Zeta and Related Functions - 25.12 Polylogarithms

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
25.12.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \dilog@{z} = -\int_{0}^{z}t^{-1}\ln@{1-t}\diff{t}}
\dilog@{z} = -\int_{0}^{z}t^{-1}\ln@{1-t}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
dilog(z) = - int((t)^(- 1)* ln(1 - t), t = 0..z)

PolyLog[2, z] == - Integrate[(t)^(- 1)* Log[1 - t], {t, 0, z}, GenerateConditions->None]
Failure Successful
Failed [6 / 7]
Result: -.8224670339-1.383979491*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: 1.644934067-2.503719574*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Successful [Tested: 7]
25.12.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \dilog@{z}+\dilog@{\frac{z}{z-1}} = -\frac{1}{2}(\ln@{1-z})^{2}}
\dilog@{z}+\dilog@{\frac{z}{z-1}} = -\frac{1}{2}(\ln@{1-z})^{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
dilog(z)+ dilog((z)/(z - 1)) = -(1)/(2)*(ln(1 - z))^(2)

PolyLog[2, z]+ PolyLog[2, Divide[z,z - 1]] == -Divide[1,2]*(Log[1 - z])^(2)
Failure Failure
Failed [1 / 1]
Result: 3.289868134-2.177586090*I
Test Values: {z = 1/2}

Successful [Tested: 1]
25.12.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \dilog@{z}+\dilog@{\frac{1}{z}} = -\frac{1}{6}\pi^{2}-\frac{1}{2}(\ln@{-z})^{2}}
\dilog@{z}+\dilog@{\frac{1}{z}} = -\frac{1}{6}\pi^{2}-\frac{1}{2}(\ln@{-z})^{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
dilog(z)+ dilog((1)/(z)) = -(1)/(6)*(Pi)^(2)-(1)/(2)*(ln(- z))^(2)

PolyLog[2, z]+ PolyLog[2, Divide[1,z]] == -Divide[1,6]*(Pi)^(2)-Divide[1,2]*(Log[- z])^(2)
Failure Failure
Failed [1 / 1]
Result: 6.579736268-4.725198502*I
Test Values: {z = -1/2}

Successful [Tested: 1]
25.12.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \dilog@{z^{m}} = m\sum_{k=0}^{m-1}\dilog@{ze^{2\pi ik/m}}}
\dilog@{z^{m}} = m\sum_{k=0}^{m-1}\dilog@{ze^{2\pi ik/m}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |z| < 1} 
dilog((z)^(m)) = m*sum(dilog(z*exp(2*Pi*I*k/m)), k = 0..m - 1)

PolyLog[2, (z)^(m)] == m*Sum[PolyLog[2, z*Exp[2*Pi*I*k/m]], {k, 0, m - 1}, GenerateConditions->None]
Failure Failure
Failed [1 / 1]
Result: -8.968925063+0.*I
Test Values: {z = 1/2, m = 3}

Successful [Tested: 1]
25.12.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \dilog@{x}+\dilog@{1-x} = \frac{1}{6}\pi^{2}-(\ln@@{x})\ln@{1-x}}
\dilog@{x}+\dilog@{1-x} = \frac{1}{6}\pi^{2}-(\ln@@{x})\ln@{1-x}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < x, x < 1} 
dilog(x)+ dilog(1 - x) = (1)/(6)*(Pi)^(2)-(ln(x))*ln(1 - x)

PolyLog[2, x]+ PolyLog[2, 1 - x] == Divide[1,6]*(Pi)^(2)-(Log[x])*Log[1 - x]
Successful Successful - Successful [Tested: 1]
25.12.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \dilog@{e^{i\theta}} = \sum_{n=1}^{\infty}\frac{\cos@{n\theta}}{n^{2}}+i\sum_{n=1}^{\infty}\frac{\sin@{n\theta}}{n^{2}}}
\dilog@{e^{i\theta}} = \sum_{n=1}^{\infty}\frac{\cos@{n\theta}}{n^{2}}+i\sum_{n=1}^{\infty}\frac{\sin@{n\theta}}{n^{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
dilog(exp(I*theta)) = sum((cos(n*theta))/((n)^(2)), n = 1..infinity)+ I*sum((sin(n*theta))/((n)^(2)), n = 1..infinity)

PolyLog[2, Exp[I*\[Theta]]] == Sum[Divide[Cos[n*\[Theta]],(n)^(2)], {n, 1, Infinity}, GenerateConditions->None]+ I*Sum[Divide[Sin[n*\[Theta]],(n)^(2)], {n, 1, Infinity}, GenerateConditions->None]
Failure Successful Skipped - Because timed out Successful [Tested: 10]
25.12.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=1}^{\infty}\frac{\cos@{n\theta}}{n^{2}} = \frac{\pi^{2}}{6}-\frac{\pi\theta}{2}+\frac{\theta^{2}}{4}}
\sum_{n=1}^{\infty}\frac{\cos@{n\theta}}{n^{2}} = \frac{\pi^{2}}{6}-\frac{\pi\theta}{2}+\frac{\theta^{2}}{4}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
sum((cos(n*theta))/((n)^(2)), n = 1..infinity) = ((Pi)^(2))/(6)-(Pi*theta)/(2)+((theta)^(2))/(4)

Sum[Divide[Cos[n*\[Theta]],(n)^(2)], {n, 1, Infinity}, GenerateConditions->None] == Divide[(Pi)^(2),6]-Divide[Pi*\[Theta],2]+Divide[\[Theta]^(2),4]
Failure Failure Skipped - Because timed out
Failed [5 / 10]
Result: Complex[-1.5707963267948957, 2.720699046351327]
Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-2.720699046351327, -1.5707963267948966]
Test Values: {Rule[θ, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}

