Results of Gamma Function
| DLMF | Formula | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|
| 5.2.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{z} = \int_{0}^{\infty}e^{-t}t^{z-1}\diff{t}} | GAMMA(z)= int(exp(- t)*(t)^(z - 1), t = 0..infinity) |
Gamma[z]= Integrate[Exp[- t]*(t)^(z - 1), {t, 0, Infinity}] |
Successful | Successful | - | - |
| 5.2.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{z} = \EulerGamma'@{z}/\EulerGamma@{z}} | Psi(z)= subs( temp=z, diff( GAMMA(temp), temp$(1) ) )/ GAMMA(z) |
PolyGamma[z]= (D[Gamma[temp], {temp, 1}]/.temp-> z)/ Gamma[z] |
Successful | Successful | - | - |
| 5.2#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Pochhammersym{a}{0} = 1} | pochhammer(a, 0)= 1 |
Pochhammer[a, 0]= 1 |
Successful | Successful | - | - |
| 5.2.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Pochhammersym{a}{n} = \EulerGamma@{a+n}/\EulerGamma@{a}} | pochhammer(a, n)= GAMMA(a + n)/ GAMMA(a) |
Pochhammer[a, n]= Gamma[a + n]/ Gamma[a] |
Successful | Successful | - | - |
| 5.2.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Pochhammersym{-a}{n} = (-1)^{n}\Pochhammersym{a-n+1}{n}} | pochhammer(- a, n)=(- 1)^(n)* pochhammer(a - n + 1, n) |
Pochhammer[- a, n]=(- 1)^(n)* Pochhammer[a - n + 1, n] |
Failure | Failure | Successful | Successful |
| 5.2.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Pochhammersym{-m}{n} = \begin{cases}\frac{(-1)^{n}m!}{(m-n)!},&0} | pochhammer(- m, n)= |
Pochhammer[- m, n]= |
Error | Failure | - | - |
| 5.2#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Pochhammersym{a}{2n} = 2^{2n}\Pochhammersym{\frac{a}{2}}{n}\Pochhammersym{\frac{a+1}{2}}{n}} | pochhammer(a, 2*n)= (2)^(2*n)* pochhammer((a)/(2), n)*pochhammer((a + 1)/(2), n) |
Pochhammer[a, 2*n]= (2)^(2*n)* Pochhammer[Divide[a,2], n]*Pochhammer[Divide[a + 1,2], n] |
Successful | Successful | - | - |
| 5.2#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Pochhammersym{a}{2n+1} = 2^{2n+1}\Pochhammersym{\frac{a}{2}}{n+1}\Pochhammersym{\frac{a+1}{2}}{n}} | pochhammer(a, 2*n + 1)= (2)^(2*n + 1)* pochhammer((a)/(2), n + 1)*pochhammer((a + 1)/(2), n) |
Pochhammer[a, 2*n + 1]= (2)^(2*n + 1)* Pochhammer[Divide[a,2], n + 1]*Pochhammer[Divide[a + 1,2], n] |
Successful | Successful | - | - |
| 5.4#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{1} = 1} | GAMMA(1)= 1 |
Gamma[1]= 1 |
Successful | Successful | - | - |
| 5.4#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n! = \EulerGamma@{n+1}} | factorial(n)= GAMMA(n + 1) |
(n)!= Gamma[n + 1] |
Successful | Successful | - | - |
| 5.4.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\EulerGamma@{iy}| = \left(\frac{\pi}{y\sinh@{\pi y}}\right)^{1/2}} | abs(GAMMA(I*y))=((Pi)/(y*sinh(Pi*y)))^(1/ 2) |
Abs[Gamma[I*y]]=(Divide[Pi,y*Sinh[Pi*y]])^(1/ 2) |
Failure | Failure | Successful | Successful |
| 5.4.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{\tfrac{1}{2}+\iunit y}\EulerGamma@{\tfrac{1}{2}-\iunit y} = \left|\EulerGamma@{\tfrac{1}{2}+\iunit y}\right|^{2}} | GAMMA((1)/(2)+ I*y)*GAMMA((1)/(2)- I*y)=(abs(GAMMA((1)/(2)+ I*y)))^(2) |
Gamma[Divide[1,2]+ I*y]*Gamma[Divide[1,2]- I*y]=(Abs[Gamma[Divide[1,2]+ I*y]])^(2) |
Failure | Failure | Successful | Successful |
| 5.4.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \left|\EulerGamma@{\tfrac{1}{2}+\iunit y}\right|^{2} = \frac{\pi}{\cosh@{\pi y}}} | (abs(GAMMA((1)/(2)+ I*y)))^(2)=(Pi)/(cosh(Pi*y)) |
(Abs[Gamma[Divide[1,2]+ I*y]])^(2)=Divide[Pi,Cosh[Pi*y]] |
Failure | Failure | Successful | Successful |
| 5.4.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{\tfrac{1}{4}+\iunit y}\EulerGamma@{\tfrac{3}{4}-\iunit y} = \frac{\pi\sqrt{2}}{\cosh@{\pi y}+\iunit\sinh@{\pi y}}} | GAMMA((1)/(4)+ I*y)*GAMMA((3)/(4)- I*y)=(Pi*sqrt(2))/(cosh(Pi*y)+ I*sinh(Pi*y)) |
Gamma[Divide[1,4]+ I*y]*Gamma[Divide[3,4]- I*y]=Divide[Pi*Sqrt[2],Cosh[Pi*y]+ I*Sinh[Pi*y]] |
Failure | Successful | Successful | - |
| 5.4.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma'@{1} = -\EulerConstant} | subs( temp=1, diff( GAMMA(temp), temp$(1) ) )= - gamma |
(D[Gamma[temp], {temp, 1}]/.temp-> 1)= - EulerGamma |
Successful | Successful | - | - |
| 5.4#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{1} = -\EulerConstant} | Psi(1)= - gamma |
PolyGamma[1]= - EulerGamma |
Successful | Successful | - | - |
| 5.4#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma'@{1} = \tfrac{1}{6}\pi^{2}} | subs( temp=1, diff( Psi(temp), temp$(1) ) )=(1)/(6)*(Pi)^(2) |
(D[PolyGamma[temp], {temp, 1}]/.temp-> 1)=Divide[1,6]*(Pi)^(2) |
Successful | Successful | - | - |
| 5.4#Ex5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{\tfrac{1}{2}} = -\EulerConstant-2\ln@@{2}} | Psi((1)/(2))= - gamma - 2*ln(2) |
PolyGamma[Divide[1,2]]= - EulerGamma - 2*Log[2] |
Successful | Successful | - | - |
| 5.4#Ex6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma'@{\tfrac{1}{2}} = \tfrac{1}{2}\pi^{2}} | subs( temp=(1)/(2), diff( Psi(temp), temp$(1) ) )=(1)/(2)*(Pi)^(2) |
(D[PolyGamma[temp], {temp, 1}]/.temp-> Divide[1,2])=Divide[1,2]*(Pi)^(2) |
Successful | Successful | - | - |
| 5.4.E14 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{n+1} = \sum_{k=1}^{n}\frac{1}{k}-\EulerConstant} | Psi(n + 1)= sum((1)/(k), k = 1..n)- gamma |
PolyGamma[n + 1]= Sum[Divide[1,k], {k, 1, n}]- EulerGamma |
Successful | Successful | - | - |
| 5.4.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \imagpart@@{\digamma@{iy}} = \frac{1}{2y}+\frac{\pi}{2}\coth@{\pi y}} | Im(Psi(I*y))=(1)/(2*y)+(Pi)/(2)*coth(Pi*y) |
Im[PolyGamma[I*y]]=Divide[1,2*y]+Divide[Pi,2]*Coth[Pi*y] |
Failure | Failure | Successful | Successful |
| 5.4.E17 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \imagpart@@{\digamma@{\tfrac{1}{2}+iy}} = \frac{\pi}{2}\tanh@{\pi y}} | Im(Psi((1)/(2)+ I*y))=(Pi)/(2)*tanh(Pi*y) |
Im[PolyGamma[Divide[1,2]+ I*y]]=Divide[Pi,2]*Tanh[Pi*y] |
