PolyLog

The polylogarithm $\operatorname{Li}_s(z) = \sum_{n=1}^{\infty} \dfrac{z^n}{n^s}$, the $a = 1$ slice of the Lerch transcendent.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaPolylogarithm↗MathWorldPolylogarithm↗WikidataQ1238449↗nLabpolylogarithm↗DLMF25.12.E10↗FungrimPolyLog✓↗Fungrim entry208da7✓↗52ea5f↗541e2e↗82b410↗d1a0ec↗Wolfram LanguagePolyLog!↗

PolyLog(s, z)the polylogarithm $\operatorname{Li}_s(z)$ at integer order $s$.

SageMathpolylog(s, z)✓↗SymPypolylog(s, z)↗mpmathpolylog(s, z)↗

Typed signatures

Type	Package	Applies
(complex | infinity, complex | infinity) -> number	compute-engine	always
(number, number, number?) -> number	analyticreplaces compute-engine	always

Domain: Special functions

Details

The polylogarithm $\operatorname{Li}_s(z) = \sum_{n=1}^{\infty} z^n/n^s$ (DLMF 25.12.10) generalizes $-\ln(1-z)$ (its $s = 1$ case) to arbitrary order $s$. It is the $a = 1$ slice of the Lerch transcendent, $\operatorname{Li}_s(z) = z\,\Phi(z, s, 1)$, and reduces to Zeta on its $z = 1$ edge.

• $\operatorname{Li}_s(z) = z\,\Phi(z, s, 1)$ in terms of the Lerch transcendent — see LerchPhi, which is how the non-integer orders are computed.
• $\operatorname{Li}_s(1) = \zeta(s)$: the ordinary zeta lives on the $z = 1$ edge. See Zeta.
• Low orders are elementary: $\operatorname{Li}_0(z) = z/(1-z)$, $\operatorname{Li}_1(z) = -\ln(1-z)$, $\operatorname{Li}_{-1}(z) = z/(1-z)^2$, and every negative integer order is a rational function.
• The dilogarithm $\operatorname{Li}_2$ has the special values $\operatorname{Li}_2(1) = \pi^2/6$, $\operatorname{Li}_2(-1) = -\pi^2/12$, and $\operatorname{Li}_2(\tfrac12) = \pi^2/12 - (\ln 2)^2/2$.
• Convergence: the series converges for $|z| < 1$, and on $|z| = 1$ for $\operatorname{Re}(s) > 1$. Outside the disk the function continues analytically, with a branch cut along $[1, \infty)$ — so $\operatorname{Li}_2(2) = \pi^2/4 - i\pi\ln 2$ is complex.
• Integer orders (including the continuation past $|z| = 1$) are compute-engine's own. @enumeratio/analytic adds non-integer and complex $s$ by summing the Lerch series, which reaches $|z| \leq 1$ only; outside that disk a non-integer order is left unevaluated rather than guessed.
• Certified digits: for exact real $s$ and $z$ (integers, rationals or decimals) with $|z| < 1$, $\mathrm{N}(\operatorname{Li}_s(z), d)$ is proven, not just agreed on. The Lerch series is summed in ball arithmetic with its tail bounded, and the d digits are those both ends of the enclosure round to. See N.
• As with the other numeric heads, an exact argument stays symbolic under plain evaluation — pair with N(), or pass an inexact argument.
• Differs from Wolfram: $PolyLog(2, 2)$ and other special values reduce on evaluation; Wolfram keeps PolyLog[2, 2] and needs FunctionExpand to get there. Both agree as identities.

Examples

79 more kept as data, checked against the oracles

Basic #

$\operatorname{Li}_2(1) = \zeta(2) = \pi^2/6$

live ↗

In$\operatorname{Li}_{2}(1)$Out$\frac{π^2}{6}$

$\operatorname{Li}_2(-1) = -\eta(2) = -\pi^2/12$

live ↗

In$\operatorname{Li}_{2}(-1)$Out$-\frac{π^2}{12}$

$\operatorname{Li}_4(1) = \zeta(4) = \pi^4/90$

live ↗

In$\operatorname{Li}_{4}(1)$Out$\frac{π^4}{90}$

A machine-precision argument

live ↗

In$\operatorname{Li}_{2}(0.9)$Out$1.299\,714\,723\,004\,958\,8$

$\operatorname{Li}_0(z) = z/(1-z)$, past the unit disk

live ↗

In$\operatorname{Li}_{0}(5.5)$Out$-1.\overline{2}$

$\operatorname{Li}_3(1/2) = \frac78\zeta(3) - \frac{\pi^2}{12}\ln 2 + \frac16\ln^3 2$

