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 Li 2 ⁡ ( z ) = - ∫ 0 z t - 1 ⁢ ln ⁡ ( 1 - t ) ⁢ d t dilogarithm 𝑧 superscript subscript 0 𝑧 superscript 𝑡 1 1 𝑡 𝑡 {\displaystyle{\displaystyle\mathrm{Li}_{2}\left(z\right)=-\int_{0}^{z}t^{-1}% \ln\left(1-t\right)\mathrm{d}t}}
\dilog@{z} = -\int_{0}^{z}t^{-1}\ln@{1-t}\diff{t}		 

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 Li 2 ⁡ ( z ) + Li 2 ⁡ ( z z - 1 ) = - 1 2 ⁢ ( ln ⁡ ( 1 - z ) ) 2 dilogarithm 𝑧 dilogarithm 𝑧 𝑧 1 1 2 superscript 1 𝑧 2 {\displaystyle{\displaystyle\mathrm{Li}_{2}\left(z\right)+\mathrm{Li}_{2}\left% (\frac{z}{z-1}\right)=-\frac{1}{2}(\ln\left(1-z\right))^{2}}}
\dilog@{z}+\dilog@{\frac{z}{z-1}} = -\frac{1}{2}(\ln@{1-z})^{2}		 

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 Li 2 ⁡ ( z ) + Li 2 ⁡ ( 1 z ) = - 1 6 ⁢ π 2 - 1 2 ⁢ ( ln ⁡ ( - z ) ) 2 dilogarithm 𝑧 dilogarithm 1 𝑧 1 6 superscript 𝜋 2 1 2 superscript 𝑧 2 {\displaystyle{\displaystyle\mathrm{Li}_{2}\left(z\right)+\mathrm{Li}_{2}\left% (\frac{1}{z}\right)=-\frac{1}{6}\pi^{2}-\frac{1}{2}(\ln\left(-z\right))^{2}}}
\dilog@{z}+\dilog@{\frac{1}{z}} = -\frac{1}{6}\pi^{2}-\frac{1}{2}(\ln@{-z})^{2}		 

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 Li 2 ⁡ ( z m ) = m ⁢ ∑ k = 0 m - 1 Li 2 ⁡ ( z ⁢ e 2 ⁢ π ⁢ i ⁢ k / m ) dilogarithm superscript 𝑧 𝑚 𝑚 superscript subscript 𝑘 0 𝑚 1 dilogarithm 𝑧 superscript 𝑒 2 𝜋 𝑖 𝑘 𝑚 {\displaystyle{\displaystyle\mathrm{Li}_{2}\left(z^{m}\right)=m\sum_{k=0}^{m-1% }\mathrm{Li}_{2}\left(ze^{2\pi ik/m}\right)}}
\dilog@{z^{m}} = m\sum_{k=0}^{m-1}\dilog@{ze^{2\pi ik/m}}		 | z | < 1 𝑧 1 {\displaystyle{\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 Li 2 ⁡ ( x ) + Li 2 ⁡ ( 1 - x ) = 1 6 ⁢ π 2 - ( ln ⁡ x ) ⁢ ln ⁡ ( 1 - x ) dilogarithm 𝑥 dilogarithm 1 𝑥 1 6 superscript 𝜋 2 𝑥 1 𝑥 {\displaystyle{\displaystyle\mathrm{Li}_{2}\left(x\right)+\mathrm{Li}_{2}\left% (1-x\right)=\frac{1}{6}\pi^{2}-(\ln x)\ln\left(1-x\right)}}
\dilog@{x}+\dilog@{1-x} = \frac{1}{6}\pi^{2}-(\ln@@{x})\ln@{1-x}		 0 < x , x < 1 formulae-sequence 0 𝑥 𝑥 1 {\displaystyle{\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 Li 2 ⁡ ( e i ⁢ θ ) = ∑ n = 1 ∞ cos ⁡ ( n ⁢ θ ) n 2 + i ⁢ ∑ n = 1 ∞ sin ⁡ ( n ⁢ θ ) n 2 dilogarithm superscript 𝑒 𝑖 𝜃 superscript subscript 𝑛 1 𝑛 𝜃 superscript 𝑛 2 𝑖 superscript subscript 𝑛 1 𝑛 𝜃 superscript 𝑛 2 {\displaystyle{\displaystyle\mathrm{Li}_{2}\left(e^{i\theta}\right)=\sum_{n=1}% ^{\infty}\frac{\cos\left(n\theta\right)}{n^{2}}+i\sum_{n=1}^{\infty}\frac{\sin% \left(n\theta\right)}{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}}		 

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 ∑ n = 1 ∞ cos ⁡ ( n ⁢ θ ) n 2 = π 2 6 - π ⁢ θ 2 + θ 2 4 superscript subscript 𝑛 1 𝑛 𝜃 superscript 𝑛 2 superscript 𝜋 2 6 𝜋 𝜃 2 superscript 𝜃 2 4 {\displaystyle{\displaystyle\sum_{n=1}^{\infty}\frac{\cos\left(n\theta\right)}% {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}		 

