LaTeX to CAS translator

This mockup demonstrates the concept of TeX to Computer Algebra System (CAS) conversion.

The demo-application converts LaTeX functions which directly translate to CAS counterparts.

Functions without explicit CAS support are available for translation via a DRMF package (under development).

The following LaTeX input ...

{\displaystyle \Phi(z,s,a)}

${\displaystyle \Phi (z,s,a)}$

... is translated to the CAS output ...

Semantic latex: \Phi(z,s,a)

Confidence: 0

Mathematica

Translation: \[CapitalPhi][z , s , a]

Information

Sub Equations

\[CapitalPhi][z , s , a]

Free variables

\[CapitalPhi]

a

s

z

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Numeric

SymPy

Translation: Symbol('Phi')(z , s , a)

Information

Sub Equations

Symbol('Phi')(z , s , a)

Free variables

Symbol('Phi')

a

s

z

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Numeric

Maple

Translation: Phi(z , s , a)

Information

Sub Equations

Phi(z , s , a)

Free variables

Phi

a

s

z

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$z$

$s$

$a$

Is part of

$\,\chi _{n}(z)=2^{-n}z\Phi (z^{2},n,1/2)$

$\Phi (z,s+1,a)=-\,{\frac {1}{s}}{\frac {\partial }{\partial a}}\Phi (z,s,a)$

$\,\eta (s)=\Phi (-1,s,1)$

$\Phi (z,s,a)={\frac {1}{a^{s}}}+\sum _{m=0}^{\infty }(1-m-s)_{m}\operatorname {Li} _{s+m}(z){\frac {a^{m}}{m!}};|a|<1$

$\Phi (z,s,a)=-{\frac {\Gamma (1-s)}{2\pi i}}\int _{0}^{(+\infty )}{\frac {(-t)^{s-1}e^{-at}}{1-ze^{-t}}}\,dt$

$\,{\textrm {Li}}_{s}(z)=z\Phi (z,s,1)$

$\Phi (z,s,a)={\frac {1}{2a^{s}}}+{\frac {1}{z^{a}}}\sum _{k=1}^{\infty }{\frac {e^{-2\pi i(k-1)a}\Gamma (1-s,a(-2\pi i(k-1)-\log(z)))}{(-2\pi i(k-1)-\log(z))^{1-s}}}+{\frac {e^{2\pi ika}\Gamma (1-s,a(2\pi ik-\log(z)))}{(2\pi ik-\log(z))^{1-s}}}$

$\Phi (z,s,a)=z^{n}\Phi (z,s,a+n)+\sum _{k=0}^{n-1}{\frac {z^{k}}{(k+a)^{s}}}$

$\Phi (-z,s,a)=z^{-a}\Gamma (1-s)\sum _{k=-\infty }^{\infty }[(2k+1)\pi i-\log(z)]^{s-1}e^{(2k+1)\pi ai}$

$\Phi (z,s-1,a)=\left(a+z{\frac {\partial }{\partial z}}\right)\Phi (z,s,a)$

$\Phi (z,s,a)={\frac {1}{2a^{s}}}+\int _{0}^{\infty }{\frac {\cos(t\log z)\sin {\Big (}s\arctan {\tfrac {t}{a}}{\Big )}-\sin(t\log z)\cos {\Big (}s\arctan {\tfrac {t}{a}}{\Big )}}{{\big (}a^{2}+t^{2}{\big )}^{\frac {s}{2}}\tanh \pi t}}\,dt$

$\Phi (z,s,\alpha )=\sum _{n=0}^{\infty }{\frac {z^{n}}{(n+\alpha )^{s}}}$

$\Phi (z,s,a)={\frac {1}{2a^{s}}}+\int _{0}^{\infty }{\frac {z^{t}}{(a+t)^{s}}}\,dt+{\frac {2}{a^{s-1}}}\int _{0}^{\infty }{\frac {\sin(s\arctan(t)-ta\log(z))}{(1+t^{2})^{s/2}(e^{2\pi at}-1)}}\,dt$

$\,\Phi (e^{2\pi i\lambda },s,\alpha )=L(\lambda ,\alpha ,s)$

$\Phi (z,n,a)=z^{-a}\left\{\sum _{{k=0} \atop k\neq n-1}^{\infty }\zeta (n-k,a){\frac {\log ^{k}(z)}{k!}}+\left[\psi (n)-\psi (a)-\log(-\log(z))\right]{\frac {\log ^{n-1}(z)}{(n-1)!}}\right\}$

