tfrac{1}{2},z^{2}\right)}} erf(z)=(1)/(sqrt(Pi))*GAMMA((1)/(2))-GAMMA((1)/(2), (z)^(2)) Erf[z]=Divide[1,Sqrt[Pi]]*Gamma[Divide[1,2], 0, (z)^(2)] Failure Failure
Fail
-.796532174e-2+.2115950078*I <- {z = 2^(1/2)+I*2^(1/2)}
-.796532174e-2-.2115950078*I <- {z = 2^(1/2)-I*2^(1/2)}
-2.028588748+.7594465268*I <- {z = -2^(1/2)-I*2^(1/2)}
-2.028588748-.7594465268*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-2.020623424050978, 0.547851518927081] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.020623424050978, -0.547851518927081] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
7.11.E2 erfc ⁡ z = 1 π ⁢ Γ ⁡ ( 1 2 , z 2 ) complementary-error-function 𝑧 1 𝜋 incomplete-Gamma 1 2 superscript 𝑧 2 {\displaystyle{\displaystyle\operatorname{erfc}z=\frac{1}{\sqrt{\pi}}\Gamma% \left(\tfrac{1}{2},z^{2}\right)}} erfc(z)=(1)/(sqrt(Pi))*GAMMA((1)/(2), (z)^(2)) Erfc[z]=Divide[1,Sqrt[Pi]]*Gamma[Divide[1,2], (z)^(2)] Failure Failure
Fail
2.020623426-.5478515190*I <- {z = -2^(1/2)-I*2^(1/2)}
2.020623426+.5478515190*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[2.0206234240509775, -0.547851518927081] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.0206234240509775, 0.547851518927081] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
7.11.E3 erfc ⁡ z = z π ⁢ E 1 2 ⁡ ( z 2 ) complementary-error-function 𝑧 𝑧 𝜋 exponential-integral-En 1 2 superscript 𝑧 2 {\displaystyle{\displaystyle\operatorname{erfc}z=\frac{z}{\sqrt{\pi}}E_{\frac{% 1}{2}}\left(z^{2}\right)}} erfc(z)=(z)/(sqrt(Pi))*Ei((1)/(2), (z)^(2)) Erfc[z]=Divide[z,Sqrt[Pi]]*ExpIntegralE[Divide[1,2], (z)^(2)] Failure Failure
Fail
2.000000000-.1e-9*I <- {z = -2^(1/2)-I*2^(1/2)}
2.000000000+.1e-9*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[1.9999999999999996, -1.1102230246251565*^-16] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.9999999999999996, 1.1102230246251565*^-16] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
7.11.E4 erf ⁡ z = 2 ⁢ z π ⁢ M ⁡ ( 1 2 , 3 2 , - z 2 ) error-function 𝑧 2 𝑧 𝜋 Kummer-confluent-hypergeometric-M 1 2 3 2 superscript 𝑧 2 {\displaystyle{\displaystyle\operatorname{erf}z=\frac{2z}{\sqrt{\pi}}M\left(% \tfrac{1}{2},\tfrac{3}{2},-z^{2}\right)}} erf(z)=(2*z)/(sqrt(Pi))*KummerM((1)/(2), (3)/(2), - (z)^(2)) Erf[z]=Divide[2*z,Sqrt[Pi]]*Hypergeometric1F1[Divide[1,2], Divide[3,2], - (z)^(2)] Successful Successful - -
7.11.E4 2 ⁢ z π ⁢ M ⁡ ( 1 2 , 3 2 , - z 2 ) = 2 ⁢ z π ⁢ e - z 2 ⁢ M ⁡ ( 1 , 3 2 , z 2 ) 2 𝑧 𝜋 Kummer-confluent-hypergeometric-M 1 2 3 2 superscript 𝑧 2 2 𝑧 𝜋 superscript 𝑒 superscript 𝑧 2 Kummer-confluent-hypergeometric-M 1 3 2 superscript 𝑧 2 {\displaystyle{\displaystyle\frac{2z}{\sqrt{\pi}}M\left(\tfrac{1}{2},\tfrac{3}% {2},-z^{2}\right)=\frac{2z}{\sqrt{\pi}}e^{-z^{2}}M\left(1,\tfrac{3}{2},z^{2}% \right)}} (2*z)/(sqrt(Pi))*KummerM((1)/(2), (3)/(2), - (z)^(2))=(2*z)/(sqrt(Pi))*exp(- (z)^(2))*KummerM(1, (3)/(2), (z)^(2)) Divide[2*z,Sqrt[Pi]]*Hypergeometric1F1[Divide[1,2], Divide[3,2], - (z)^(2)]=Divide[2*z,Sqrt[Pi]]*Exp[- (z)^(2)]*Hypergeometric1F1[1, Divide[3,2], (z)^(2)] Successful Successful - -
7.11.E5 erfc ⁡ z = 1 π ⁢ e - z 2 ⁢ U ⁡ ( 1 2 , 1 2 , z 2 ) complementary-error-function 𝑧 1 𝜋 superscript 𝑒 superscript 𝑧 2 Kummer-confluent-hypergeometric-U 1 2 1 2 superscript 𝑧 2 {\displaystyle{\displaystyle\operatorname{erfc}z=\frac{1}{\sqrt{\pi}}e^{-z^{2}% }U\left(\tfrac{1}{2},\tfrac{1}{2},z^{2}\right)}} erfc(z)=(1)/(sqrt(Pi))*exp(- (z)^(2))*KummerU((1)/(2), (1)/(2), (z)^(2)) Erfc[z]=Divide[1,Sqrt[Pi]]*Exp[- (z)^(2)]*HypergeometricU[Divide[1,2], Divide[1,2], (z)^(2)] Failure Failure
Fail
2.020623426-.5478515190*I <- {z = -2^(1/2)-I*2^(1/2)}
2.020623426+.5478515190*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[2.0206234240509775, -0.547851518927081] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[2.0206234240509775, 0.547851518927081] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