Failure | Failure | Successful | Successful |
| 5.4.E18 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \imagpart@@{\digamma@{1+iy}} = -\frac{1}{2y}+\frac{\pi}{2}\coth@{\pi y}} | Im(Psi(1 + I*y))= -(1)/(2*y)+(Pi)/(2)*coth(Pi*y) |
Im[PolyGamma[1 + I*y]]= -Divide[1,2*y]+Divide[Pi,2]*Coth[Pi*y] |
Failure | Failure | Successful | Successful |
| 5.4.E19 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{\frac{p}{q}} = -\EulerConstant-\ln@@{q}-\frac{\pi}{2}\cot@{\frac{\pi p}{q}}+\frac{1}{2}\sum_{k=1}^{q-1}\cos@{\frac{2\pi kp}{q}}\ln@{2-2\cos@{\frac{2\pi k}{q}}}} | Psi((p)/(q))= - gamma - ln(q)-(Pi)/(2)*cot((Pi*p)/(q))+(1)/(2)*sum(cos((2*Pi*k*p)/(q))*ln(2 - 2*cos((2*Pi*k)/(q))), k = 1..q - 1) |
PolyGamma[Divide[p,q]]= - EulerGamma - Log[q]-Divide[Pi,2]*Cot[Divide[Pi*p,q]]+Divide[1,2]*Sum[Cos[Divide[2*Pi*k*p,q]]*Log[2 - 2*Cos[Divide[2*Pi*k,q]]], {k, 1, q - 1}] |
Failure | Failure | Skip | Fail
DirectedInfinity[] <- {Rule[k, 1], Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}DirectedInfinity[] <- {Rule[k, 2], Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}DirectedInfinity[] <- {Rule[k, 3], Rule[p, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}DirectedInfinity[] <- {Rule[k, 1], Rule[p, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[q, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 5.5.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{z+1} = z\EulerGamma@{z}} | GAMMA(z + 1)= z*GAMMA(z) |
Gamma[z + 1]= z*Gamma[z] |
Successful | Successful | - | - |
| 5.5.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{z+1} = \digamma@{z}+\frac{1}{z}} | Psi(z + 1)= Psi(z)+(1)/(z) |
PolyGamma[z + 1]= PolyGamma[z]+Divide[1,z] |
Successful | Successful | - | - |
| 5.5.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{z}\EulerGamma@{1-z} = \pi/\sin@{\pi z}} | GAMMA(z)*GAMMA(1 - z)= Pi/ sin(Pi*z) |
Gamma[z]*Gamma[1 - z]= Pi/ Sin[Pi*z] |
Successful | Successful | - | - |
| 5.5.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{z}-\digamma@{1-z} = -\pi/\tan@{\pi z}} | Psi(z)- Psi(1 - z)= - Pi/ tan(Pi*z) |
PolyGamma[z]- PolyGamma[1 - z]= - Pi/ Tan[Pi*z] |
Successful | Successful | - | - |
| 5.5.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{2z} = \pi^{-1/2}2^{2z-1}\EulerGamma@{z}\EulerGamma@{z+\tfrac{1}{2}}} | GAMMA(2*z)= (Pi)^(- 1/ 2)* (2)^(2*z - 1)* GAMMA(z)*GAMMA(z +(1)/(2)) |
Gamma[2*z]= (Pi)^(- 1/ 2)* (2)^(2*z - 1)* Gamma[z]*Gamma[z +Divide[1,2]] |
Successful | Successful | - | - |
| 5.5.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{nz} = (2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\EulerGamma@{z+\frac{k}{n}}} | GAMMA(n*z)=(2*Pi)^((1 - n)/ 2)* (n)^(n*z -(1/ 2))* product(GAMMA(z +(k)/(n)), k = 0..n - 1) |
Gamma[n*z]=(2*Pi)^((1 - n)/ 2)* (n)^(n*z -(1/ 2))* Product[Gamma[z +Divide[k,n]], {k, 0, n - 1}] |
Failure | Successful | Skip | - |
| 5.5.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \prod_{k=1}^{n-1}\EulerGamma@{\frac{k}{n}} = (2\pi)^{(n-1)/2}n^{-1/2}} | product(GAMMA((k)/(n)), k = 1..n - 1)=(2*Pi)^((n - 1)/ 2)* (n)^(- 1/ 2) |
Product[Gamma[Divide[k,n]], {k, 1, n - 1}]=(2*Pi)^((n - 1)/ 2)* (n)^(- 1/ 2) |
Failure | Failure | Skip | Fail
Complex[1.4142135623730951, 1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[1.4142135623730951, -1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[Indeterminate, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 5.5.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{2z} = \tfrac{1}{2}\left(\digamma@{z}+\digamma@{z+\tfrac{1}{2}}\right)+\ln@@{2}} | Psi(2*z)=(1)/(2)*(Psi(z)+ Psi(z +(1)/(2)))+ ln(2) |
PolyGamma[2*z]=Divide[1,2]*(PolyGamma[z]+ PolyGamma[z +Divide[1,2]])+ Log[2] |
Successful | Successful | - | - |
| 5.5.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{nz} = \frac{1}{n}\sum_{k=0}^{n-1}\digamma@{z+\frac{k}{n}}+\ln@@{n}} | Psi(n*z)=(1)/(n)*sum(Psi(z +(k)/(n)), k = 0..n - 1)+ ln(n) |
PolyGamma[n*z]=Divide[1,n]*Sum[PolyGamma[z +Divide[k,n]], {k, 0, n - 1}]+ Log[n] |
Failure | Successful | Skip | - |
| 5.6.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 1 < (2\pi)^{-1/2}x^{(1/2)-x}e^{x}\EulerGamma@{x}} | 1 <(2*Pi)^(- 1/ 2)* (x)^((1/ 2)- x)* exp(x)*GAMMA(x) |
1 <(2*Pi)^(- 1/ 2)* (x)^((1/ 2)- x)* Exp[x]*Gamma[x] |
Failure | Failure | Successful | Successful |
| 5.6.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (2\pi)^{-1/2}x^{(1/2)-x}e^{x}\EulerGamma@{x} < e^{1/(12x)}} | (2*Pi)^(- 1/ 2)* (x)^((1/ 2)- x)* exp(x)*GAMMA(x)< exp(1/(12*x)) |
(2*Pi)^(- 1/ 2)* (x)^((1/ 2)- x)* Exp[x]*Gamma[x]< Exp[1/(12*x)] |
Failure | Failure | Successful | Successful |
| 5.6.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\EulerGamma@{x}}+\frac{1}{\EulerGamma@{1/x}} <= 2} | (1)/(GAMMA(x))+(1)/(GAMMA(1/ x))< = 2 |
Divide[1,Gamma[x]]+Divide[1,Gamma[1/ x]]< = 2 |
Failure | Failure | Successful | Successful |
| 5.6.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{(\EulerGamma@{x})^{2}}+\frac{1}{(\EulerGamma@{1/x})^{2}} <= 2} | (1)/((GAMMA(x))^(2))+(1)/((GAMMA(1/ x))^(2))< = 2 |
Divide[1,(Gamma[x])^(2)]+Divide[1,(Gamma[1/ x])^(2)]< = 2 |
Failure | Failure | Successful | Successful |
| 5.6.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x^{1-s} < \frac{\EulerGamma@{x+1}}{\EulerGamma@{x+s}}} | (x)^(1 - s)<(GAMMA(x + 1))/(GAMMA(x + s)) |
(x)^(1 - s)<Divide[Gamma[x + 1],Gamma[x + s]] |
Failure | Failure | Successful | Successful |
| 5.6.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\EulerGamma@{x+1}}{\EulerGamma@{x+s}} < (x+1)^{1-s}} | (GAMMA(x + 1))/(GAMMA(x + s))<(x + 1)^(1 - s) |
Divide[Gamma[x + 1],Gamma[x + s]]<(x + 1)^(1 - s) |
Failure | Failure | Successful | Successful |
| 5.6.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \exp@{(1-s)\digamma@{x+s^{1/2}}} <= \frac{\EulerGamma@{x+1}}{\EulerGamma@{x+s}}} | exp((1 - s)* Psi(x + (s)^(1/ 2)))< =(GAMMA(x + 1))/(GAMMA(x + s)) |