$\Phi (-z,s,a)={\frac {1}{2a^{s}}}+\int _{0}^{\infty }{\frac {\cos(t\log z)\sin {\Big (}s\arctan {\tfrac {t}{a}}{\Big )}-\sin(t\log z)\cos {\Big (}s\arctan {\tfrac {t}{a}}{\Big )}}{{\big (}a^{2}+t^{2}{\big )}^{\frac {s}{2}}\sinh \pi t}}\,dt$

$\Phi (\omega ,s,\alpha )=\sum _{n=0}^{\infty }{\frac {\omega ^{n}}{(n+\alpha )^{s}}}=\sum _{m=0}^{q-1}\sum _{n=0}^{\infty }{\frac {\omega ^{qn+m}}{(qn+m+\alpha )^{s}}}=\sum _{m=0}^{q-1}\omega ^{m}q^{-s}\zeta (s,{\frac {m+\alpha }{q}})$

$\,\zeta (s,\alpha )=L(0,\alpha ,s)=\Phi (1,s,\alpha )$

$\Phi (z,s,a)=z^{-a}\left[\Gamma (1-s)\left(-\log(z)\right)^{s-1}+\sum _{k=0}^{\infty }\zeta (s-k,a){\frac {\log ^{k}(z)}{k!}}\right]$

$\Phi (e^{i\varphi },s,a)=L{\big (}{\tfrac {\varphi }{2\pi }},a,s{\big )}={\frac {1}{a^{s}}}+{\frac {1}{2\Gamma (s)}}\int _{0}^{\infty }{\frac {t^{s-1}e^{-at}{\big (}e^{i\varphi }-e^{-t}{\big )}}{\cosh {t}-\cos {\varphi }}}\,dt$

$\Phi (z,s,a)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {t^{s-1}e^{-at}}{1-ze^{-t}}}\,dt$

$\Phi (z,s,a)={\frac {1}{2a^{s}}}+{\frac {\log ^{s-1}(1/z)}{z^{a}}}\Gamma (1-s,a\log(1/z))+{\frac {2}{a^{s-1}}}\int _{0}^{\infty }{\frac {\sin(s\arctan(t)-ta\log(z))}{(1+t^{2})^{s/2}(e^{2\pi at}-1)}}\,dt$

$\,\zeta (s)=\Phi (1,s,1)$

$\Phi (z,s,q)={\frac {1}{1-z}}\sum _{n=0}^{\infty }\left({\frac {-z}{1-z}}\right)^{n}\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(q+k)^{-s}$

$\Phi (z,s,a+x)=\sum _{k=0}^{\infty }\Phi (z,s+k,a)(s)_{k}{\frac {(-x)^{k}}{k!}};|x|<\Re (a)$

$\Phi (z,s,a)=\sum _{k=0}^{n}{\frac {z^{k}}{(a+k)^{s}}}+z^{n}\sum _{m=0}^{\infty }(1-m-s)_{m}\operatorname {Li} _{s+m}(z){\frac {(a+n)^{m}}{m!}};\ a\rightarrow -n$

$\Phi (z,s,a)=z^{-a}\Gamma (1-s)\sum _{k=-\infty }^{\infty }[2k\pi i-\log(z)]^{s-1}e^{2k\pi ai}$

$\Phi (z,s,a)={\frac {1}{1-z}}{\frac {1}{a^{s}}}+\sum _{n=1}^{N-1}{\frac {(-1)^{n}\mathrm {Li} _{-n}(z)}{n!}}{\frac {(s)_{n}}{a^{n+s}}}+O(a^{-N-s})$

$f(z,x,a)\equiv {\frac {1-(ze^{-x})^{1-a}}{1-ze^{-x}}}$

$\Phi (z,s,a)-{\frac {\mathrm {Li} _{s}(z)}{z^{a}}}=\sum _{n=0}^{N-1}C_{n}(z,a){\frac {(s)_{n}}{a^{n+s}}}+O\left((\Re a)^{1-N-s}+az^{-\Re a}\right)$