7.11.E5 1 π ⁢ e - z 2 ⁢ U ⁡ ( 1 2 , 1 2 , z 2 ) = z π ⁢ e - z 2 ⁢ U ⁡ ( 1 , 3 2 , z 2 ) 1 𝜋 superscript 𝑒 superscript 𝑧 2 Kummer-confluent-hypergeometric-U 1 2 1 2 superscript 𝑧 2 𝑧 𝜋 superscript 𝑒 superscript 𝑧 2 Kummer-confluent-hypergeometric-U 1 3 2 superscript 𝑧 2 {\displaystyle{\displaystyle\frac{1}{\sqrt{\pi}}e^{-z^{2}}U\left(\tfrac{1}{2},% \tfrac{1}{2},z^{2}\right)=\frac{z}{\sqrt{\pi}}e^{-z^{2}}U\left(1,\tfrac{3}{2},% z^{2}\right)}} (1)/(sqrt(Pi))*exp(- (z)^(2))*KummerU((1)/(2), (1)/(2), (z)^(2))=(z)/(sqrt(Pi))*exp(- (z)^(2))*KummerU(1, (3)/(2), (z)^(2)) Divide[1,Sqrt[Pi]]*Exp[- (z)^(2)]*HypergeometricU[Divide[1,2], Divide[1,2], (z)^(2)]=Divide[z,Sqrt[Pi]]*Exp[- (z)^(2)]*HypergeometricU[1, Divide[3,2], (z)^(2)] Failure Failure
Fail
-.2062342514e-1+.5478515190*I <- {z = -2^(1/2)-I*2^(1/2)}
-.2062342514e-1-.5478515190*I <- {z = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-0.02062342405097809, 0.5478515189270807] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.020623424050978133, -0.5478515189270807] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
7.11.E6 C ⁡ ( z ) + i ⁢ S ⁡ ( z ) = z ⁢ M ⁡ ( 1 2 , 3 2 , 1 2 ⁢ π ⁢ i ⁢ z 2 ) Fresnel-cosine-integral 𝑧 𝑖 Fresnel-sine-integral 𝑧 𝑧 Kummer-confluent-hypergeometric-M 1 2 3 2 1 2 𝜋 𝑖 superscript 𝑧 2 {\displaystyle{\displaystyle C\left(z\right)+iS\left(z\right)=zM\left(\tfrac{1% }{2},\tfrac{3}{2},\tfrac{1}{2}\pi iz^{2}\right)}} FresnelC(z)+ I*FresnelS(z)= z*KummerM((1)/(2), (3)/(2), (1)/(2)*Pi*I*(z)^(2)) FresnelC[z]+ I*FresnelS[z]= z*Hypergeometric1F1[Divide[1,2], Divide[3,2], Divide[1,2]*Pi*I*(z)^(2)] Failure Successful Successful -
7.11.E6 z ⁢ M ⁡ ( 1 2 , 3 2 , 1 2 ⁢ π ⁢ i ⁢ z 2 ) = z ⁢ e π ⁢ i ⁢ z 2 / 2 ⁢ M ⁡ ( 1 , 3 2 , - 1 2 ⁢ π ⁢ i ⁢ z 2 ) 𝑧 Kummer-confluent-hypergeometric-M 1 2 3 2 1 2 𝜋 𝑖 superscript 𝑧 2 𝑧 superscript 𝑒 𝜋 𝑖 superscript 𝑧 2 2 Kummer-confluent-hypergeometric-M 1 3 2 1 2 𝜋 𝑖 superscript 𝑧 2 {\displaystyle{\displaystyle zM\left(\tfrac{1}{2},\tfrac{3}{2},\tfrac{1}{2}\pi iz% ^{2}\right)=ze^{\pi iz^{2}/2}M\left(1,\tfrac{3}{2},-\tfrac{1}{2}\pi iz^{2}% \right)}} z*KummerM((1)/(2), (3)/(2), (1)/(2)*Pi*I*(z)^(2))= z*exp(Pi*I*(z)^(2)/ 2)*KummerM(1, (3)/(2), -(1)/(2)*Pi*I*(z)^(2)) z*Hypergeometric1F1[Divide[1,2], Divide[3,2], Divide[1,2]*Pi*I*(z)^(2)]= z*Exp[Pi*I*(z)^(2)/ 2]*Hypergeometric1F1[1, Divide[3,2], -Divide[1,2]*Pi*I*(z)^(2)] Successful Successful - -
7.11.E7 C ⁡ ( z ) = z ⁢ F 2 1 ⁡ ( 1 4 ; 5 4 , 1 2 ; - 1 16 ⁢ π 2 ⁢ z 4 ) Fresnel-cosine-integral 𝑧 𝑧 Gauss-hypergeometric-pFq 1 2 1 4 5 4 1 2 1 16 superscript 𝜋 2 superscript 𝑧 4 {\displaystyle{\displaystyle C\left(z\right)=z{{}_{1}F_{2}}\left(\tfrac{1}{4};% \tfrac{5}{4},\tfrac{1}{2};-\tfrac{1}{16}\pi^{2}z^{4}\right)}} FresnelC(z)= z*hypergeom([(1)/(4)], [(5)/(4),(1)/(2)], -(1)/(16)*(Pi)^(2)* (z)^(4)) FresnelC[z]= z*HypergeometricPFQ[{Divide[1,4]}, {Divide[5,4],Divide[1,2]}, -Divide[1,16]*(Pi)^(2)* (z)^(4)] Successful Successful - -
7.11.E8 S ⁡ ( z ) = 1 6 ⁢ π ⁢ z 3 ⁢ F 2 1 ⁡ ( 3 4 ; 7 4 , 3 2 ; - 1 16 ⁢ π 2 ⁢ z 4 ) Fresnel-sine-integral 𝑧 1 6 𝜋 superscript 𝑧 3 Gauss-hypergeometric-pFq 1 2 3 4 7 4 3 2 1 16 superscript 𝜋 2 superscript 𝑧 4 {\displaystyle{\displaystyle S\left(z\right)=\tfrac{1}{6}\pi z^{3}{{}_{1}F_{2}% }\left(\tfrac{3}{4};\tfrac{7}{4},\tfrac{3}{2};-\tfrac{1}{16}\pi^{2}z^{4}\right% )}} FresnelS(z)=(1)/(6)*Pi*(z)^(3)* hypergeom([(3)/(4)], [(7)/(4),(3)/(2)], -(1)/(16)*(Pi)^(2)* (z)^(4)) FresnelS[z]=Divide[1,6]*Pi*(z)^(3)* HypergeometricPFQ[{Divide[3,4]}, {Divide[7,4],Divide[3,2]}, -Divide[1,16]*(Pi)^(2)* (z)^(4)] Successful Successful - -