Exp[(1 - s)* PolyGamma[x + (s)^(1/ 2)]]< =Divide[Gamma[x + 1],Gamma[x + s]] |
Failure | Failure | Successful | Successful |
| 5.6.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\EulerGamma@{x+1}}{\EulerGamma@{x+s}} <= \exp@{(1-s)\digamma@{x+\tfrac{1}{2}(s+1)}}} | (GAMMA(x + 1))/(GAMMA(x + s))< = exp((1 - s)* Psi(x +(1)/(2)*(s + 1))) |
Divide[Gamma[x + 1],Gamma[x + s]]< = Exp[(1 - s)* PolyGamma[x +Divide[1,2]*(s + 1)]] |
Failure | Failure | Successful | Successful |
| 5.6.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\EulerGamma@{x+\iunit y}| <= |\EulerGamma@{x}|} | abs(GAMMA(x + I*y))< =abs(GAMMA(x)) |
Abs[Gamma[x + I*y]]< =Abs[Gamma[x]] |
Failure | Failure | Successful | Successful |
| 5.6.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\EulerGamma@{x+\iunit y}| >= (\sech@{\pi y})^{1/2}\EulerGamma@{x}} | abs(GAMMA(x + I*y))> =(sech(Pi*y))^(1/ 2)* GAMMA(x) |
Abs[Gamma[x + I*y]]> =(Sech[Pi*y])^(1/ 2)* Gamma[x] |
Failure | Failure | Skip | Successful |
| 5.6.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \left|\frac{\EulerGamma@{z+a}}{\EulerGamma@{z+b}}\right| <= \frac{1}{|z|^{b-a}}} | abs((GAMMA(z + a))/(GAMMA(z + b)))< =(1)/((abs(z))^(b - a)) |
Abs[Divide[Gamma[z + a],Gamma[z + b]]]< =Divide[1,(Abs[z])^(b - a)] |
Failure | Failure | Error | Successful |
| 5.6.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\EulerGamma@{z}| <= (2\pi)^{1/2}|z|^{x-(1/2)}e^{-\pi|y|/2}\exp@{\tfrac{1}{6}|z|^{-1}}} | abs(GAMMA(z))< =(2*Pi)^(1/ 2)*(abs(z))^(x -(1/ 2))* exp(- Pi*abs(y)/ 2)*exp((1)/(6)*(abs(z))^(- 1)) |
Abs[Gamma[z]]< =(2*Pi)^(1/ 2)*(Abs[z])^(x -(1/ 2))* Exp[- Pi*Abs[y]/ 2]*Exp[Divide[1,6]*(Abs[z])^(- 1)] |
Failure | Failure | Fail .3896047846 <= .1665021267 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 2}.3896047846 <= .3461239156e-1 <- {z = 2^(1/2)+I*2^(1/2), x = 1, y = 3}.3896047846 <= .3330042534 <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 2}.3896047846 <= .6922478312e-1 <- {z = 2^(1/2)+I*2^(1/2), x = 2, y = 3}... skip entries to safe data |
Successful |
| 5.7.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\EulerGamma@{z}} = \sum_{k=1}^{\infty}c_{k}z^{k}} | (1)/(GAMMA(z))= sum(c[k]*(z)^(k), k = 1..infinity) |
Divide[1,Gamma[z]]= Sum[Subscript[c, k]*(z)^(k), {k, 1, Infinity}] |
Failure | Failure | Skip | Skip |
| 5.7.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ln@@{\EulerGamma@{1+z}} = -\ln@{1+z}+z(1-\EulerConstant)+\sum_{k=2}^{\infty}(-1)^{k}(\Riemannzeta@{k}-1)\frac{z^{k}}{k}} | ln(GAMMA(1 + z))= - ln(1 + z)+ z*(1 - gamma)+ sum((- 1)^(k)*(Zeta(k)- 1)*((z)^(k))/(k), k = 2..infinity) |
Log[Gamma[1 + z]]= - Log[1 + z]+ z*(1 - EulerGamma)+ Sum[(- 1)^(k)*(Zeta[k]- 1)*Divide[(z)^(k),k], {k, 2, Infinity}] |
Failure | Successful | Skip | - |
| 5.7.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{1+z} = -\EulerConstant+\sum_{k=2}^{\infty}(-1)^{k}\Riemannzeta@{k}z^{k-1}} | Psi(1 + z)= - gamma + sum((- 1)^(k)* Zeta(k)*(z)^(k - 1), k = 2..infinity) |
PolyGamma[1 + z]= - EulerGamma + Sum[(- 1)^(k)* Zeta[k]*(z)^(k - 1), {k, 2, Infinity}] |
Failure | Successful | Skip | - |
| 5.7.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{1+z} = \frac{1}{2z}-\frac{\pi}{2}\cot@{\pi z}+\frac{1}{z^{2}-1}+1-\EulerConstant-\sum_{k=1}^{\infty}(\Riemannzeta@{2k+1}-1)z^{2k}} | Psi(1 + z)=(1)/(2*z)-(Pi)/(2)*cot(Pi*z)+(1)/((z)^(2)- 1)+ 1 - gamma - sum((Zeta(2*k + 1)- 1)* (z)^(2*k), k = 1..infinity) |
PolyGamma[1 + z]=Divide[1,2*z]-Divide[Pi,2]*Cot[Pi*z]+Divide[1,(z)^(2)- 1]+ 1 - EulerGamma - Sum[(Zeta[2*k + 1]- 1)* (z)^(2*k), {k, 1, Infinity}] |
Failure | Successful | Skip | - |
| 5.7.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{z} = -\EulerConstant-\frac{1}{z}+\sum_{k=1}^{\infty}\frac{z}{k(k+z)}} | Psi(z)= - gamma -(1)/(z)+ sum((z)/(k*(k + z)), k = 1..infinity) |
PolyGamma[z]= - EulerGamma -Divide[1,z]+ Sum[Divide[z,k*(k + z)], {k, 1, Infinity}] |
Successful | Successful | - | - |
| 5.7.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -\EulerConstant-\frac{1}{z}+\sum_{k=1}^{\infty}\frac{z}{k(k+z)} = -\EulerConstant+\sum_{k=0}^{\infty}\left(\frac{1}{k+1}-\frac{1}{k+z}\right)} | - gamma -(1)/(z)+ sum((z)/(k*(k + z)), k = 1..infinity)= - gamma + sum((1)/(k + 1)-(1)/(k + z), k = 0..infinity) |
- EulerGamma -Divide[1,z]+ Sum[Divide[z,k*(k + z)], {k, 1, Infinity}]= - EulerGamma + Sum[Divide[1,k + 1]-Divide[1,k + z], {k, 0, Infinity}] |
Successful | Successful | - | - |
| 5.7.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{\frac{z+1}{2}}-\digamma@{\frac{z}{2}} = 2\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k+z}} | Psi((z + 1)/(2))- Psi((z)/(2))= 2*sum(((- 1)^(k))/(k + z), k = 0..infinity) |
PolyGamma[Divide[z + 1,2]]- PolyGamma[Divide[z,2]]= 2*Sum[Divide[(- 1)^(k),k + z], {k, 0, Infinity}] |
Successful | Successful | - | - |
| 5.7.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \imagpart@@{\digamma@{1+\iunit y}} = \sum_{k=1}^{\infty}\frac{y}{k^{2}+y^{2}}} | Im(Psi(1 + I*y))= sum((y)/((k)^(2)+ (y)^(2)), k = 1..infinity) |
Im[PolyGamma[1 + I*y]]= Sum[Divide[y,(k)^(2)+ (y)^(2)], {k, 1, Infinity}] |
Failure | Failure | Skip | Successful |
| 5.8.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\EulerGamma@{z}} = ze^{\EulerConstant z}\prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right)e^{-z/k}} | (1)/(GAMMA(z))= z*exp(gamma*z)*product((1 +(z)/(k))* exp(- z/ k), k = 1..infinity) |
Divide[1,Gamma[z]]= z*Exp[EulerGamma*z]*Product[(1 +Divide[z,k])* Exp[- z/ k], {k, 1, Infinity}] |
Successful | Failure | - | Successful |
| 5.8.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \left|\frac{\EulerGamma@{x}}{\EulerGamma@{x+\iunit y}}\right|^{2} = \prod_{k=0}^{\infty}\left(1+\frac{y^{2}}{(x+k)^{2}}\right)} | (abs((GAMMA(x))/(GAMMA(x + I*y))))^(2)= product(1 +((y)^(2))/((x + k)^(2)), k = 0..infinity) |