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 ∑ n = 1 ∞ sin ⁡ ( n ⁢ θ ) n 2 = - ∫ 0 θ ln ⁡ ( 2 ⁢ sin ⁡ ( 1 2 ⁢ x ) ) ⁢ d x superscript subscript 𝑛 1 𝑛 𝜃 superscript 𝑛 2 superscript subscript 0 𝜃 2 1 2 𝑥 𝑥 {\displaystyle{\displaystyle\sum_{n=1}^{\infty}\frac{\sin\left(n\theta\right)}% {n^{2}}=-\int_{0}^{\theta}\ln\left(2\sin\left(\tfrac{1}{2}x\right)\right)% \mathrm{d}x}}
\sum_{n=1}^{\infty}\frac{\sin@{n\theta}}{n^{2}} = -\int_{0}^{\theta}\ln@{2\sin@{\tfrac{1}{2}x}}\diff{x}		 

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 Li s ⁡ ( z ) = ∑ n = 1 ∞ z n n s polylogarithm 𝑠 𝑧 superscript subscript 𝑛 1 superscript 𝑧 𝑛 superscript 𝑛 𝑠 {\displaystyle{\displaystyle\mathrm{Li}_{s}\left(z\right)=\sum_{n=1}^{\infty}% \frac{z^{n}}{n^{s}}}}
\polylog{s}@{z} = \sum_{n=1}^{\infty}\frac{z^{n}}{n^{s}}		 

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 Li s ⁡ ( z ) = Γ ⁡ ( 1 - s ) ⁢ ( ln ⁡ 1 z ) s - 1 + ∑ n = 0 ∞ ζ ⁡ ( s - n ) ⁢ ( ln ⁡ z ) n n ! polylogarithm 𝑠 𝑧 Euler-Gamma 1 𝑠 superscript 1 𝑧 𝑠 1 superscript subscript 𝑛 0 Riemann-zeta 𝑠 𝑛 superscript 𝑧 𝑛 𝑛 {\displaystyle{\displaystyle\mathrm{Li}_{s}\left(z\right)=\Gamma\left(1-s% \right)\left(\ln\frac{1}{z}\right)^{s-1}+\sum_{n=0}^{\infty}\zeta\left(s-n% \right)\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!}		 | ln ⁡ z | < 2 ⁢ π , ℜ ⁡ ( 1 - s ) > 0 formulae-sequence 𝑧 2 𝜋 1 𝑠 0 {\displaystyle{\displaystyle|\ln z|<2\pi,\Re(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 Li s ⁡ ( e 2 ⁢ π ⁢ i ⁢ a ) + e π ⁢ i ⁢ s ⁢ Li s ⁡ ( e - 2 ⁢ π ⁢ i ⁢ a ) = ( 2 ⁢ π ) s ⁢ e π ⁢ i ⁢ s / 2 Γ ⁡ ( s ) ⁢ ζ ⁡ ( 1 - s , a ) polylogarithm 𝑠 superscript 𝑒 2 𝜋 𝑖 𝑎 superscript 𝑒 𝜋 𝑖 𝑠 polylogarithm 𝑠 superscript 𝑒 2 𝜋 𝑖 𝑎 superscript 2 𝜋 𝑠 superscript 𝑒 𝜋 𝑖 𝑠 2 Euler-Gamma 𝑠 Hurwitz-zeta 1 𝑠 𝑎 {\displaystyle{\displaystyle\mathrm{Li}_{s}\left(e^{2\pi ia}\right)+e^{\pi is}% \mathrm{Li}_{s}\left(e^{-2\pi ia}\right)=\frac{(2\pi)^{s}e^{\pi is/2}}{\Gamma% \left(s\right)}\zeta\left(1-s,a\right)}}
\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}		 ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re 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 F s ⁢ ( x ) = - Li s + 1 ⁡ ( - e x ) subscript 𝐹 𝑠 𝑥 polylogarithm 𝑠 1 superscript 𝑒 𝑥 {\displaystyle{\displaystyle F_{s}(x)=-\mathrm{Li}_{s+1}\left(-e^{x}\right)}}
F_{s}(x) = -\polylog{s+1}@{-e^{x}}		 s > - 1 , ℜ ⁡ ( s + 1 ) > 0 , x < 0 , s > 0 , x ≤ 0 formulae-sequence 𝑠 1 formulae-sequence 𝑠 1 0 formulae-sequence 𝑥 0 formulae-sequence 𝑠 0 𝑥 0 {\displaystyle{\displaystyle s>-1,\Re(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 G s ⁢ ( x ) = Li s + 1 ⁡ ( e x ) subscript 𝐺 𝑠 𝑥 polylogarithm 𝑠 1 superscript 𝑒 𝑥 {\displaystyle{\displaystyle G_{s}(x)=\mathrm{Li}_{s+1}\left(e^{x}\right)}}
G_{s}(x) = \polylog{s+1}@{e^{x}}		 s > - 1 , ℜ ⁡ ( s + 1 ) > 0 , x < 0 , s > 0 , x ≤ 0 formulae-sequence 𝑠 1 formulae-sequence 𝑠 1 0 formulae-sequence 𝑥 0 formulae-sequence 𝑠 0 𝑥 0 {\displaystyle{\displaystyle s>-1,\Re(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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