7.13#Ex4 μ = ln ⁡ ( λ ⁢ 2 ⁢ π ) 𝜇 𝜆 2 𝜋 {\displaystyle{\displaystyle\mu=\ln\left(\lambda\sqrt{2\pi}\right)}} mu = ln(lambda*sqrt(2*Pi)) \[Mu]= Log[\[Lambda]*Sqrt[2*Pi]] Failure Failure
Fail
-.197872151+.6288153986*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2)}
-.197872151-2.199611725*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)-I*2^(1/2)}
-3.026299275-2.199611725*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = -2^(1/2)-I*2^(1/2)}
-3.026299275+.6288153986*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.1978721513915227, 0.6288153989756469] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.1978721513915227, -2.199611725770543] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-3.026299276137713, -2.199611725770543] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-3.026299276137713, 0.6288153989756469] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
7.13#Ex8 μ = ln ⁡ ( 2 ⁢ λ ⁢ 2 ⁢ π ) 𝜇 2 𝜆 2 𝜋 {\displaystyle{\displaystyle\mu=\ln\left(2\lambda\sqrt{2\pi}\right)}} mu = ln(2*lambda*sqrt(2*Pi)) \[Mu]= Log[2*\[Lambda]*Sqrt[2*Pi]] Failure Failure
Fail
-.891019332+.6288153986*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2)}
-.891019332-2.199611725*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)-I*2^(1/2)}
-3.719446456-2.199611725*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = -2^(1/2)-I*2^(1/2)}
-3.719446456+.6288153986*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.8910193319514683, 0.6288153989756469] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.8910193319514683, -2.199611725770543] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-3.719446456697659, -2.199611725770543] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-3.719446456697659, 0.6288153989756469] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
7.13#Ex12 α = ( 2 / π ) ⁢ ln ⁡ ( π ⁢ λ ) 𝛼 2 𝜋 𝜋 𝜆 {\displaystyle{\displaystyle\alpha=(2/\pi)\ln\left(\pi\lambda\right)}} alpha =(2/ Pi)* ln(Pi*lambda) \[Alpha]=(2/ Pi)* Log[Pi*\[Lambda]] Failure Failure
Fail
.244184683+.9142135621*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2)}
.244184683+1.914213562*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)-I*2^(1/2)}
.244184683+2.914213561*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = -2^(1/2)-I*2^(1/2)}
.244184683-.85786437e-1*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[0.24418468271597948, 0.9142135623730951] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.24418468271597948, 1.9142135623730951] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.24418468271597948, 2.914213562373095] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.24418468271597948, -0.08578643762690485] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
7.13#Ex14 α = ( 2 / π ) ⁢ ln ⁡ ( π ⁢ λ ) 𝛼 2 𝜋 𝜋 𝜆 {\displaystyle{\displaystyle\alpha=(2/\pi)\ln\left(\pi\lambda\right)}} alpha =(2/ Pi)* ln(Pi*lambda) \[Alpha]=(2/ Pi)* Log[Pi*\[Lambda]] Failure Failure
Fail
.244184683+.9142135621*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2)}
.244184683+1.914213562*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)-I*2^(1/2)}
.244184683+2.914213561*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = -2^(1/2)-I*2^(1/2)}
.244184683-.85786437e-1*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[0.24418468271597948, 0.9142135623730951] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.24418468271597948, 1.9142135623730951] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.24418468271597948, 2.914213562373095] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.24418468271597948, -0.08578643762690485] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