(Abs[Divide[Gamma[x],Gamma[x + I*y]]])^(2)= Product[1 +Divide[(y)^(2),(x + k)^(2)], {k, 0, Infinity}] |
Failure | Failure | Skip | Successful |
| 5.9.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\mu}\EulerGamma@{\frac{\nu}{\mu}}\frac{1}{z^{\nu/\mu}} = \int_{0}^{\infty}\exp@{-zt^{\mu}}t^{\nu-1}\diff{t}} | (1)/(mu)*GAMMA((nu)/(mu))*(1)/((z)^(nu/ mu))= int(exp(- z*(t)^(mu))*(t)^(nu - 1), t = 0..infinity) |
Divide[1,\[Mu]]*Gamma[Divide[\[Nu],\[Mu]]]*Divide[1,(z)^(\[Nu]/ \[Mu])]= Integrate[Exp[- z*(t)^(\[Mu])]*(t)^(\[Nu]- 1), {t, 0, Infinity}] |
Failure | Failure | Skip | Successful |
| 5.9.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\EulerGamma@{z}} = \frac{1}{2\pi i}\int_{-\infty}^{(0+)}e^{t}t^{-z}\diff{t}} | (1)/(GAMMA(z))=(1)/(2*Pi*I)*int(exp(t)*(t)^(- z), t = - infinity..(0 +)) |
Divide[1,Gamma[z]]=Divide[1,2*Pi*I]*Integrate[Exp[t]*(t)^(- z), {t, - Infinity, (0 +)}] |
Error | Failure | - | Error |
| 5.9.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{z} = \int_{1}^{\infty}t^{z-1}e^{-t}\diff{t}+\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(z+k)k!}} | GAMMA(z)= int((t)^(z - 1)* exp(- t), t = 1..infinity)+ sum(((- 1)^(k))/((z + k)* factorial(k)), k = 0..infinity) |
Gamma[z]= Integrate[(t)^(z - 1)* Exp[- t], {t, 1, Infinity}]+ Sum[Divide[(- 1)^(k),(z + k)* (k)!], {k, 0, Infinity}] |
Failure | Successful | Skip | - |
| 5.9.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{z} = \int_{0}^{\infty}t^{z-1}\left(e^{-t}-\sum_{k=0}^{n}\frac{(-1)^{k}t^{k}}{k!}\right)\diff{t}} | GAMMA(z)= int((t)^(z - 1)*(exp(- t)- sum(((- 1)^(k)* (t)^(k))/(factorial(k)), k = 0..n)), t = 0..infinity) |
Gamma[z]= Integrate[(t)^(z - 1)*(Exp[- t]- Sum[Divide[(- 1)^(k)* (t)^(k),(k)!], {k, 0, n}]), {t, 0, Infinity}] |
Failure | Failure | Skip | Skip |
| 5.9.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{z}\cos@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\cos@@{t}\diff{t}} | GAMMA(z)*cos((1)/(2)*Pi*z)= int((t)^(z - 1)* cos(t), t = 0..infinity) |
Gamma[z]*Cos[Divide[1,2]*Pi*z]= Integrate[(t)^(z - 1)* Cos[t], {t, 0, Infinity}] |
Successful | Failure | - | Successful |
| 5.9.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{z}\sin@{\tfrac{1}{2}\pi z} = \int_{0}^{\infty}t^{z-1}\sin@@{t}\diff{t}} | GAMMA(z)*sin((1)/(2)*Pi*z)= int((t)^(z - 1)* sin(t), t = 0..infinity) |
Gamma[z]*Sin[Divide[1,2]*Pi*z]= Integrate[(t)^(z - 1)* Sin[t], {t, 0, Infinity}] |
Successful | Failure | - | Successful |
| 5.9.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{1+\frac{1}{n}}\cos@{\frac{\pi}{2n}} = \int_{0}^{\infty}\cos@{t^{n}}\diff{t}} | GAMMA(1 +(1)/(n))*cos((Pi)/(2*n))= int(cos((t)^(n)), t = 0..infinity) |
Gamma[1 +Divide[1,n]]*Cos[Divide[Pi,2*n]]= Integrate[Cos[(t)^(n)], {t, 0, Infinity}] |
Successful | Failure | - | Successful |
| 5.9.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{1+\frac{1}{n}}\sin@{\frac{\pi}{2n}} = \int_{0}^{\infty}\sin@{t^{n}}\diff{t}} | GAMMA(1 +(1)/(n))*sin((Pi)/(2*n))= int(sin((t)^(n)), t = 0..infinity) |
Gamma[1 +Divide[1,n]]*Sin[Divide[Pi,2*n]]= Integrate[Sin[(t)^(n)], {t, 0, Infinity}] |
Successful | Failure | - | Successful |
| 5.9.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Ln@@{\EulerGamma@{z}} = \left(z-\tfrac{1}{2}\right)\ln@@{z}-z+\tfrac{1}{2}\ln@{2\pi}+2\int_{0}^{\infty}\frac{\atan@{t/z}}{e^{2\pi t}-1}\diff{t}} | ln(GAMMA(z))=(z -(1)/(2))* ln(z)- z +(1)/(2)*ln(2*Pi)+ 2*int((arctan(t/ z))/(exp(2*Pi*t)- 1), t = 0..infinity) |
Log[Gamma[z]]=(z -Divide[1,2])* Log[z]- z +Divide[1,2]*Log[2*Pi]+ 2*Integrate[Divide[ArcTan[t/ z],Exp[2*Pi*t]- 1], {t, 0, Infinity}] |
Failure | Failure | Skip | Skip |
| 5.9.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{z} = \int_{0}^{\infty}\left(\frac{e^{-t}}{t}-\frac{e^{-zt}}{1-e^{-t}}\right)\diff{t}} | Psi(z)= int((exp(- t))/(t)-(exp(- z*t))/(1 - exp(- t)), t = 0..infinity) |
PolyGamma[z]= Integrate[Divide[Exp[- t],t]-Divide[Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}] |
Failure | Failure | Skip | Successful |
| 5.9.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{z} = \ln@@{z}+\int_{0}^{\infty}\left(\frac{1}{t}-\frac{1}{1-e^{-t}}\right)e^{-tz}\diff{t}} | Psi(z)= ln(z)+ int(((1)/(t)-(1)/(1 - exp(- t)))* exp(- t*z), t = 0..infinity) |
PolyGamma[z]= Log[z]+ Integrate[(Divide[1,t]-Divide[1,1 - Exp[- t]])* Exp[- t*z], {t, 0, Infinity}] |
Failure | Failure | Skip | Error |
| 5.9.E14 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{z} = \int_{0}^{\infty}\left(e^{-t}-\frac{1}{(1+t)^{z}}\right)\frac{\diff{t}}{t}} | Psi(z)= int((exp(- t)-(1)/((1 + t)^(z)))*(1)/(t), t = 0..infinity) |
PolyGamma[z]= Integrate[(Exp[- t]-Divide[1,(1 + t)^(z)])*Divide[1,t], {t, 0, Infinity}] |
Failure | Failure | Skip | Successful |
| 5.9.E15 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{z} = \ln@@{z}-\frac{1}{2z}-2\int_{0}^{\infty}\frac{t\diff{t}}{(t^{2}+z^{2})(e^{2\pi t}-1)}} | Psi(z)= ln(z)-(1)/(2*z)- 2*int((t)/(((t)^(2)+ (z)^(2))*(exp(2*Pi*t)- 1)), t = 0..infinity) |
PolyGamma[z]= Log[z]-Divide[1,2*z]- 2*Integrate[Divide[t,((t)^(2)+ (z)^(2))*(Exp[2*Pi*t]- 1)], {t, 0, Infinity}] |
Failure | Failure | Skip | Skip |
| 5.9.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{z}+\EulerConstant = \int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t}} | Psi(z)+ gamma = int((exp(- t)- exp(- z*t))/(1 - exp(- t)), t = 0..infinity) |
PolyGamma[z]+ EulerGamma = Integrate[Divide[Exp[- t]- Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}] |
Failure | Failure | Skip | Successful |
| 5.9.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}\frac{e^{-t}-e^{-zt}}{1-e^{-t}}\diff{t} = \int_{0}^{1}\frac{1-t^{z-1}}{1-t}\diff{t}} | int((exp(- t)- exp(- z*t))/(1 - exp(- t)), t = 0..infinity)= int((1 - (t)^(z - 1))/(1 - t), t = 0..1) |
Integrate[Divide[Exp[- t]- Exp[- z*t],1 - Exp[- t]], {t, 0, Infinity}]= Integrate[Divide[1 - (t)^(z - 1),1 - t], {t, 0, 1}] |
