25.12

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
25.12.E2	${\displaystyle{\displaystyle\mathrm{Li}_{2}\left(z\right)=-\int_{0}^{z}t^{-1}\ln\left(1-t\right)\mathrm{d}t}}$ \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	${\displaystyle{\displaystyle\mathrm{Li}_{2}\left(z\right)+\mathrm{Li}_{2}\left(\frac{z}{z-1}\right)=-\frac{1}{2}(\ln\left(1-z\right))^{2}}}$ \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	${\displaystyle{\displaystyle\mathrm{Li}_{2}\left(z\right)+\mathrm{Li}_{2}\left(\frac{1}{z}\right)=-\frac{1}{6}\pi^{2}-\frac{1}{2}(\ln\left(-z\right))^{2}}}$ \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	${\displaystyle{\displaystyle\mathrm{Li}_{2}\left(z^{m}\right)=m\sum_{k=0}^{m-1}\mathrm{Li}_{2}\left(ze^{2\pi ik/m}\right)}}$ \dilog@{z^{m}} = m\sum_{k=0}^{m-1}\dilog@{ze^{2\pi ik/m}}	${\displaystyle{\displaystyle|z|<1}}$	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	${\displaystyle{\displaystyle\mathrm{Li}_{2}\left(x\right)+\mathrm{Li}_{2}\left(1-x\right)=\frac{1}{6}\pi^{2}-(\ln x)\ln\left(1-x\right)}}$ \dilog@{x}+\dilog@{1-x} = \frac{1}{6}\pi^{2}-(\ln@@{x})\ln@{1-x}	${\displaystyle{\displaystyle 0<x,x<1}}$	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	${\displaystyle{\displaystyle\mathrm{Li}_{2}\left(e^{i\theta}\right)=\sum_{n=1}^{\infty}\frac{\cos\left(n\theta\right)}{n^{2}}+i\sum_{n=1}^{\infty}\frac{\sin\left(n\theta\right)}{n^{2}}}}$ \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	${\displaystyle{\displaystyle\sum_{n=1}^{\infty}\frac{\cos\left(n\theta\right)}{n^{2}}=\frac{\pi^{2}}{6}-\frac{\pi\theta}{2}+\frac{\theta^{2}}{4}}}$ \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	${\displaystyle{\displaystyle\sum_{n=1}^{\infty}\frac{\sin\left(n\theta\right)}{n^{2}}=-\int_{0}^{\theta}\ln\left(2\sin\left(\tfrac{1}{2}x\right)\right)\mathrm{d}x}}$ \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	${\displaystyle{\displaystyle\mathrm{Li}_{s}\left(z\right)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{s}}}}$ \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	${\displaystyle{\displaystyle\mathrm{Li}_{s}\left(z\right)=\Gamma\left(1-s\right)\left(\ln\frac{1}{z}\right)^{s-1}+\sum_{n=0}^{\infty}\zeta\left(s-n\right)\frac{(\ln z)^{n}}{n!}}}$ \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!}	${\displaystyle{\displaystyle|\ln z|<2\pi,\Re(1-s)>0}}$	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	${\displaystyle{\displaystyle\mathrm{Li}_{s}\left(e^{2\pi ia}\right)+e^{\pi is}\mathrm{Li}_{s}\left(e^{-2\pi ia}\right)=\frac{(2\pi)^{s}e^{\pi is/2}}{\Gamma\left(s\right)}\zeta\left(1-s,a\right)}}$ \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}	${\displaystyle{\displaystyle\Re s>0}}$	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	${\displaystyle{\displaystyle F_{s}(x)=-\mathrm{Li}_{s+1}\left(-e^{x}\right)}}$ F_{s}(x) = -\polylog{s+1}@{-e^{x}}	${\displaystyle{\displaystyle s>-1,\Re(s+1)>0,x<0,s>0,x\leq 0}}$	((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	${\displaystyle{\displaystyle G_{s}(x)=\mathrm{Li}_{s+1}\left(e^{x}\right)}}$ G_{s}(x) = \polylog{s+1}@{e^{x}}	${\displaystyle{\displaystyle s>-1,\Re(s+1)>0,x<0,s>0,x\leq 0}}$	((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]