7.14.E1 ∫ 0 ∞ e 2 ⁢ i ⁢ a ⁢ t ⁢ erfc ⁡ ( b ⁢ t ) ⁢ d t = 1 a ⁢ π ⁢ F ⁡ ( a b ) + i 2 ⁢ a ⁢ ( 1 - e - ( a / b ) 2 ) superscript subscript 0 superscript 𝑒 2 𝑖 𝑎 𝑡 complementary-error-function 𝑏 𝑡 𝑡 1 𝑎 𝜋 Dawsons-integral 𝑎 𝑏 𝑖 2 𝑎 1 superscript 𝑒 superscript 𝑎 𝑏 2 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{2iat}\operatorname{erfc}\left(% bt\right)\mathrm{d}t={\frac{1}{a\sqrt{\pi}}F\left(\frac{a}{b}\right)+\frac{i}{% 2a}\left(1-e^{-(a/b)^{2}}\right)}}} int(exp(2*I*a*t)*erfc(b*t), t = 0..infinity)=(1)/(a*sqrt(Pi))*dawson((a)/(b))+(I)/(2*a)*(1 - exp(-(a/ b)^(2))) Integrate[Exp[2*I*a*t]*Erfc[b*t], {t, 0, Infinity}]=Divide[1,a*Sqrt[Pi]]*DawsonF[Divide[a,b]]+Divide[I,2*a]*(1 - Exp[-(a/ b)^(2)]) Failure Failure Skip Error
7.14.E2 ∫ 0 ∞ e - a ⁢ t ⁢ erf ⁡ ( b ⁢ t ) ⁢ d t = 1 a ⁢ e a 2 / ( 4 ⁢ b 2 ) ⁢ erfc ⁡ ( a 2 ⁢ b ) superscript subscript 0 superscript 𝑒 𝑎 𝑡 error-function 𝑏 𝑡 𝑡 1 𝑎 superscript 𝑒 superscript 𝑎 2 4 superscript 𝑏 2 complementary-error-function 𝑎 2 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}\operatorname{erf}\left(bt% \right)\mathrm{d}t=\frac{1}{a}e^{a^{2}/(4b^{2})}\operatorname{erfc}\left(\frac% {a}{2b}\right)}} int(exp(- a*t)*erf(b*t), t = 0..infinity)=(1)/(a)*exp((a)^(2)/(4*(b)^(2)))*erfc((a)/(2*b)) Integrate[Exp[- a*t]*Erf[b*t], {t, 0, Infinity}]=Divide[1,a]*Exp[(a)^(2)/(4*(b)^(2))]*Erfc[Divide[a,2*b]] Successful Failure - Error
7.14.E4 ∫ 0 ∞ e ( a - b ) ⁢ t ⁢ erfc ⁡ ( a ⁢ t + c t ) ⁢ d t = e - 2 ⁢ ( a ⁢ c + b ⁢ c ) b ⁢ ( a + b ) superscript subscript 0 superscript 𝑒 𝑎 𝑏 𝑡 complementary-error-function 𝑎 𝑡 𝑐 𝑡 𝑡 superscript 𝑒 2 𝑎 𝑐 𝑏 𝑐 𝑏 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{(a-b)t}\operatorname{erfc}% \left(\sqrt{at}+\sqrt{\frac{c}{t}}\right)\mathrm{d}t=\frac{e^{-2(\sqrt{ac}+% \sqrt{bc})}}{\sqrt{b}(\sqrt{a}+\sqrt{b})}}} int(exp((a - b)* t)*erfc(sqrt(a*t)+sqrt((c)/(t))), t = 0..infinity)=(exp(- 2*(sqrt(a*c)+sqrt(b*c))))/(sqrt(b)*(sqrt(a)+sqrt(b))) Integrate[Exp[(a - b)* t]*Erfc[Sqrt[a*t]+Sqrt[Divide[c,t]]], {t, 0, Infinity}]=Divide[Exp[- 2*(Sqrt[a*c]+Sqrt[b*c])],Sqrt[b]*(Sqrt[a]+Sqrt[b])] Failure Failure Skip Error
7.14.E5 ∫ 0 ∞ e - a ⁢ t ⁢ C ⁡ ( t ) ⁢ d t = 1 a ⁢ f ⁡ ( a π ) superscript subscript 0 superscript 𝑒 𝑎 𝑡 Fresnel-cosine-integral 𝑡 𝑡 1 𝑎 Fresnel-auxilliary-function-f 𝑎 𝜋 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}C\left(t\right)\mathrm{d}t% =\frac{1}{a}\mathrm{f}\left(\frac{a}{\pi}\right)}} int(exp(- a*t)*FresnelC(t), t = 0..infinity)=(1)/(a)*Fresnelf((a)/(Pi)) Integrate[Exp[- a*t]*FresnelC[t], {t, 0, Infinity}]=Divide[1,a]*FresnelF[Divide[a,Pi]] Failure Failure Skip Successful
7.14.E6 ∫ 0 ∞ e - a ⁢ t ⁢ S ⁡ ( t ) ⁢ d t = 1 a ⁢ g ⁡ ( a π ) superscript subscript 0 superscript 𝑒 𝑎 𝑡 Fresnel-sine-integral 𝑡 𝑡 1 𝑎 Fresnel-auxilliary-function-g 𝑎 𝜋 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}S\left(t\right)\mathrm{d}t% =\frac{1}{a}\mathrm{g}\left(\frac{a}{\pi}\right)}} int(exp(- a*t)*FresnelS(t), t = 0..infinity)=(1)/(a)*Fresnelg((a)/(Pi)) Integrate[Exp[- a*t]*FresnelS[t], {t, 0, Infinity}]=Divide[1,a]*FresnelG[Divide[a,Pi]] Failure Failure Skip Successful
7.17#Ex1 y = inverf ⁡ x 𝑦 inverse-error-function 𝑥 {\displaystyle{\displaystyle y=\operatorname{inverf}x}} Error y = InverseErf[x] Error Failure -
Fail
DirectedInfinity[-1] <- {Rule[x, 1], Rule[y, 1]}
DirectedInfinity[-1] <- {Rule[x, 1], Rule[y, 2]}
DirectedInfinity[-1] <- {Rule[x, 1], Rule[y, 3]}
7.17#Ex2 y = inverfc ⁡ x 𝑦 inverse-complementary-error-function 𝑥 {\displaystyle{\displaystyle y=\operatorname{inverfc}x}} Error y = InverseErfc[x] Error Failure -
Fail
1.0 <- {Rule[x, 1], Rule[y, 1]}
2.0 <- {Rule[x, 1], Rule[y, 2]}
3.0 <- {Rule[x, 1], Rule[y, 3]}
DirectedInfinity[1] <- {Rule[x, 2], Rule[y, 1]}
... skip entries to safe data