Failure | Failure | Skip | Skip |
| 5.9.E17 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{z+1} = -\EulerConstant+\frac{1}{2\pi i}\int_{-c-\infty i}^{-c+\infty i}\frac{\pi z^{-s-1}}{\sin@{\pi s}}\Riemannzeta@{-s}\diff{s}} | Psi(z + 1)= - gamma +(1)/(2*Pi*I)*int((Pi*(z)^(- s - 1))/(sin(Pi*s))*Zeta(- s), s = - c - infinity*I..- c + infinity*I) |
PolyGamma[z + 1]= - EulerGamma +Divide[1,2*Pi*I]*Integrate[Divide[Pi*(z)^(- s - 1),Sin[Pi*s]]*Zeta[- s], {s, - c - Infinity*I, - c + Infinity*I}] |
Failure | Failure | Skip | Skip |
| 5.9.E18 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerConstant = -\int_{0}^{\infty}e^{-t}\ln@@{t}\diff{t}} | gamma = - int(exp(- t)*ln(t), t = 0..infinity) |
EulerGamma = - Integrate[Exp[- t]*Log[t], {t, 0, Infinity}] |
Successful | Successful | - | - |
| 5.9.E18 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -\int_{0}^{\infty}e^{-t}\ln@@{t}\diff{t} = \int_{0}^{\infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{\diff{t}}{t}} | - int(exp(- t)*ln(t), t = 0..infinity)= int(((1)/(1 + t)- exp(- t))*(1)/(t), t = 0..infinity) |
- Integrate[Exp[- t]*Log[t], {t, 0, Infinity}]= Integrate[(Divide[1,1 + t]- Exp[- t])*Divide[1,t], {t, 0, Infinity}] |
Successful | Successful | - | - |
| 5.9.E18 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}\left(\frac{1}{1+t}-e^{-t}\right)\frac{\diff{t}}{t} = \int_{0}^{1}(1-e^{-t})\frac{\diff{t}}{t}-\int_{1}^{\infty}e^{-t}\frac{\diff{t}}{t}} | int(((1)/(1 + t)- exp(- t))*(1)/(t), t = 0..infinity)= int((1 - exp(- t))*(1)/(t), t = 0..1)- int(exp(- t)*(1)/(t), t = 1..infinity) |
Integrate[(Divide[1,1 + t]- Exp[- t])*Divide[1,t], {t, 0, Infinity}]= Integrate[(1 - Exp[- t])*Divide[1,t], {t, 0, 1}]- Integrate[Exp[- t]*Divide[1,t], {t, 1, Infinity}] |
Successful | Successful | - | - |
| 5.9.E18 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{1}(1-e^{-t})\frac{\diff{t}}{t}-\int_{1}^{\infty}e^{-t}\frac{\diff{t}}{t} = \int_{0}^{\infty}\left(\frac{e^{-t}}{1-e^{-t}}-\frac{e^{-t}}{t}\right)\diff{t}} | int((1 - exp(- t))*(1)/(t), t = 0..1)- int(exp(- t)*(1)/(t), t = 1..infinity)= int((exp(- t))/(1 - exp(- t))-(exp(- t))/(t), t = 0..infinity) |
Integrate[(1 - Exp[- t])*Divide[1,t], {t, 0, 1}]- Integrate[Exp[- t]*Divide[1,t], {t, 1, Infinity}]= Integrate[Divide[Exp[- t],1 - Exp[- t]]-Divide[Exp[- t],t], {t, 0, Infinity}] |
Successful | Successful | - | - |
| 5.9.E19 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma^{(n)}@{z} = \int_{0}^{\infty}(\ln@@{t})^{n}e^{-t}t^{z-1}\diff{t}} | subs( temp=z, diff( GAMMA(temp), temp$(n) ) )= int((ln(t))^(n)* exp(- t)*(t)^(z - 1), t = 0..infinity) |
(D[Gamma[temp], {temp, n}]/.temp-> z)= Integrate[(Log[t])^(n)* Exp[- t]*(t)^(z - 1), {t, 0, Infinity}] |
Successful | Failure | - | Error |
| 5.11.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle g_{k} = \sqrt{2}\Pochhammersym{\tfrac{1}{2}}{k}a_{2k}} | g[k]=sqrt(2)*pochhammer((1)/(2), k)*a[2*k] |
Subscript[g, k]=Sqrt[2]*Pochhammer[Divide[1,2], k]*Subscript[a, 2*k] |
Failure | Failure | Fail .4142135625+.4142135625*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 1}-.8578643792e-1-.8578643792e-1*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 2}-2.335786436-2.335786436*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)+I*2^(1/2), k = 3}.4142135625-2.414213561*I <- {a[2*k] = 2^(1/2)+I*2^(1/2), g[k] = 2^(1/2)-I*2^(1/2), k = 1}... skip entries to safe data |
Successful |
| 5.11.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerGamma@{z} = e^{-z}z^{z}\left(\frac{2\pi}{z}\right)^{1/2}\left(\sum_{k=0}^{K-1}\frac{g_{k}}{z^{k}}+R_{K}(z)\right)} | GAMMA(z)= exp(- z)*(z)^(z)*((2*Pi)/(z))^(1/ 2)*(sum((g[k])/((z)^(k)), k = 0..K - 1)+ R[K]*(z)) |
Gamma[z]= Exp[- z]*(z)^(z)*(Divide[2*Pi,z])^(1/ 2)*(Sum[Divide[Subscript[g, k],(z)^(k)], {k, 0, K - 1}]+ Subscript[R, K]*(z)) |
Failure | Failure | Skip | Skip |
| 5.11#Ex10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle G_{2}(a,b) = \frac{1}{12}\binom{a-b}{2}(3(a+b-1)^{2}-(a-b+1))} | G[2]*(a , b)=(1)/(12)*binomial(a - b,2)*(3*(a + b - 1)^(2)-(a - b + 1)) |
Subscript[G, 2]*(a , b)=Divide[1,12]*Binomial[a - b,2]*(3*(a + b - 1)^(2)-(a - b + 1)) |
Failure | Failure | Error | Error |
| 5.11#Ex12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle H_{1}(a,b) = -\frac{1}{12}\binom{a-b}{2}(a-b+1)} | H[1]*(a , b)= -(1)/(12)*binomial(a - b,2)*(a - b + 1) |
Subscript[H, 1]*(a , b)= -Divide[1,12]*Binomial[a - b,2]*(a - b + 1) |
Failure | Failure | Error | Error |
| 5.11#Ex13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle H_{2}(a,b) = \frac{1}{240}\binom{a-b}{4}(2(a-b+1)+5(a-b+1)^{2})} | H[2]*(a , b)=(1)/(240)*binomial(a - b,4)*(2*(a - b + 1)+ 5*(a - b + 1)^(2)) |
Subscript[H, 2]*(a , b)=Divide[1,240]*Binomial[a - b,4]*(2*(a - b + 1)+ 5*(a - b + 1)^(2)) |
Failure | Failure | Error | Error |
| 5.12.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerBeta@{a}{b} = \int_{0}^{1}t^{a-1}(1-t)^{b-1}\diff{t}} | Beta(a, b)= int((t)^(a - 1)*(1 - t)^(b - 1), t = 0..1) |
Beta[a, b]= Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, 0, 1}] |
Failure | Failure | Skip | Successful |
| 5.12.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{1}t^{a-1}(1-t)^{b-1}\diff{t} = \frac{\EulerGamma@{a}\EulerGamma@{b}}{\EulerGamma@{a+b}}} | int((t)^(a - 1)*(1 - t)^(b - 1), t = 0..1)=(GAMMA(a)*GAMMA(b))/(GAMMA(a + b)) |
Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, 0, 1}]=Divide[Gamma[a]*Gamma[b],Gamma[a + b]] |
Successful | Failure | - | Successful |
| 5.12.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\pi/2}\sin^{2a-1}@@{\theta}\cos^{2b-1}@@{\theta}\diff{\theta} = \tfrac{1}{2}\EulerBeta@{a}{b}} | int((sin(theta))^(2*a - 1)* (cos(theta))^(2*b - 1), theta = 0..Pi/ 2)=(1)/(2)*Beta(a, b) |
Integrate[(Sin[\[Theta]])^(2*a - 1)* (Cos[\[Theta]])^(2*b - 1), {\[Theta], 0, Pi/ 2}]=Divide[1,2]*Beta[a, b] |