... skip entries to safe data
25.12.E9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=1}^{\infty}\frac{\sin@{n\theta}}{n^{2}} = -\int_{0}^{\theta}\ln@{2\sin@{\tfrac{1}{2}x}}\diff{x}}
\sum_{n=1}^{\infty}\frac{\sin@{n\theta}}{n^{2}} = -\int_{0}^{\theta}\ln@{2\sin@{\tfrac{1}{2}x}}\diff{x}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
sum((sin(n*theta))/((n)^(2)), n = 1..infinity) = - int(ln(2*sin((1)/(2)*x)), x = 0..theta)

Sum[Divide[Sin[n*\[Theta]],(n)^(2)], {n, 1, Infinity}, GenerateConditions->None] == - Integrate[Log[2*Sin[Divide[1,2]*x]], {x, 0, \[Theta]}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
25.12.E10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \polylog{s}@{z} = \sum_{n=1}^{\infty}\frac{z^{n}}{n^{s}}}
\polylog{s}@{z} = \sum_{n=1}^{\infty}\frac{z^{n}}{n^{s}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
polylog(s, z) = sum(((z)^(n))/((n)^(s)), n = 1..infinity)

PolyLog[s, z] == Sum[Divide[(z)^(n),(n)^(s)], {n, 1, Infinity}, GenerateConditions->None]
Failure Successful
Failed [12 / 42]
Result: Float(-infinity)-12.69850170*I
Test Values: {s = -3/2, z = 3/2}


Result: Float(-infinity)-3.323322953*I
Test Values: {s = -3/2, z = 2}

... skip entries to safe data
 Successful [Tested: 42]
25.12.E12 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \polylog{s}@{z} = \EulerGamma@{1-s}\left(\ln@@{\frac{1}{z}}\right)^{s-1}+\sum_{n=0}^{\infty}\Riemannzeta@{s-n}\frac{(\ln@@{z})^{n}}{n!}}
\polylog{s}@{z} = \EulerGamma@{1-s}\left(\ln@@{\frac{1}{z}}\right)^{s-1}+\sum_{n=0}^{\infty}\Riemannzeta@{s-n}\frac{(\ln@@{z})^{n}}{n!}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\ln@@{z}| < 2\pi, \realpart@@{(1-s)} > 0} 
polylog(s, z) = GAMMA(1 - s)*(ln((1)/(z)))^(s - 1)+ sum(Zeta(s - n)*((ln(z))^(n))/(factorial(n)), n = 0..infinity)

PolyLog[s, z] == Gamma[1 - s]*(Log[Divide[1,z]])^(s - 1)+ Sum[Zeta[s - n]*Divide[(Log[z])^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]
Failure Failure Successful [Tested: 7] Skipped - Because timed out
25.12.E13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \polylog{s}@{e^{2\pi ia}}+e^{\pi is}\polylog{s}@{e^{-2\pi ia}} = \frac{(2\pi)^{s}e^{\pi is/2}}{\EulerGamma@{s}}\Hurwitzzeta@{1-s}{a}}
\polylog{s}@{e^{2\pi ia}}+e^{\pi is}\polylog{s}@{e^{-2\pi ia}} = \frac{(2\pi)^{s}e^{\pi is/2}}{\EulerGamma@{s}}\Hurwitzzeta@{1-s}{a}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{s} > 0} 
polylog(s, exp(2*Pi*I*a))+ exp(Pi*I*s)*polylog(s, exp(- 2*Pi*I*a)) = ((2*Pi)^(s)* exp(Pi*I*s/2))/(GAMMA(s))*Zeta(0, 1 - s, a)

PolyLog[s, Exp[2*Pi*I*a]]+ Exp[Pi*I*s]*PolyLog[s, Exp[- 2*Pi*I*a]] == Divide[(2*Pi)^(s)* Exp[Pi*I*s/2],Gamma[s]]*HurwitzZeta[1 - s, a]
Failure Failure
Failed [15 / 18]
Result: 24.27636385+24.27636386*I
Test Values: {a = -3/2, s = 3/2}


Result: -2.230710143+2.230710142*I
Test Values: {a = -3/2, s = 1/2}

... skip entries to safe data
 Skip - No test values generated
25.12#Ex1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle F_{s}(x) = -\polylog{s+1}@{-e^{x}}}
F_{s}(x) = -\polylog{s+1}@{-e^{x}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle s > -1, \realpart@@{(s+1)} > 0, x < 0, s > 0, x \leq 0} 
((1)/(GAMMA(s + 1))*int(((t)^(s))/(exp(t - x)+ 1), t = 0..infinity)) = - polylog(s + 1, - exp(x))

(Divide[1,Gamma[s + 1]]*Integrate[Divide[(t)^(s),Exp[t - x]+ 1], {t, 0, Infinity}, GenerateConditions->None]) == - PolyLog[s + 1, - Exp[x]]
Failure Successful Error Successful [Tested: 0]
25.12#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle G_{s}(x) = \polylog{s+1}@{e^{x}}}
G_{s}(x) = \polylog{s+1}@{e^{x}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle s > -1, \realpart@@{(s+1)} > 0, x < 0, s > 0, x \leq 0} 
((1)/(GAMMA(s + 1))*int(((t)^(s))/(exp(t - x)- 1), t = 0..infinity)) = polylog(s + 1, exp(x))

(Divide[1,Gamma[s + 1]]*Integrate[Divide[(t)^(s),Exp[t - x]- 1], {t, 0, Infinity}, GenerateConditions->None]) == PolyLog[s + 1, Exp[x]]
Failure Successful Error Successful [Tested: 0]
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