live ↗

In$\operatorname{Li}_{3}(\frac{1}{2})$Out$-\frac{1}{12}(\ln(2)π^2)+\frac{\ln(2)^3}{6}+\frac{7ζ(3)}{8}$

live ↗

In$\operatorname{Normal}(\operatorname{Series}(\operatorname{Li}_{2}(x), x, ∞, 6))$Out$\operatorname{Series}(\operatorname{Li}_{2}(x), x, ∞, 6)$

Scope #

Threads over a list of arguments

live ↗

In$\operatorname{Li}_{2}(\bigl\lbrack0.5, 0.25\bigr\rbrack)$Out$\bigl\lbrack0.582\,240\,526\,465\,012\,5, 0.267\,652\,639\,082\,732\,5\bigr\rbrack$

A non-integer order continues past $|z| = 1$ to a complex value, as Wolfram does: $\mathrm{Li}_{5/2}(2) \approx 2.78966 - 1.36380i$

live ↗

In$\operatorname{Li}_{2.5}(2)$Out$2.789\,660\,332\,382\,777\,8-1.363\,803\,700\,539\,351i$

Complex order and argument

live ↗

In$\operatorname{Li}_{0.2+i}(0.5-0.5i)$Out$0.070\,326\,520\,550\,516\,03-0.563\,293\,262\,539\,874\,4i$

Past $z = 1$ the continuation is complex

live ↗

In$\operatorname{Li}_{2}(2.5)$Out$2.420\,790\,806\,565\,934-2.878\,612\,231\,808\,261i$

$\operatorname{Li}_3(-1) = -\eta(3) = -\frac34\zeta(3)$

live ↗

In$\operatorname{Li}_{3}(-1)$Out$-\frac{3ζ(3)}{4}$

$\operatorname{Li}_2(2) = \pi^2/4 - i\pi\ln 2$ exactly

live ↗

In$\operatorname{Li}_{2}(2)$Out$\frac{π^2}{4}-i π\ln(2)$

$\operatorname{Li}_n(-1) = -(1 - 2^{1-n})\zeta(n)$ for symbolic $n$

live ↗

In$\operatorname{Li}_{n}(-1)$Out$(2^{1-n}-1)ζ(n)$

$\operatorname{Li}_{-2}(z) = z(1+z)/(1-z)^3$

live ↗

In$\operatorname{Li}_{-2}(z)$Out$\frac{z×(z+1)}{(1-z)^3}$

Interval arithmetic in $z$, proven: $\operatorname{Li}_2'(z) = \Phi(z, 1, 1)$ keeps one sign, so the image is the values at the ends, rounded outward

live ↗

In$\operatorname{Li}_{2}(\mathrm{Interval}(0.7, 0.8))$Out$\mathrm{Interval}(0.889\,377\,624\,286\,038, 1.074\,794\,600\,008\,25)$

Three arguments: the Nielsen generalized polylogarithm $S_{n,p}(z)$, with $S_{1,2}(1) = \zeta(3)$

live ↗

In$\operatorname{Li}_{1}(2)$Out$ζ(3)$

$S_{2,2}(1) = \pi^4/360$

live ↗

In$\operatorname{Li}_{2}(2)$Out$\frac{π^4}{360}$

$S_{1,2}(1/2) = \zeta(3)/8 - \ln^3 2/6$

live ↗

In$\operatorname{Li}_{1}(2)$Out$0.094\,753\,004\,230\,127\,7$

To 30 significant digits, past a double's reach

live ↗

In$N(\operatorname{Li}_{2}(\frac{1}{2}), 30)$Out$0.582\,240\,526\,465\,012\,505\,902\,656\,320\,16$

$\mathrm{Li}_0(z)=\frac{z}{1-z}$, so $\mathrm{Li}_0(5)=-\frac54$

live ↗

In$\operatorname{Li}_{0}(5)$Out$-\frac{5}{4}$

A 50-digit value of $\mathrm{Li}_1(1/3) = -\ln(2/3)$

live ↗

In$N(\operatorname{Li}_{1}(\frac{1}{3}), 50)$Out$0.405\,465\,108\,108\,164\,381\,978\,013\,115\,464\,349\,136\,571\,990\,423\,462\,49$