Failure | Failure | Skip | Successful |
| 5.12.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}\frac{t^{a-1}\diff{t}}{(1+t)^{a+b}} = \EulerBeta@{a}{b}} | int(((t)^(a - 1))/((1 + t)^(a + b)), t = 0..infinity)= Beta(a, b) |
Integrate[Divide[(t)^(a - 1),(1 + t)^(a + b)], {t, 0, Infinity}]= Beta[a, b] |
Failure | Failure | Skip | Successful |
| 5.12.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{1}\frac{t^{a-1}(1-t)^{b-1}}{(t+z)^{a+b}}\diff{t} = \EulerBeta@{a}{b}(1+z)^{-a}z^{-b}} | int(((t)^(a - 1)*(1 - t)^(b - 1))/((t + z)^(a + b)), t = 0..1)= Beta(a, b)*(1 + z)^(- a)* (z)^(- b) |
Integrate[Divide[(t)^(a - 1)*(1 - t)^(b - 1),(t + z)^(a + b)], {t, 0, 1}]= Beta[a, b]*(1 + z)^(- a)* (z)^(- b) |
Failure | Failure | Skip | Successful |
| 5.12.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\pi/2}(\cos@@{t})^{a-1}\cos@{bt}\diff{t} = \frac{\pi}{2^{a}}\frac{1}{a\EulerBeta@{\frac{1}{2}(a+b+1)}{\frac{1}{2}(a-b+1)}}} | int((cos(t))^(a - 1)* cos(b*t), t = 0..Pi/ 2)=(Pi)/((2)^(a))*(1)/(a*Beta((1)/(2)*(a + b + 1), (1)/(2)*(a - b + 1))) |
Integrate[(Cos[t])^(a - 1)* Cos[b*t], {t, 0, Pi/ 2}]=Divide[Pi,(2)^(a)]*Divide[1,a*Beta[Divide[1,2]*(a + b + 1), Divide[1,2]*(a - b + 1)]] |
Failure | Failure | Skip | Error |
| 5.12.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\pi}(\sin@@{t})^{a-1}e^{ibt}\diff{t} = \frac{\pi}{2^{a-1}}\frac{e^{i\pi b/2}}{a\EulerBeta@{\frac{1}{2}(a+b+1)}{\frac{1}{2}(a-b+1)}}} | int((sin(t))^(a - 1)* exp(I*b*t), t = 0..Pi)=(Pi)/((2)^(a - 1))*(exp(I*Pi*b/ 2))/(a*Beta((1)/(2)*(a + b + 1), (1)/(2)*(a - b + 1))) |
Integrate[(Sin[t])^(a - 1)* Exp[I*b*t], {t, 0, Pi}]=Divide[Pi,(2)^(a - 1)]*Divide[Exp[I*Pi*b/ 2],a*Beta[Divide[1,2]*(a + b + 1), Divide[1,2]*(a - b + 1)]] |
Failure | Failure | Skip | Skip |
| 5.12.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}\frac{\cosh@{2bt}}{(\cosh@@{t})^{2a}}\diff{t} = 4^{a-1}\EulerBeta@{a+b}{a-b}} | int((cosh(2*b*t))/((cosh(t))^(2*a)), t = 0..infinity)= (4)^(a - 1)* Beta(a + b, a - b) |
Integrate[Divide[Cosh[2*b*t],(Cosh[t])^(2*a)], {t, 0, Infinity}]= (4)^(a - 1)* Beta[a + b, a - b] |
Failure | Failure | Skip | Skip |
| 5.12.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{e^{2\pi ia}-1}\int_{\infty}^{(0+)}t^{a-1}(1+t)^{-a-b}\diff{t} = \EulerBeta@{a}{b}} | (1)/(exp(2*Pi*I*a)- 1)*int((t)^(a - 1)*(1 + t)^(- a - b), t = infinity..(0 +))= Beta(a, b) |
Divide[1,Exp[2*Pi*I*a]- 1]*Integrate[(t)^(a - 1)*(1 + t)^(- a - b), {t, Infinity, (0 +)}]= Beta[a, b] |
Error | Failure | - | Error |
| 5.12.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{P}^{(1+,0+,1-,0-)}t^{a-1}(1-t)^{b-1}\diff{t} = -4e^{\pi i(a+b)}\sin@{\pi a}\sin@{\pi b}\EulerBeta@{a}{b}} | int((t)^(a - 1)*(1 - t)^(b - 1), t = P..(1 + , 0 + , 1 - , 0 -))= - 4*exp(Pi*I*(a + b))*sin(Pi*a)*sin(Pi*b)*Beta(a, b) |
Integrate[(t)^(a - 1)*(1 - t)^(b - 1), {t, P, (1 + , 0 + , 1 - , 0 -)}]= - 4*Exp[Pi*I*(a + b)]*Sin[Pi*a]*Sin[Pi*b]*Beta[a, b] |
Error | Failure | - | Error |
| 5.13.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{2\pi}\int_{-\infty}^{\infty}\EulerGamma@{a+it}\EulerGamma@{b+it}\EulerGamma@{c-it}\EulerGamma@{d-it}\diff{t} = \frac{\EulerGamma@{a+c}\EulerGamma@{a+d}\EulerGamma@{b+c}\EulerGamma@{b+d}}{\EulerGamma@{a+b+c+d}}} | (1)/(2*Pi)*int(GAMMA(a + I*t)*GAMMA(b + I*t)*GAMMA(c - I*t)*GAMMA(d - I*t), t = - infinity..infinity)=(GAMMA(a + c)*GAMMA(a + d)*GAMMA(b + c)*GAMMA(b + d))/(GAMMA(a + b + c + d)) |
Divide[1,2*Pi]*Integrate[Gamma[a + I*t]*Gamma[b + I*t]*Gamma[c - I*t]*Gamma[d - I*t], {t, - Infinity, Infinity}]=Divide[Gamma[a + c]*Gamma[a + d]*Gamma[b + c]*Gamma[b + d],Gamma[a + b + c + d]] |
Error | Failure | - | Successful |
| 5.13.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{-\infty}^{\infty}\frac{\diff{t}}{\EulerGamma@{a+t}\EulerGamma@{b+t}\EulerGamma@{c-t}\EulerGamma@{d-t}} = \frac{\EulerGamma@{a+b+c+d-3}}{\EulerGamma@{a+c-1}\EulerGamma@{a+d-1}\EulerGamma@{b+c-1}\EulerGamma@{b+d-1}}} | int((1)/(GAMMA(a + t)*GAMMA(b + t)*GAMMA(c - t)*GAMMA(d - t)), t = - infinity..infinity)=(GAMMA(a + b + c + d - 3))/(GAMMA(a + c - 1)*GAMMA(a + d - 1)*GAMMA(b + c - 1)*GAMMA(b + d - 1)) |
Integrate[Divide[1,Gamma[a + t]*Gamma[b + t]*Gamma[c - t]*Gamma[d - t]], {t, - Infinity, Infinity}]=Divide[Gamma[a + b + c + d - 3],Gamma[a + c - 1]*Gamma[a + d - 1]*Gamma[b + c - 1]*Gamma[b + d - 1]] |
Failure | Failure | Skip | Error |
| 5.13.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{4\pi}\int_{-\infty}^{\infty}\frac{\prod_{k=1}^{4}\EulerGamma@{a_{k}+it}\EulerGamma@{a_{k}-it}}{\EulerGamma@{2it}\EulerGamma@{-2it}}\diff{t} = \frac{\prod_{1\leq j<k\leq 4}\EulerGamma@{a_{j}+a_{k}}}{\EulerGamma@{a_{1}+a_{2}+a_{3}+a_{4}}}} | (1)/(4*Pi)*int((product(GAMMA(a[k]+ I*t)*GAMMA(a[k]- I*t), k = 1..4))/(GAMMA(2*I*t)*GAMMA(- 2*I*t)), t = - infinity..infinity)=(product(product(GAMMA(a[j]+ a[k]), k = j + 1..4), j = 1..k - 1))/(GAMMA(a[1]+ a[2]+ a[3]+ a[4])) |
Divide[1,4*Pi]*Integrate[Divide[Product[Gamma[Subscript[a, k]+ I*t]*Gamma[Subscript[a, k]- I*t], {k, 1, 4}],Gamma[2*I*t]*Gamma[- 2*I*t]], {t, - Infinity, Infinity}]=Divide[Product[Product[Gamma[Subscript[a, j]+ Subscript[a, k]], {k, j + 1, 4}], {j, 1, k - 1}],Gamma[Subscript[a, 1]+ Subscript[a, 2]+ Subscript[a, 3]+ Subscript[a, 4]]] |
Failure | Failure | Skip | Error |
| 5.15.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma'@{z} = \sum_{k=0}^{\infty}\frac{1}{(k+z)^{2}}} | subs( temp=z, diff( Psi(temp), temp$(1) ) )= sum((1)/((k + z)^(2)), k = 0..infinity) |
(D[PolyGamma[temp], {temp, 1}]/.temp-> z)= Sum[Divide[1,(k + z)^(2)], {k, 0, Infinity}] |
Successful | Successful | - | - |
| 5.15.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \polygamma{n}@{1} = (-1)^{n+1}n!\Riemannzeta@{n+1}} | Psi(n, 1)=(- 1)^(n + 1)* factorial(n)*Zeta(n + 1) |
PolyGamma[n, 1]=(- 1)^(n + 1)* (n)!*Zeta[n + 1] |
