LerchPhi

The Lerch transcendent $\Phi(z, s, a) = \sum_{n=0}^{\infty} \dfrac{z^n}{(n+a)^s}$ — a common generalization of the polylogarithm and the Hurwitz zeta. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaLerch transcendent↗MathWorldLerchTranscendent↗WikidataQ1546155↗DLMF25.14.E1↗FungrimLerchPhi↗Fungrim entryd6fbc8↗Wolfram LanguageLerchPhi!↗

LerchPhi(z, s, a)the Lerch transcendent $\Phi(z, s, a)$.

SymPylerchphi(z, s, a)↗mpmathlerchphi(z, s, a)↗Juliaenumeratio_lerch(z, s, a)

Typed signatures

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

Domain: Special functions

Details

• At $z = 1$ it is the Hurwitz zeta: $\Phi(1, s, a) = \zeta(s, a)$, so $\Phi(1, s, 1) = \zeta(s)$. See HurwitzZeta.
• It carries the polylogarithm: $\operatorname{Li}_s(z) = z\,\Phi(z, s, 1)$ — e.g. $\operatorname{Li}_2(1/2) = \tfrac12\Phi(1/2, 2, 1) = \pi^2/12 - (\ln 2)^2/2$.
• At $z = -1$ it is the Dirichlet eta: $\Phi(-1, s, 1) = \eta(s) = \sum_{n\ge1} (-1)^{n-1} n^{-s} = (1 - 2^{1-s})\zeta(s)$, so $\Phi(-1, 1, 1) = \ln 2$.
• And the Dirichlet beta / Catalan's constant: $\Phi(-1, 2, \tfrac12) = 4\beta(2) = 4G = 3.6638\ldots$
• $\Phi(z, 0, a) = 1/(1 - z)$ — the geometric series (and its continuation), independent of $a$, for all $z \ne 1$.
• Convergence: the series converges for $|z| < 1$ (any $s$, $a$), and on $|z| = 1$ only for $\operatorname{Re}(s) > 1$; elsewhere it is defined by analytic continuation in $z$.
• Poles at $a = 0, -1, -2, \ldots$, from the singular $(n+a) = 0$ term, as for HurwitzZeta.
• Numeric evaluation sums the series directly for $|z| < 1$ (geometric convergence), routes $z = 1$ through the Euler–Maclaurin Hurwitz kernel, and sums real $z < 0$ (the $z = -1$ rim included) by a van Wijngaarden Euler transform, so the alternating $\eta(s)$ and Catalan cases reach machine precision. Past $|z| = 1$ it is continued by the Hermite-type integral representation, $\Phi = \tfrac{1}{2a^s} + z^{-a}(-\ln z)^{s-1}\Gamma(1-s, -a\ln z) - 2\int_0^\infty \frac{\sin(t\ln z - s\arctan(t/a))}{(a^2+t^2)^{s/2}(e^{2\pi t}-1)}\,dt$, with other $a$ shifted by $\Phi(z, s, a) = a^{-s} + z\Phi(z, s, a+1)$; on the cut, real $z > 1$, it takes the side below, as mpmath and Wolfram do. Where the terms cancel below double precision, or the incomplete gamma can't be trusted (near the negative real axis past $|x| \approx 20$), it stays unevaluated rather than guess.
• Certified digits: for exact real $z$, $s$ and $a$ (integers, rationals or decimals) with $|z| < 1$ and $a > 0$, $\mathrm{N}(\Phi(z, s, a), d)$ is proven. The series is summed in ball arithmetic, and its tail after a term $t$ is bounded by $|t| R/(1-R)$, where $R = |z|\,e^{\max(0, -s)/(n+a)}$ bounds every later ratio of terms; the d digits are those both ends of the enclosure round to. See N.

Plot

ComplexPlot

real slice

Examples

385 more kept as data, checked against the oracles

Basic #

$\Phi(1, 2, 1) = \zeta(2) = \pi^2/6$ — reduces to the ordinary zeta

live ↗

In$\mathrm{LerchPhi}(1, 2, 1)$Out$\frac{π^2}{6}$

$\Phi(1, 2, 2) = \zeta(2, 2) = \pi^2/6 - 1$

live ↗

In$\mathrm{LerchPhi}(1, 2, 2)$Out$\frac{π^2}{6}-1$

Scope #

past the unit disk, by the integral representation (mpmath: 0.00202500995700991 + 0.00332789753681356i)

live ↗

In$N(\mathrm{LerchPhi}(1+2i, 3-i, 4+2i))$Out$0.002\,025\,009\,957\,009\,905\,3+0.003\,327\,897\,536\,813\,557i$