7.18#Ex1 i - 1 ⁢ erfc ⁡ ( z ) = 2 π ⁢ e - z 2 repeated-integral-complementary-error-function 1 𝑧 2 𝜋 superscript 𝑒 superscript 𝑧 2 {\displaystyle{\displaystyle\mathop{\mathrm{i}^{-1}\mathrm{erfc}}\left(z\right% )=\frac{2}{\sqrt{\pi}}e^{-z^{2}}}} erfc(- 1, z)=(2)/(sqrt(Pi))*exp(- (z)^(2)) I^(- 1)*Erfc[z]=Divide[2,Sqrt[Pi]]*Exp[- (z)^(2)] Successful Failure -
Fail
Complex[1.011483603950918, -0.8436484572858769] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.46363208502383696, 0.8642718813368542] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.46363208502383696, -2.864271881336854] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.011483603950918, -1.1563515427141229] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
7.18#Ex2 i 0 ⁢ erfc ⁡ ( z ) = erfc ⁡ z repeated-integral-complementary-error-function 0 𝑧 complementary-error-function 𝑧 {\displaystyle{\displaystyle\mathop{\mathrm{i}^{0}\mathrm{erfc}}\left(z\right)% =\operatorname{erfc}z}} erfc(0, z)= erfc(z) I^(0)*Erfc[z]= Erfc[z] Successful Successful - -
7.18.E2 i n ⁢ erfc ⁡ ( z ) = ∫ z ∞ i n - 1 ⁢ erfc ⁡ ( t ) ⁢ d t repeated-integral-complementary-error-function 𝑛 𝑧 superscript subscript 𝑧 repeated-integral-complementary-error-function 𝑛 1 𝑡 𝑡 {\displaystyle{\displaystyle\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)% =\int_{z}^{\infty}\mathop{\mathrm{i}^{n-1}\mathrm{erfc}}\left(t\right)\mathrm{% d}t}} erfc(n, z)= int(erfc(n - 1, t), t = z..infinity) I^(n)*Erfc[z]= Integrate[I^(n - 1)*Erfc[t], {t, z, Infinity}] Failure Failure Skip
Fail
Complex[-0.30711932433427286, -0.06448523556221403] <- {Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.06448523556221401, -0.30711932433427286] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.30711932433427286, 0.06448523556221403] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.2407321945928082, 0.04386181151123664] <- {Rule[n, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
7.18.E2 ∫ z ∞ i n - 1 ⁢ erfc ⁡ ( t ) ⁢ d t = 2 π ⁢ ∫ z ∞ ( t - z ) n n ! ⁢ e - t 2 ⁢ d t superscript subscript 𝑧 repeated-integral-complementary-error-function 𝑛 1 𝑡 𝑡 2 𝜋 superscript subscript 𝑧 superscript 𝑡 𝑧 𝑛 𝑛 superscript 𝑒 superscript 𝑡 2 𝑡 {\displaystyle{\displaystyle\int_{z}^{\infty}\mathop{\mathrm{i}^{n-1}\mathrm{% erfc}}\left(t\right)\mathrm{d}t=\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}\frac{(t-% z)^{n}}{n!}e^{-t^{2}}\mathrm{d}t}} int(erfc(n - 1, t), t = z..infinity)=(2)/(sqrt(Pi))*int(((t - z)^(n))/(factorial(n))*exp(- (t)^(2)), t = z..infinity) Integrate[I^(n - 1)*Erfc[t], {t, z, Infinity}]=Divide[2,Sqrt[Pi]]*Integrate[Divide[(t - z)^(n),(n)!]*Exp[- (t)^(2)], {t, z, Infinity}] Failure Failure Skip
Fail
Complex[-0.06643066657209085, 0.02648998567028575] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.036107850584238765, -0.054264273946754926] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.04191638050136022, 0.039897144071178225] <- {Rule[n, 2], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.03610785058423853, 0.054264273946755606] <- {Rule[n, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
7.18.E3 d d z ⁡ i n ⁢ erfc ⁡ ( z ) = - i n - 1 ⁢ erfc ⁡ ( z ) derivative 𝑧 repeated-integral-complementary-error-function 𝑛 𝑧 repeated-integral-complementary-error-function 𝑛 1 𝑧 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{\mathrm{i}^{% n}\mathrm{erfc}}\left(z\right)=-\mathop{\mathrm{i}^{n-1}\mathrm{erfc}}\left(z% \right)}} diff(erfc(n, z), z)= - erfc(n - 1, z) D[I^(n)*Erfc[z], z]= - I^(n - 1)*Erfc[z] Successful Failure -
Fail
Complex[-0.8436484572858769, -1.011483603950918] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.8642718813368542, -0.46363208502383696] <- {Rule[n, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.864271881336854, -0.46363208502383696] <- {Rule[n, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-1.1563515427141229, -1.011483603950918] <- {Rule[n, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