Failure | Failure | Successful | Successful |
| 5.15.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \polygamma{n}@{\tfrac{1}{2}} = (-1)^{n+1}n!(2^{n+1}-1)\Riemannzeta@{n+1}} | Psi(n, (1)/(2))=(- 1)^(n + 1)* factorial(n)*((2)^(n + 1)- 1)* Zeta(n + 1) |
PolyGamma[n, Divide[1,2]]=(- 1)^(n + 1)* (n)!*((2)^(n + 1)- 1)* Zeta[n + 1] |
Failure | Failure | Successful | Successful |
| 5.15.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma'@{n-\tfrac{1}{2}} = \tfrac{1}{2}\pi^{2}-4\sum_{k=1}^{n-1}\frac{1}{(2k-1)^{2}}} | subs( temp=n -(1)/(2), diff( Psi(temp), temp$(1) ) )=(1)/(2)*(Pi)^(2)- 4*sum((1)/((2*k - 1)^(2)), k = 1..n - 1) |
(D[PolyGamma[temp], {temp, 1}]/.temp-> n -Divide[1,2])=Divide[1,2]*(Pi)^(2)- 4*Sum[Divide[1,(2*k - 1)^(2)], {k, 1, n - 1}] |
Successful | Successful | - | - |
| 5.15.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma^{(n)}@{z+1} = \digamma^{(n)}@{z}+(-1)^{n}n!z^{-n-1}} | subs( temp=z + 1, diff( Psi(temp), temp$(n) ) )= subs( temp=z, diff( Psi(temp), temp$(n) ) )+(- 1)^(n)* factorial(n)*(z)^(- n - 1) |
(D[PolyGamma[temp], {temp, n}]/.temp-> z + 1)= (D[PolyGamma[temp], {temp, n}]/.temp-> z)+(- 1)^(n)* (n)!*(z)^(- n - 1) |
Failure | Failure | Successful | Successful |
| 5.15.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma^{(n)}@{1-z}+(-1)^{n-1}\digamma^{(n)}@{z} = (-1)^{n}\pi\deriv[n]{}{z}\cot@{\pi z}} | subs( temp=1 - z, diff( Psi(temp), temp$(n) ) )+(- 1)^(n - 1)* subs( temp=z, diff( Psi(temp), temp$(n) ) )=(- 1)^(n)* Pi*diff(cot(Pi*z), [z$(n)]) |
(D[PolyGamma[temp], {temp, n}]/.temp-> 1 - z)+(- 1)^(n - 1)* (D[PolyGamma[temp], {temp, n}]/.temp-> z)=(- 1)^(n)* Pi*D[Cot[Pi*z], {z, n}] |
Failure | Failure | Successful | Successful |
| 5.15.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma^{(n)}@{mz} = \frac{1}{m^{n+1}}\sum_{k=0}^{m-1}\digamma^{(n)}@{z+\frac{k}{m}}} | subs( temp=m*z, diff( Psi(temp), temp$(n) ) )=(1)/((m)^(n + 1))*sum(subs( temp=z +(k)/(m), diff( Psi(temp), temp$(n) ) ), k = 0..m - 1) |
(D[PolyGamma[temp], {temp, n}]/.temp-> m*z)=Divide[1,(m)^(n + 1)]*Sum[D[PolyGamma[temp], {temp, n}]/.temp-> z +Divide[k,m], {k, 0, m - 1}] |
Failure | Failure | Skip | Successful |
| 5.16.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=1}^{\infty}(-1)^{k}\digamma'@{k} = -\frac{\pi^{2}}{8}} | sum((- 1)^(k)* subs( temp=k, diff( Psi(temp), temp$(1) ) ), k = 1..infinity)= -((Pi)^(2))/(8) |
Sum[(- 1)^(k)* (D[PolyGamma[temp], {temp, 1}]/.temp-> k), {k, 1, Infinity}]= -Divide[(Pi)^(2),8] |
Failure | Successful | Skip | - |
| 5.16.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{k=1}^{\infty}\frac{1}{k}\digamma'@{k+1} = \Riemannzeta@{3}} | sum((1)/(k)*subs( temp=k + 1, diff( Psi(temp), temp$(1) ) ), k = 1..infinity)= Zeta(3) |
Sum[Divide[1,k]*(D[PolyGamma[temp], {temp, 1}]/.temp-> k + 1), {k, 1, Infinity}]= Zeta[3] |
Failure | Successful | Skip | - |
| 5.16.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{3} = -\frac{1}{2}\digamma''@{1}} | Zeta(3)= -(1)/(2)*subs( temp=1, diff( Psi(temp), temp$(2) ) ) |
Zeta[3]= -Divide[1,2]*(D[PolyGamma[temp], {temp, 2}]/.temp-> 1) |
Successful | Successful | - | - |
| 5.17#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BarnesG@{z+1} = \EulerGamma@{z}\BarnesG@{z}} | Error |
BarnesG[z + 1]= Gamma[z]*BarnesG[z] |
Error | Failure | - | Successful |
| 5.17#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BarnesG@{1} = 1} | Error |
BarnesG[1]= 1 |
Error | Successful | - | - |
| 5.17.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BarnesG@{z+1} = (2\pi)^{z/2}\exp@{-\tfrac{1}{2}z(z+1)-\tfrac{1}{2}\EulerConstant z^{2}}\*\prod_{k=1}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp@{-z+\frac{z^{2}}{2k}}\right)} | Error |
BarnesG[z + 1]=(2*Pi)^(z/ 2)* Exp[-Divide[1,2]*z*(z + 1)-Divide[1,2]*EulerGamma*(z)^(2)]* Product[(1 +Divide[z,k])^(k)* Exp[- z +Divide[(z)^(2),2*k]], {k, 1, Infinity}] |
Error | Successful | - | - |
| 5.17.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Ln@@{\BarnesG@{z+1}} = \tfrac{1}{2}z\ln@{2\pi}-\tfrac{1}{2}z(z+1)+z\Ln@@{\EulerGamma@{z+1}}-\int_{0}^{z}\Ln@@{\EulerGamma@{t+1}}\diff{t}} | Error |
Log[BarnesG[z + 1]]=Divide[1,2]*z*Log[2*Pi]-Divide[1,2]*z*(z + 1)+ z*Log[Gamma[z + 1]]- Integrate[Log[Gamma[t + 1]], {t, 0, z}] |
Error | Failure | - | Successful |
| 5.17.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle C = \lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln@@{k}-\left(\tfrac{1}{2}n^{2}+\tfrac{1}{2}n+\tfrac{1}{12}\right)\ln@@{n}+\tfrac{1}{4}n^{2}\right)} | C = limit(sum(k*ln(k), k = 1..n)-((1)/(2)*(n)^(2)+(1)/(2)*n +(1)/(12))* ln(n)+(1)/(4)*(n)^(2), n = infinity) |
C = Limit[Sum[k*Log[k], {k, 1, n}]-(Divide[1,2]*(n)^(2)+Divide[1,2]*n +Divide[1,12])* Log[n]+Divide[1,4]*(n)^(2), n -> Infinity] |
Failure | Failure | Skip | Fail
Complex[1.165459085339311, 1.4142135623730951] <- {Rule[C, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[1.165459085339311, -1.4142135623730951] <- {Rule[C, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.6629680394068793, -1.4142135623730951] <- {Rule[C, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.6629680394068793, 1.4142135623730951] <- {Rule[C, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 5.17.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln@@{k}-\left(\tfrac{1}{2}n^{2}+\tfrac{1}{2}n+\tfrac{1}{12}\right)\ln@@{n}+\tfrac{1}{4}n^{2}\right) = \frac{\EulerConstant+\ln@{2\pi}}{12}-\frac{\Riemannzeta'@{2}}{2\pi^{2}}} | limit(sum(k*ln(k), k = 1..n)-((1)/(2)*(n)^(2)+(1)/(2)*n +(1)/(12))* ln(n)+(1)/(4)*(n)^(2), n = infinity)=(gamma + ln(2*Pi))/(12)-(subs( temp=2, diff( Zeta(temp), temp$(1) ) ))/(2*(Pi)^(2)) |
Limit[Sum[k*Log[k], {k, 1, n}]-(Divide[1,2]*(n)^(2)+Divide[1,2]*n +Divide[1,12])* Log[n]+Divide[1,4]*(n)^(2), n -> Infinity]=Divide[EulerGamma + Log[2*Pi],12]-Divide[D[Zeta[temp], {temp, 1}]/.temp-> 2,2*(Pi)^(2)] |