Applies to a diagonal matrix entry by entry

live ↗

In$\operatorname{Li}_{2}(\bigl\lbrack\bigl\lbrack\frac{1}{2}, 0\bigr\rbrack, \bigl\lbrack0, \frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\operatorname{Li}_{2}(\frac{1}{2}), 0\bigr\rbrack, \bigl\lbrack0, \operatorname{Li}_{2}(\frac{1}{2})\bigr\rbrack\bigr\rbrack$

$\mathrm{Li}_s(0)=0$ for any order

live ↗

In$\operatorname{Li}_{1}(0)$Out$0$

Differentiation lowers the order: $\partial_z \mathrm{Li}_n(z)=\mathrm{Li}_{n-1}(z)/z$

live ↗

In$\frac{\mathrm{d}\operatorname{Li}_{n}(z)}{\mathrm{d}z}$Out$\frac{\operatorname{Li}_{n-1}(z)}{z}$

MatrixFunction agrees with direct threading for a diagonal matrix

live ↗

In$\mathrm{MatrixFunction}(()↦\operatorname{Li}_{2}(\operatorname{\_1}), \bigl\lbrack\bigl\lbrack\frac{1}{2}, 0\bigr\rbrack, \bigl\lbrack0, \frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\operatorname{Li}_{2}(\frac{1}{2}), 0\bigr\rbrack, \bigl\lbrack0, \operatorname{Li}_{2}(\frac{1}{2})\bigr\rbrack\bigr\rbrack$

live ↗

In$\mathrm{FullSimplify}(\operatorname{Series}(\operatorname{Li}_{n}(x), x, \mathrm{x0}, 2))$Out$\operatorname{Li}_{n}(\mathrm{x0})+\frac{1}{\mathrm{x0}}((x-\mathrm{x0})\operatorname{Li}_{n-1}(\mathrm{x0}))+\frac{(\operatorname{Li}_{n-2}(\mathrm{x0})-\operatorname{Li}_{n-1}(\mathrm{x0}))(x-\mathrm{x0})^2}{2\mathrm{x0}^2}+O\left((x-\mathrm{x0})^3\right)$

Applications #

$\operatorname{Li}_{1/2}(1) = \zeta(1/2)$ — a non-integer order, from @enumeratio/analytic

live ↗

In$\operatorname{Li}_{0.5}(1)$Out$-1.460\,354\,508\,809\,586\,812\,89$

$\operatorname{Li}_{1/2}(1/2) = 0.80612672\ldots$, via the Lerch series

live ↗

In$\operatorname{Li}_{0.5}(0.5)$Out$0.806\,126\,723\,042\,852\,2$

$\operatorname{Li}_{5/2}(-1/2) = -0.46229778\ldots$ — the alternating side, Euler-transformed

live ↗

In$\operatorname{Li}_{2.5}(-0.5)$Out$-0.462\,297\,782\,190\,063\,46$

$\operatorname{Li}_2(1/3)$ to 40 digits, proven: at an exact argument the series bounds its own error, and both ends of the enclosure round to these digits

live ↗

In$N(\operatorname{Li}_{2}(\frac{1}{3}), 40)$Out$0.366\,213\,229\,977\,063\,487\,616\,746\,297\,664\,262\,763\,802\,1$

$\operatorname{Li}_{5/2}(-1/2)$ to 40 digits, proven the same way at a non-integer order

live ↗

In$N(\operatorname{Li}_{\frac{5}{2}}(-\frac{1}{2}), 40)$Out$-0.462\,297\,782\,190\,063\,438\,189\,164\,323\,274\,546\,489\,886\,9$

