Zeta and Related Functions - 25.14 Lerch’s Transcendent
| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 25.14.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Hurwitzzeta@{s}{a} = \LerchPhi@{1}{s}{a}}
\Hurwitzzeta@{s}{a} = \LerchPhi@{1}{s}{a} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{s} > 1} | Zeta(0, s, a) = LerchPhi(1, s, a)
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HurwitzZeta[s, a] == LerchPhi[1, s, a]
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Successful | Failure | - | Successful [Tested: 2] |
| 25.14.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \polylog{s}@{z} = z\LerchPhi@{z}{s}{1}}
\polylog{s}@{z} = z\LerchPhi@{z}{s}{1} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{s} > 1, |z| \leq 1} | polylog(s, z) = z*LerchPhi(z, s, 1)
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PolyLog[s, z] == z*LerchPhi[z, s, 1]
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Successful | Successful | - | Successful [Tested: 10] |
| 25.14.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LerchPhi@{z}{s}{a} = z^{m}\LerchPhi@{z}{s}{a+m}+\sum_{n=0}^{m-1}\frac{z^{n}}{(a+n)^{s}}}
\LerchPhi@{z}{s}{a} = z^{m}\LerchPhi@{z}{s}{a+m}+\sum_{n=0}^{m-1}\frac{z^{n}}{(a+n)^{s}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | LerchPhi(z, s, a) = (z)^(m)* LerchPhi(z, s, a + m)+ sum(((z)^(n))/((a + n)^(s)), n = 0..m - 1)
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LerchPhi[z, s, a] == (z)^(m)* LerchPhi[z, s, a + m]+ Sum[Divide[(z)^(n),(a + n)^(s)], {n, 0, m - 1}, GenerateConditions->None]
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Failure | Successful | Failed [6 / 300] Result: .27656730e-2-.27656730e-2*I
Test Values: {a = -3/2, s = -2, z = 1/2*3^(1/2)+1/2*I, m = 2}
Result: 0.+.228647547e-1*I
Test Values: {a = -3/2, s = -2, z = -1/2+1/2*I*3^(1/2), m = 2}
... skip entries to safe data |
Successful [Tested: 300] |
| 25.14.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LerchPhi@{z}{s}{a} = \frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1-ze^{-x}}\diff{x}}
\LerchPhi@{z}{s}{a} = \frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1-ze^{-x}}\diff{x} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{s} > 0, \realpart@@{a} > 0} | LerchPhi(x + y*I, s, a) = (1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(1 -(x + y*I)*exp(- x)), x = 0..infinity)
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LerchPhi[x + y*I, s, a] == Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],1 -(x + y*I)*Exp[- x]], {x, 0, Infinity}, GenerateConditions->None]
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Failure | Aborted | Skipped - Because timed out | Failed [162 / 162]
Result: Plus[Complex[0.29818646299224294, -0.45270555517796296], Times[-1.1283791670955126, NIntegrate[Complex[0.15484016278663867, -0.07789552790412994]
Test Values: {1.5, 0, DirectedInfinity[1]}]]], {Rule[a, 1.5], Rule[s, 1.5], Rule[x, 1.5], Rule[y, -1.5], Rule[z, Rational[1, 2]]}
Result: Plus[Complex[0.29818646299224244, 0.45270555517796246], Times[-1.1283791670955126, NIntegrate[Complex[0.15484016278663867, 0.07789552790412994]
Test Values: {1.5, 0, DirectedInfinity[1]}]]], {Rule[a, 1.5], Rule[s, 1.5], Rule[x, 1.5], Rule[y, 1.5], Rule[z, Rational[1, 2]]}
... skip entries to safe data |
| 25.14.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LerchPhi@{z}{s}{a} = \frac{1}{2}a^{-s}+\int_{0}^{\infty}\frac{z^{x}}{(a+x)^{s}}\diff{x}-2\int_{0}^{\infty}\frac{\sin@{x\ln@@{z}-s\atan@{x/a}}}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\diff{x}}
\LerchPhi@{z}{s}{a} = \frac{1}{2}a^{-s}+\int_{0}^{\infty}\frac{z^{x}}{(a+x)^{s}}\diff{x}-2\int_{0}^{\infty}\frac{\sin@{x\ln@@{z}-s\atan@{x/a}}}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\diff{x} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{s} > 0, |z| < 1, \realpart@@{s} > 1, |z| = 1, \realpart@@{a} > 0} | LerchPhi(x + y*I, s, a) = (1)/(2)*(a)^(- s)+ int(((x + y*I)^(x))/((a + x)^(s)), x = 0..infinity)- 2*int((sin(x*ln(x + y*I)- s*arctan(x/a)))/(((a)^(2)+ (x)^(2))^(s/2)*(exp(2*Pi*x)- 1)), x = 0..infinity)
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LerchPhi[x + y*I, s, a] == Divide[1,2]*(a)^(- s)+ Integrate[Divide[(x + y*I)^(x),(a + x)^(s)], {x, 0, Infinity}, GenerateConditions->None]- 2*Integrate[Divide[Sin[x*Log[x + y*I]- s*ArcTan[x/a]],((a)^(2)+ (x)^(2))^(s/2)*(Exp[2*Pi*x]- 1)], {x, 0, Infinity}, GenerateConditions->None]
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Error | Aborted | - | Skipped - Because timed out |