Gamma Function - 5.17 Barnes’ -Function (Double Gamma Function)

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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}}
\BarnesG@{z+1} = \EulerGamma@{z}\BarnesG@{z}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{z} > 0} 
Error

BarnesG[z + 1] == Gamma[z]*BarnesG[z]
Missing Macro Error Failure - Successful [Tested: 5]
5.17#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BarnesG@{1} = 1}
\BarnesG@{1} = 1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

BarnesG[1] == 1
Missing Macro Error Successful - Successful [Tested: 1]
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)}
\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)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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}, GenerateConditions->None]
Missing Macro Error Successful - Successful [Tested: 7]
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}}
\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(z+1)} > 0, \realpart@@{(t+1)} > 0} 
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}, GenerateConditions->None]
Missing Macro Error Failure - Successful [Tested: 7]
5.17.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A = e^{C}}
A = e^{C}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
A = exp(C)
 
A == Exp[C]
 Skipped - no semantic math Skipped - no semantic math - -
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 = \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)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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}, GenerateConditions->None]-(Divide[1,2]*(n)^(2)+Divide[1,2]*n +Divide[1,12])*Log[n]+Divide[1,4]*(n)^(2), n -> Infinity, GenerateConditions->None]
Failure Failure
Failed [10 / 10]
Result: .6172709270+.5000000000*I
Test Values: {C = 1/2*3^(1/2)+1/2*I}


Result: -.7487544770+.8660254040*I
Test Values: {C = -1/2+1/2*I*3^(1/2)}


Result: .2512455230-.8660254040*I
Test Values: {C = 1/2-1/2*I*3^(1/2)}


Result: -1.114779881-.5000000000*I
Test Values: {C = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [10 / 10]
Result: Complex[0.6172709267506544, 0.49999999999999994]
Test Values: {Rule[C, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.7487544770337841, 0.8660254037844387]
Test Values: {Rule[C, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
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}}}
\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}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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}, GenerateConditions->None]-(Divide[1,2]*(n)^(2)+Divide[1,2]*n +Divide[1,12])*Log[n]+Divide[1,4]*(n)^(2), n -> Infinity, GenerateConditions->None] == Divide[EulerGamma + Log[2*Pi],12]-Divide[D[Zeta[temp], {temp, 1}]/.temp-> 2,2*(Pi)^(2)]
Failure Successful Successful [Tested: 0] Successful [Tested: 1]
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}}
\frac{\EulerConstant+\ln@{2\pi}}{12}-\frac{\Riemannzeta'@{2}}{2\pi^{2}} = \frac{1}{12}-\Riemannzeta'@{-1}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(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 Successful [Tested: 0] Successful [Tested: 1]
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