Failure | Successful | Skip | - |
| 5.17.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\EulerConstant+\ln@{2\pi}}{12}-\frac{\Riemannzeta'@{2}}{2\pi^{2}} = \frac{1}{12}-\Riemannzeta'@{-1}} | (gamma + ln(2*Pi))/(12)-(subs( temp=2, diff( Zeta(temp), temp$(1) ) ))/(2*(Pi)^(2))=(1)/(12)- subs( temp=- 1, diff( Zeta(temp), temp$(1) ) ) |
Divide[EulerGamma + Log[2*Pi],12]-Divide[D[Zeta[temp], {temp, 1}]/.temp-> 2,2*(Pi)^(2)]=Divide[1,12]- (D[Zeta[temp], {temp, 1}]/.temp-> - 1) |
Failure | Successful | Skip | - |
| 5.19.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle S = \digamma@{\tfrac{1}{2}}-2\digamma@{\tfrac{2}{3}}-\EulerConstant} | S = Psi((1)/(2))- 2*Psi((2)/(3))- gamma |
S = PolyGamma[Divide[1,2]]- 2*PolyGamma[Divide[2,3]]- EulerGamma |
Failure | Failure | Fail 1.318470421+1.414213562*I <- {S = 2^(1/2)+I*2^(1/2)}1.318470421-1.414213562*I <- {S = 2^(1/2)-I*2^(1/2)}-1.509956703-1.414213562*I <- {S = -2^(1/2)-I*2^(1/2)}-1.509956703+1.414213562*I <- {S = -2^(1/2)+I*2^(1/2)} |
Fail
Complex[1.3184704217228749, 1.4142135623730951] <- {Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[1.3184704217228749, -1.4142135623730951] <- {Rule[S, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.5099567030233154, -1.4142135623730951] <- {Rule[S, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.5099567030233154, 1.4142135623730951] <- {Rule[S, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 5.19.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \digamma@{\tfrac{1}{2}}-2\digamma@{\tfrac{2}{3}}-\EulerConstant = 3\ln@@{3}-2\ln@@{2}-\tfrac{1}{3}\pi\sqrt{3}} | Psi((1)/(2))- 2*Psi((2)/(3))- gamma = 3*ln(3)- 2*ln(2)-(1)/(3)*Pi*sqrt(3) |
PolyGamma[Divide[1,2]]- 2*PolyGamma[Divide[2,3]]- EulerGamma = 3*Log[3]- 2*Log[2]-Divide[1,3]*Pi*Sqrt[3] |
Successful | Successful | - | - |
| 5.19#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle V = \frac{\pi^{\frac{1}{2}n}r^{n}}{\EulerGamma@{\frac{1}{2}n+1}}} | V =((Pi)^((1)/(2)*n)* (r)^(n))/(GAMMA((1)/(2)*n + 1)) |
V =Divide[(Pi)^(Divide[1,2]*n)* (r)^(n),Gamma[Divide[1,2]*n + 1]] |
Failure | Failure | Fail -1.414213562-1.414213562*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}1.414213562-11.15215705*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}25.10958922-22.28116210*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}-1.414213562+4.242640686*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}... skip entries to safe data |
Fail
Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[1.4142135623730951, -11.152157051986077] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[25.10958923255105, -22.281162107804857] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-1.4142135623730951, -4.242640687119286] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 5.19#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle S = \frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\EulerGamma@{\frac{1}{2}n}}} | S =(2*(Pi)^((1)/(2)*n)* (r)^(n - 1))/(GAMMA((1)/(2)*n)) |
S =Divide[2*(Pi)^(Divide[1,2]*n)* (r)^(n - 1),Gamma[Divide[1,2]*n]] |
Failure | Failure | Fail -.585786438+1.414213562*I <- {S = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}-7.471552313-7.471552313*I <- {S = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}1.414213562-48.85126889*I <- {S = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}-.585786438+1.414213562*I <- {S = 2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}... skip entries to safe data |
Fail
Complex[-0.5857864376269049, 1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-7.471552313943637, -7.471552313943637] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[1.4142135623730951, -48.8512688950636] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-0.5857864376269049, -1.4142135623730951] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[S, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 5.19#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{2\pi^{\frac{1}{2}n}r^{n-1}}{\EulerGamma@{\frac{1}{2}n}} = \frac{n}{r}V} | (2*(Pi)^((1)/(2)*n)* (r)^(n - 1))/(GAMMA((1)/(2)*n))=(n)/(r)*V |
Divide[2*(Pi)^(Divide[1,2]*n)* (r)^(n - 1),Gamma[Divide[1,2]*n]]=Divide[n,r]*V |
Failure | Failure | Fail 1.000000000 <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 1}6.885765875+8.885765875*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 2}-3.+50.26548245*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)+I*2^(1/2), n = 3}2.000000000-1.000000000*I <- {V = 2^(1/2)+I*2^(1/2), r = 2^(1/2)-I*2^(1/2), n = 1}... skip entries to safe data |
Fail
1.0 <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[6.885765876316732, 8.885765876316732] <- {Rule[n, 2], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-3.0, 50.26548245743669] <- {Rule[n, 3], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[2.0, 1.0] <- {Rule[n, 1], Rule[r, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[V, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 5.20.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle W = \frac{1}{2}\sum_{\ell=1}^{n}x_{\ell}^{2}-\sum_{1\leq\ell<j\leq n}\ln@@{|x_{\ell}-x_{j}|}} | W =sum(x(x[ell])^(2), ell = 1..n)- sum(sum(ln(abs(x[ell]- x[j])), j = ell + 1..n), ell = 1..j - 1) |
W =Sum[x(Subscript[x, \[ScriptL]])^(2), {\[ScriptL], 1, n}]- Sum[Sum[Log[Abs[Subscript[x, \[ScriptL]]- Subscript[x, j]]], {j, [ScriptL] + 1, n}], {\[ScriptL], 1, j - 1}] |
Failure | Failure | Skip | Error |
| 5.20.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle W = -\sum_{1\leq\ell<j\leq n}\ln@@{|e^{i\theta_{\ell}}-e^{i\theta_{j}}|}} | W = - sum(sum(ln(abs(exp(I*theta[ell])- exp(I*theta[j]))), j = ell + 1..n), ell = 1..j - 1) |
W = - Sum[Sum[Log[Abs[Exp[I*Subscript[\[Theta], \[ScriptL]]]- Exp[I*Subscript[\[Theta], j]]]], {j, [ScriptL] + 1, n}], {\[ScriptL], 1, j - 1}] |
Failure | Failure | Skip | Error |