7.18.E4 d n d z n ⁡ ( e z 2 ⁢ erfc ⁡ z ) = ( - 1 ) n ⁢ 2 n ⁢ n ! ⁢ e z 2 ⁢ i n ⁢ erfc ⁡ ( z ) derivative 𝑧 𝑛 superscript 𝑒 superscript 𝑧 2 complementary-error-function 𝑧 superscript 1 𝑛 superscript 2 𝑛 𝑛 superscript 𝑒 superscript 𝑧 2 repeated-integral-complementary-error-function 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^% {z^{2}}\operatorname{erfc}z\right)=(-1)^{n}2^{n}n!e^{z^{2}}\mathop{\mathrm{i}^% {n}\mathrm{erfc}}\left(z\right)}} diff(exp((z)^(2))*erfc(z), [z$(n)])=(- 1)^(n)* (2)^(n)* factorial(n)*exp((z)^(2))*erfc(n, z) D[Exp[(z)^(2)]*Erfc[z], {z, n}]=(- 1)^(n)* (2)^(n)* (n)!*Exp[(z)^(2)]*I^(n)*Erfc[z] Failure Failure Successful
Fail
Complex[-8.131664243641417, -10.165585245606788] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[8.307941760049161, -10.383011529763138] <- {Rule[n, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-195.50543578827103, 111.66805229196896] <- {Rule[n, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-66.63896269283943, 34.38011968921443] <- {Rule[n, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
7.18.E5 d 2 W d z 2 + 2 ⁢ z ⁢ d W d z - 2 ⁢ n ⁢ W = 0 derivative 𝑊 𝑧 2 2 𝑧 derivative 𝑊 𝑧 2 𝑛 𝑊 0 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}z}^{2}}+2z% \frac{\mathrm{d}W}{\mathrm{d}z}-2nW=0}} diff(W, [z$(2)])+ 2*z*diff(W, z)- 2*n*W = 0 D[W, {z, 2}]+ 2*z*D[W, z]- 2*n*W = 0 Failure Failure Skip Successful
7.18.E6 i n ⁢ erfc ⁡ ( z ) = ∑ k = 0 ∞ ( - 1 ) k ⁢ z k 2 n - k ⁢ k ! ⁢ Γ ⁡ ( 1 + 1 2 ⁢ ( n - k ) ) repeated-integral-complementary-error-function 𝑛 𝑧 superscript subscript 𝑘 0 superscript 1 𝑘 superscript 𝑧 𝑘 superscript 2 𝑛 𝑘 𝑘 Euler-Gamma 1 1 2 𝑛 𝑘 {\displaystyle{\displaystyle\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)% =\sum_{k=0}^{\infty}\frac{(-1)^{k}z^{k}}{2^{n-k}k!\Gamma\left(1+\frac{1}{2}(n-% k)\right)}}} erfc(n, z)= sum(((- 1)^(k)* (z)^(k))/((2)^(n - k)* factorial(k)*GAMMA(1 +(1)/(2)*(n - k))), k = 0..infinity) I^(n)*Erfc[z]= Sum[Divide[(- 1)^(k)* (z)^(k),(2)^(n - k)* (k)!*Gamma[1 +Divide[1,2]*(n - k)]], {k, 0, Infinity}] Failure Failure Skip
Fail
Complex[-0.3071193243342728, -0.06448523556221496] <- {Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.0019454310098768, -0.280629338663987] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.2710114737500341, 0.01022096161545949] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.24073219459280773, 0.043861811511237594] <- {Rule[n, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
7.18.E7 i n ⁢ erfc ⁡ ( z ) = - z n ⁢ i n - 1 ⁢ erfc ⁡ ( z ) + 1 2 ⁢ n ⁢ i n - 2 ⁢ erfc ⁡ ( z ) repeated-integral-complementary-error-function 𝑛 𝑧 𝑧 𝑛 repeated-integral-complementary-error-function 𝑛 1 𝑧 1 2 𝑛 repeated-integral-complementary-error-function 𝑛 2 𝑧 {\displaystyle{\displaystyle\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)% =-\frac{z}{n}\mathop{\mathrm{i}^{n-1}\mathrm{erfc}}\left(z\right)+\frac{1}{2n}% \mathop{\mathrm{i}^{n-2}\mathrm{erfc}}\left(z\right)}} erfc(n, z)= -(z)/(n)*erfc(n - 1, z)+(1)/(2*n)*erfc(n - 2, z) I^(n)*Erfc[z]= -Divide[z,n]*I^(n - 1)*Erfc[z]+Divide[1,2*n]*I^(n - 2)*Erfc[z] Successful Failure -
Fail
Complex[0.45357088174560434, -0.11223852300991567] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.18558922366932362, 0.1362991844027226] <- {Rule[n, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.7572198179127398, -1.5268234761539925] <- {Rule[n, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.3963799233276677, -3.163903851905481] <- {Rule[n, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