Properties #

$\operatorname{Li}_1(z) = -\ln(1 - z)$, symbolically

live ↗

In$\operatorname{Li}_{1}(z)$Out$-\ln(1-z)$

$\operatorname{Li}_{-1}(z) = z/(1-z)^2$ — negative orders are rational

live ↗

In$\operatorname{Li}_{-1}(z)$Out$\frac{z}{(1-z)^2}$

$\operatorname{Li}_2(1/2) = \pi^2/12 - (\ln 2)^2/2$

live ↗

In$\operatorname{Li}_{2}(0.5)+0.5\ln(2)^2=\frac{π^2}{12}$Out$\mathrm{True}$

$\operatorname{Li}_s(1) = \zeta(s)$ for symbolic $s$

live ↗

In$\operatorname{Li}_{s}(1)$Out$ζ(s)$

$\frac{d}{dx}\operatorname{Li}_n(x) = \operatorname{Li}_{n-1}(x)/x$

live ↗

In$\frac{\mathrm{d}\operatorname{Li}_{n}(x)}{\mathrm{d}x}$Out$\frac{\operatorname{Li}_{n-1}(x)}{x}$

An argument on the unit circle is rewritten in Cartesian form

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In$\operatorname{Li}_{3}(\exp(\frac{1}{3}i π))$Out$\operatorname{Li}_{3}(\frac{1}{2}+\frac{\sqrt{3}}{2}i)$

Known value · Wolfram MathematicalFunctionData, PolyLog

The conjugate point on the unit circle, $e^{-i\pi/3}$

live ↗

In$\operatorname{Li}_{3}(\exp((-\frac{1}{3}i)π))$Out$\operatorname{Li}_{3}(\frac{1}{2}-\frac{\sqrt{3}}{2}i)$

Known value · Wolfram MathematicalFunctionData, PolyLog

A point at angle $2\pi/3$ on the unit circle

live ↗

In$\operatorname{Li}_{3}(\exp(\frac{2}{3}i π))$Out$\operatorname{Li}_{3}(\frac{3^{1/2}}{2}i-\frac{1}{2})$

Known value · Wolfram MathematicalFunctionData, PolyLog

FunctionExpand only normalises the argument, the dilogarithm stays

live ↗

In$\mathrm{FunctionExpand}(\operatorname{Li}_{2}(\exp(1/3i π)))$Out$\operatorname{Li}_{2}(\frac{1}{2}+\frac{\sqrt{3}}{2}i)$

live ↗

In$\operatorname{Li}_{2}(\frac{1}{2})-\operatorname{Li}_{2}(-1)$Out$\frac{π^2}{12}+\operatorname{Li}_{2}(\frac{1}{2})$

Known value · Wolfram MathematicalFunctionData, PolyLog, FunctionalEquations

live ↗

In$-\frac{\operatorname{Li}_{2}(-\frac{1}{3})}{2}+\operatorname{Li}_{2}(-1)+\operatorname{Li}_{2}(\frac{1}{3})$Out$-\frac{\operatorname{Li}_{2}(-\frac{1}{3})}{2}-\frac{π^2}{12}+\operatorname{Li}_{2}(\frac{1}{3})$

Known value · Wolfram MathematicalFunctionData, PolyLog, FunctionalEquations

live ↗

In$\operatorname{Li}_{2}(\frac{1}{4})-2(\operatorname{Li}_{2}(-\frac{1}{2})+\operatorname{Li}_{2}(\frac{1}{2}))$Out$-2\operatorname{Li}_{2}(-\frac{1}{2})-2\operatorname{Li}_{2}(\frac{1}{2})+\operatorname{Li}_{2}(\frac{1}{4})$

Known value · Wolfram MathematicalFunctionData, PolyLog, FunctionalEquations

live ↗

In$\operatorname{Li}_{2}(-1)+\operatorname{Li}_{2}(0)+\operatorname{Li}_{2}(\frac{1}{2})$Out$\operatorname{Li}_{2}(\frac{1}{2})-\frac{π^2}{12}$

Known value · Wolfram MathematicalFunctionData, PolyLog, FunctionalEquations

Possible issues #

An exact argument stays symbolic under plain evaluation -- pair with N() or pass 0.5 instead

live ↗

In$\operatorname{Li}_{2}(\frac{1}{2})$Out$\operatorname{Li}_{2}(\frac{1}{2})$

About

Examples	125
Checked against	Wolfram Language 106 agree15 differ2 unevaluated1 errorSageMath 88 agree4 unevaluatedSymPy 107 agree8 differ3 unevaluatedmpmath 89 agree3 unevaluated
Crosswalk	WikipediaMathWorldWikidatanLabDLMFFungrimFungrim entry ×5Wolfram LanguageSageMathSymPympmath

Implementation

mappedwolframpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

See also: LerchPhi, Zeta, HurwitzZeta, Ln