Inexact arguments evaluate numerically without N() — a float operand means a float answer, as HurwitzZeta and Zeta already do

live ↗

In$\mathrm{LerchPhi}(0.5, 2, 3.5)$Out$0.119\,386\,226\,869\,820\,48$

$s = 0$ far outside the unit disk: $1/(1 - 49.5)$

live ↗

In$\mathrm{LerchPhi}(49.5, 0, 2)$Out$-0.020\,618\,556\,701\,030\,927\,835\,1$

$|z| > 1$ by analytic continuation: $\Phi(z, 1, 2) = (-\ln(1-z) - z)/z^2$ at $z = 7.5$, via the Hermite-integral continuation (lerch-continuation.ts)

live ↗

In$\mathrm{LerchPhi}(7.5, 1, 2)$Out$-0.166\,609\,816\,478\,250\,43-0.055\,850\,536\,063\,818\,53i$

Uncertainty in $z$

live ↗

In$\mathrm{LerchPhi}(\mathrm{Around}(0.5, 0.01), 1, 2)$Out$\mathrm{Around}(0.772\,588\,722\,239\,781\,1, 0.009\,096\,451\,110\,408\,75)$

To 30 significant digits, past a double's reach

live ↗

In$N(\mathrm{LerchPhi}(\frac{1}{2}, 2, \frac{1}{3}), 30)$Out$9.343\,474\,659\,375\,939\,518\,555\,549\,658\,04$

Past $|z| = 1$ it is continued; on the cut, real $z > 1$, it takes the side below, as mpmath and Wolfram do

live ↗

In$N(\mathrm{LerchPhi}(2.809, 2, 2))$Out$-0.056\,587\,701\,973\,223\,09-0.411\,220\,377\,971\,662\,5i$

Applications #

$\Phi(1, 3, 1) = \zeta(3)$, Apéry's constant

live ↗

In$\mathrm{LerchPhi}(1, 3, 1)=ζ(3)$Out$\mathrm{True}$

$\Phi(-1, 2, 1) = \eta(2) = \pi^2/12$ — the alternating rim, via the Euler transform

live ↗

In$\mathrm{LerchPhi}(-1, 2, 1)=\frac{π^2}{12}$Out$\mathrm{True}$

$\Phi(1/2, 2, 1) = 2\operatorname{Li}_2(1/2)$

live ↗

In$N(\mathrm{LerchPhi}(\frac{1}{2}, 2, 1))$Out$1.164\,481\,052\,930\,025$

$\Phi(1/2, 1/2, 1/3)$ to 40 digits, proven: the series' tail is bounded, and both ends of the enclosure round to these digits

live ↗

In$N(\mathrm{LerchPhi}(\frac{1}{2}, \frac{1}{2}, \frac{1}{3}), 40)$Out$2.452\,417\,145\,481\,526\,111\,742\,144\,156\,526\,818\,292\,226$

Properties #

$\Phi(1, 4, 1) = \zeta(4) = \pi^4/90$

live ↗

In$\mathrm{LerchPhi}(1, 4, 1)$Out$\frac{π^4}{90}$

$\Phi(1, -1, 1) = \zeta(-1) = -1/12$

live ↗

In$\mathrm{LerchPhi}(1, -1, 1)$Out$-\frac{1}{12}$

$\Phi(1, s, a) = \zeta(s, a)$ for symbolic $s$, $a$

live ↗

In$\mathrm{LerchPhi}(1, s, a)$Out$\mathrm{HurwitzZeta}(s, a)$

$\Phi(z, 0, a) = 1/(1 - z)$, independent of $a$

live ↗

In$\mathrm{LerchPhi}(z, 0, a)$Out$\frac{1}{1-z}$

$\Phi(2, 0, 7) = 1/(1 - 2) = -1$: the $s = 0$ closed form holds past the unit disk

live ↗

In$\mathrm{LerchPhi}(2, 0, 7)$Out$-1$

$\Phi(0, s, a) = a^{-s}$: only the $n = 0$ term survives

live ↗

In$\mathrm{LerchPhi}(0, s, a)$Out$a^{-s}$

$\Phi(z, s, 1) = \operatorname{Li}_s(z)/z$ symbolically. See PolyLog

live ↗

In$\mathrm{LerchPhi}(z, s, 1)$Out$\frac{\operatorname{Li}_{s}(z)}{z}$

$\Phi(z, 1, 1) = -\ln(1-z)/z$ -- the $s = 1$ instance, through $\operatorname{Li}_1(z) = -\ln(1-z)$