7.18.E8 ( - 1 ) n ⁢ i n ⁢ erfc ⁡ ( z ) + i n ⁢ erfc ⁡ ( - z ) = i - n 2 n - 1 ⁢ n ! ⁢ H n ⁡ ( i ⁢ z ) superscript 1 𝑛 repeated-integral-complementary-error-function 𝑛 𝑧 repeated-integral-complementary-error-function 𝑛 𝑧 superscript 𝑖 𝑛 superscript 2 𝑛 1 𝑛 Hermite-polynomial-H 𝑛 𝑖 𝑧 {\displaystyle{\displaystyle(-1)^{n}\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(% z\right)+\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(-z\right)=\frac{i^{-n}}{2^{% n-1}n!}H_{n}\left(iz\right)}} (- 1)^(n)* erfc(n, z)+ erfc(n, - z)=((I)^(- n))/((2)^(n - 1)* factorial(n))*HermiteH(n, I*z) (- 1)^(n)* I^(n)*Erfc[z]+ I^(n)*Erfc[- z]=Divide[(I)^(- n),(2)^(n - 1)* (n)!]*HermiteH[n, I*z] Failure Failure Successful
Fail
Complex[-2.280575605819109, -0.8078037006952131] <- {Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-2.5, -4.0] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.6306597830504983, -4.613348288401651] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-3.3762786436732717, 4.849050548797168] <- {Rule[n, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
7.18.E9 i n ⁢ erfc ⁡ ( z ) = e - z 2 ⁢ ( 1 2 n ⁢ Γ ⁡ ( 1 2 ⁢ n + 1 ) ⁢ M ⁡ ( 1 2 ⁢ n + 1 2 , 1 2 , z 2 ) - z 2 n - 1 ⁢ Γ ⁡ ( 1 2 ⁢ n + 1 2 ) ⁢ M ⁡ ( 1 2 ⁢ n + 1 , 3 2 , z 2 ) ) repeated-integral-complementary-error-function 𝑛 𝑧 superscript 𝑒 superscript 𝑧 2 1 superscript 2 𝑛 Euler-Gamma 1 2 𝑛 1 Kummer-confluent-hypergeometric-M 1 2 𝑛 1 2 1 2 superscript 𝑧 2 𝑧 superscript 2 𝑛 1 Euler-Gamma 1 2 𝑛 1 2 Kummer-confluent-hypergeometric-M 1 2 𝑛 1 3 2 superscript 𝑧 2 {\displaystyle{\displaystyle\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)% =e^{-z^{2}}\left(\frac{1}{2^{n}\Gamma\left(\tfrac{1}{2}n+1\right)}M\left(% \tfrac{1}{2}n+\tfrac{1}{2},\tfrac{1}{2},z^{2}\right)-\frac{z}{2^{n-1}\Gamma% \left(\tfrac{1}{2}n+\tfrac{1}{2}\right)}M\left(\tfrac{1}{2}n+1,\tfrac{3}{2},z^% {2}\right)\right)}} erfc(n, z)= exp(- (z)^(2))*((1)/((2)^(n)* GAMMA((1)/(2)*n + 1))*KummerM((1)/(2)*n +(1)/(2), (1)/(2), (z)^(2))-(z)/((2)^(n - 1)* GAMMA((1)/(2)*n +(1)/(2)))*KummerM((1)/(2)*n + 1, (3)/(2), (z)^(2))) I^(n)*Erfc[z]= Exp[- (z)^(2)]*(Divide[1,(2)^(n)* Gamma[Divide[1,2]*n + 1]]*Hypergeometric1F1[Divide[1,2]*n +Divide[1,2], Divide[1,2], (z)^(2)]-Divide[z,(2)^(n - 1)* Gamma[Divide[1,2]*n +Divide[1,2]]]*Hypergeometric1F1[Divide[1,2]*n + 1, Divide[3,2], (z)^(2)]) Failure Failure Successful
Fail
Complex[-0.3071193243342726, -0.06448523556221446] <- {Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.0019454310098766699, -0.28062933866398737] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.27101147375003376, 0.010220961615459446] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.2407321945928081, 0.04386181151123732] <- {Rule[n, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
7.18.E10 i n ⁢ erfc ⁡ ( z ) = e - z 2 2 n ⁢ π ⁢ U ⁡ ( 1 2 ⁢ n + 1 2 , 1 2 , z 2 ) repeated-integral-complementary-error-function 𝑛 𝑧 superscript 𝑒 superscript 𝑧 2 superscript 2 𝑛 𝜋 Kummer-confluent-hypergeometric-U 1 2 𝑛 1 2 1 2 superscript 𝑧 2 {\displaystyle{\displaystyle\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)% =\frac{e^{-z^{2}}}{2^{n}\sqrt{\pi}}U\left(\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{1}% {2},z^{2}\right)}} erfc(n, z)=(exp(- (z)^(2)))/((2)^(n)*sqrt(Pi))*KummerU((1)/(2)*n +(1)/(2), (1)/(2), (z)^(2)) I^(n)*Erfc[z]=Divide[Exp[- (z)^(2)],(2)^(n)*Sqrt[Pi]]*HypergeometricU[Divide[1,2]*n +Divide[1,2], Divide[1,2], (z)^(2)] Failure Failure
Fail
2.828427124+2.828427124*I <- {z = -2^(1/2)-I*2^(1/2), n = 1}
.4754857140+3.986592840*I <- {z = -2^(1/2)-I*2^(1/2), n = 2}
-1.178511301+2.592724863*I <- {z = -2^(1/2)-I*2^(1/2), n = 3}
2.828427124-2.828427124*I <- {z = -2^(1/2)+I*2^(1/2), n = 1}
... skip entries to safe data
Fail