live ↗

In$\mathrm{LerchPhi}(z, 1, 1)$Out$-\frac{\ln(1-z)}{z}$

$\Phi(1, 2, 1/4) = \zeta(2, 1/4) = \pi^2 + 8G$, through HurwitzZeta's own closed form at that argument

live ↗

In$\mathrm{LerchPhi}(1, 2, \frac{1}{4})$Out$8\operatorname{G}+π^2$

$\Phi(-1, 1, 1) = \eta(1) = \ln 2$, the alternating harmonic series -- via $\Phi(z,1,1) = -\ln(1-z)/z$

live ↗

In$\mathrm{LerchPhi}(-1, 1, 1)$Out$\ln(2)$

$\Phi(1/2, 1, 1) = 2\ln 2$

live ↗

In$\mathrm{LerchPhi}(\frac{1}{2}, 1, 1)$Out$2\ln(2)$

Phi(z, s, 1) = Li_s(z)/z for any s, here as (1+i)*Li_2((1-i)/2) -- confirmed numerically against Wolframs closed form in Catalans constant

live ↗

In$\mathrm{LerchPhi}(\frac{1}{2}-\frac{1}{2}i, 2, 1)$Out$(1+i)\operatorname{Li}_{2}(\frac{1}{2}-\frac{1}{2}i)$

$\Phi(-1, 2, 1/2) = 4G$, via $\Phi(-1, s, 1/2) = 2^s\beta(s)$ and $\beta(2) = G$

live ↗

In$\mathrm{LerchPhi}(-1, 2, \frac{1}{2})$Out$4\operatorname{G}$

...though N() gets $4G = 3.66386\ldots$

live ↗

In$N(\mathrm{LerchPhi}(-1, 2, \frac{1}{2}))$Out$3.663\,862\,376\,708\,875$

$\Phi(-1, 3, \tfrac12) = 8\beta(3) = \pi^3/4$ — the Dirichlet beta at 3, in closed form

live ↗

In$\mathrm{LerchPhi}(-1, 3, \frac{1}{2})$Out$\frac{π^3}{4}$

$\Phi(-1, 3, 1) = \eta(3) = \tfrac34\zeta(3)$

live ↗

In$\mathrm{LerchPhi}(-1, 3, 1)=\frac{3ζ(3)}{4}$Out$\mathrm{True}$

Possible issues #

Where the continuation's terms cancel below double precision it stays unevaluated rather than guess

live ↗

In$N(\mathrm{LerchPhi}(10, 10, 10))$Out$-4.462\,130\,727\,102\,148\,4\cdot10^{-11}-1.575\,172\,198\,981\,023\,8\cdot10^{-12}i$

About

Examples	415
Checked against	Wolfram Language 381 agree31 differ1 unevaluatedSymPy 405 agree3 differ3 unevaluated3 errormpmath 400 agree3 differ3 unevaluated3 errorJulia 294 agree115 error
Crosswalk	WikipediaMathWorldWikidataDLMFFungrimFungrim entryWolfram LanguageSymPympmathJulia

Implementation

referenceenginenotatio

The defining series, converging for |z| < 1 (and for |z| = 1 with Re(s) > 1); ζ(s, a) is its z = 1 edge and Li_s(z) its a = 1 slice.

nativeenginetypescriptpackages/ce-patches/src/compute-engine/numerics/lerch-phi.ts

The direct series, which covers |z| ≤ 1; z = 1 hands off to HurwitzZeta and inherits its closed forms.

compiledgpuwgslpackages/symbols/analysis/analytic/src/shader.ts:zetaWGSL

Produces a colour per pixel — complex Φ, domain-coloured.

mappedwolframpackages/oracle/src/mappings.ts

For a + k < 0, Wolfram sums ((a+k)²)^(−s/2), the generalized-zeta convention; mpmath, SymPy and ours take (a+k)^(−s), principal.

mappedsympypackages/oracle/src/mappings.ts

For a + k < 0, Wolfram sums ((a+k)²)^(−s/2), the generalized-zeta convention; mpmath, SymPy and ours take (a+k)^(−s), principal.

mappedmpmathpackages/oracle/src/mappings.ts

For a + k < 0, Wolfram sums ((a+k)²)^(−s/2), the generalized-zeta convention; mpmath, SymPy and ours take (a+k)^(−s), principal.

mappedjuliapackages/oracle/src/mappings.ts

See also: HurwitzZeta, Zeta, Gamma, PolyLog