Complex[-0.30711932433427297, -0.06448523556221639] <- {Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.0019454310098774158, -0.28062933866398587] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.2710114737500343, 0.010220961615459385] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.24073219459280773, 0.04386181151123868] <- {Rule[n, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
7.18.E11 i n ⁢ erfc ⁡ ( z ) = e - z 2 / 2 2 n - 1 ⁢ π ⁢ U ⁡ ( n + 1 2 , z ⁢ 2 ) repeated-integral-complementary-error-function 𝑛 𝑧 superscript 𝑒 superscript 𝑧 2 2 superscript 2 𝑛 1 𝜋 parabolic-U 𝑛 1 2 𝑧 2 {\displaystyle{\displaystyle\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)% =\frac{e^{-z^{2}/2}}{\sqrt{2^{n-1}\pi}}U\left(n+\tfrac{1}{2},z\sqrt{2}\right)}} erfc(n, z)=(exp(- (z)^(2)/ 2))/(sqrt((2)^(n - 1)* Pi))*CylinderU(n +(1)/(2), z*sqrt(2)) I^(n)*Erfc[z]=Divide[Exp[- (z)^(2)/ 2],Sqrt[(2)^(n - 1)* Pi]]*ParabolicCylinderD[-n +Divide[1,2] - 1/2, z*Sqrt[2]] Failure Failure Successful
Fail
Complex[-0.26663427796467404, -0.20400647408383285] <- {Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.013159682786361896, -0.3122322253171296] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.2652586505052597, 0.005571565714632675] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.281217240962407, 0.18338305003285552] <- {Rule[n, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
7.20.E1 1 σ ⁢ 2 ⁢ π ⁢ ∫ - ∞ x e - ( t - m ) 2 / ( 2 ⁢ σ 2 ) ⁢ d t = 1 2 ⁢ erfc ⁡ ( m - x σ ⁢ 2 ) 1 𝜎 2 𝜋 superscript subscript 𝑥 superscript 𝑒 superscript 𝑡 𝑚 2 2 superscript 𝜎 2 𝑡 1 2 complementary-error-function 𝑚 𝑥 𝜎 2 {\displaystyle{\displaystyle\frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{x}e^{-(% t-m)^{2}/(2\sigma^{2})}\mathrm{d}t=\frac{1}{2}\operatorname{erfc}\left(\frac{m% -x}{\sigma\sqrt{2}}\right)}} (1)/(sigma*sqrt(2*Pi))*int(exp(-(t - m)^(2)/(2*(sigma)^(2))), t = - infinity..x)=(1)/(2)*erfc((m - x)/(sigma*sqrt(2))) Divide[1,\[Sigma]*Sqrt[2*Pi]]*Integrate[Exp[-(t - m)^(2)/(2*(\[Sigma])^(2))], {t, - Infinity, x}]=Divide[1,2]*Erfc[Divide[m - x,\[Sigma]*Sqrt[2]]] Failure Failure Skip Successful
7.20.E1 1 2 ⁢ erfc ⁡ ( m - x σ ⁢ 2 ) = Q ⁢ ( m - x σ ) 1 2 complementary-error-function 𝑚 𝑥 𝜎 2 𝑄 𝑚 𝑥 𝜎 {\displaystyle{\displaystyle\frac{1}{2}\operatorname{erfc}\left(\frac{m-x}{% \sigma\sqrt{2}}\right)=Q\left(\frac{m-x}{\sigma}\right)}} (1)/(2)*erfc((m - x)/(sigma*sqrt(2)))= Q*((m - x)/(sigma)) Divide[1,2]*Erfc[Divide[m - x,\[Sigma]*Sqrt[2]]]= Q*(Divide[m - x,\[Sigma]]) Failure Failure
Fail
.5000000000 <- {Q = 2^(1/2)+I*2^(1/2), sigma = 2^(1/2)+I*2^(1/2), m = 1, x = 1}
1.646697588-.1349567498*I <- {Q = 2^(1/2)+I*2^(1/2), sigma = 2^(1/2)+I*2^(1/2), m = 1, x = 2}
2.821306458-.2289406972*I <- {Q = 2^(1/2)+I*2^(1/2), sigma = 2^(1/2)+I*2^(1/2), m = 1, x = 3}
-.6466975876+.1349567498*I <- {Q = 2^(1/2)+I*2^(1/2), sigma = 2^(1/2)+I*2^(1/2), m = 2, x = 1}
... skip entries to safe data
Fail
0.5 <- {Rule[m, 1], Rule[Q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[σ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[1.6466975876615073, -0.13495674973157074] <- {Rule[m, 1], Rule[Q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2], Rule[σ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[2.8213064574274105, -0.22894069721759613] <- {Rule[m, 1], Rule[Q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3], Rule[σ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-0.6466975876615073, 0.13495674973157074] <- {Rule[m, 2], Rule[Q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[σ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
7.20.E1 Q ⁢ ( m - x σ ) = P ⁢ ( x - m σ ) 𝑄 𝑚 𝑥 𝜎 𝑃 𝑥 𝑚 𝜎 {\displaystyle{\displaystyle Q\left(\frac{m-x}{\sigma}\right)=P\left(\frac{x-m% }{\sigma}\right)}} Q*((m - x)/(sigma))= P*((x - m)/(sigma)) Q*(Divide[m - x,\[Sigma]])= P*(Divide[x - m,\[Sigma]]) Failure Failure Skip Skip
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