Description

asymptotic expansion

Pochhammer symbol

series

asymptotic series in the incomplete gamma function

special case

polylogarithm

asymptotic series

integral representation

contour

Various identity

positive integer

special case of the Lerch Zeta

last formula

Taylor series in the third variable

digamma function

n

series representation for the Lerch

Dirichlet eta function

Riemann zeta function

Legendre chi function

Hurwitz zeta function

Hermite-like integral representation

Lipschitz formula

related function

Similar representation

integral

point

Complete translation information:

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  "formula" : "\\Phi(z,s,a)",
  "semanticFormula" : "\\Phi(z,s,a)",
  "confidence" : 0.0,
  "translations" : {
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        "tokenTranslations" : {
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        "tokenTranslations" : {
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      "translation" : "Phi(z , s , a)",
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        "freeVariables" : [ "Phi", "a", "s", "z" ],
        "tokenTranslations" : {
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  "includes" : [ "z", "s", "a" ],
  "isPartOf" : [ "\\,\\chi_n(z)=2^{-n}z \\Phi (z^2,n,1/2)", "\\Phi(z,s+1,a)=-\\,\\frac{1}{s}\\frac{\\partial}{\\partial a} \\Phi(z,s,a)", "\\,\\eta(s)=\\Phi (-1,s,1)", "\\Phi(z,s,a)=\\frac{1}{a^s}+\\sum_{m=0}^\\infty (1-m-s)_m \\operatorname{Li}_{s+m}(z)\\frac{a^m}{m!}; |a|<1", "\\Phi(z,s,a)=-\\frac{\\Gamma(1-s)}{2\\pi i}\\int_0^{(+\\infty)}\\frac{(-t)^{s-1}e^{-at}}{1-ze^{-t}}\\,dt", "\\,\\textrm{Li}_s(z)=z\\Phi(z,s,1)", "\\Phi(z,s,a)=\\frac{1}{2a^s}+\\frac{1}{z^a}\\sum_{k=1}^\\infty\\frac{e^{-2\\pi i(k-1)a}\\Gamma(1-s,a(-2\\pi i(k-1)-\\log(z)))}     {(-2\\pi i(k-1)-\\log(z))^{1-s}}+\\frac{e^{2\\pi ika}\\Gamma(1-s,a(2\\pi ik-\\log(z)))}{(2\\pi ik-\\log(z))^{1-s}}", "\\Phi(z,s,a)=z^n \\Phi(z,s,a+n) + \\sum_{k=0}^{n-1} \\frac {z^k}{(k+a)^s}", "\\Phi(-z,s,a)=z^{-a}\\Gamma(1-s)\\sum_{k=-\\infty}^\\infty[(2k+1)\\pi i-\\log(z)]^{s-1}e^{(2k+1)\\pi ai}", "\\Phi(z,s-1,a)=\\left(a+z\\frac{\\partial}{\\partial z}\\right) \\Phi(z,s,a)", "\\Phi(z,s,a)= \\frac{1}{2a^s} + \\int_{0}^{\\infty}\\frac{\\cos(t\\log z)\\sin\\Big(s\\arctan\\tfrac{t}{a}\\Big) - \\sin(t\\log z)\\cos\\Big(s\\arctan\\tfrac{t}{a}\\Big)}{\\big(a^2 + t^2\\big)^{\\frac{s}{2}} \\tanh\\pi t }\\,dt", "\\Phi(z, s, \\alpha) = \\sum_{n=0}^\\infty\\frac { z^n} {(n+\\alpha)^s}", "\\Phi(z,s,a)=\\frac{1}{2a^s}+\\int_0^\\infty \\frac{z^t}{(a+t)^s}\\,dt+\\frac{2}{a^{s-1}}\\int_0^\\infty\\frac{\\sin(s\\arctan(t)-ta\\log(z))}{(1+t^2)^{s/2}(e^{2\\pi at}-1)}\\,dt", "\\,\\Phi(e^{2\\pi i\\lambda}, s,\\alpha)=L(\\lambda, \\alpha,s)", "\\Phi(z,n,a)=z^{-a}\\left\\{\\sum_{{k=0}\\atop k\\neq n-1}^ \\infty \\zeta(n-k,a)\\frac{\\log^k (z)}{k!}+\\left[\\psi(n)-\\psi(a)-\\log(-\\log(z))\\right]\\frac{\\log^{n-1}(z)}{(n-1)!}\\right\\}", "\\Phi(-z,s,a)= \\frac{1}{2a^s} + \\int_{0}^{\\infty}\\frac{\\cos(t\\log z)\\sin\\Big(s\\arctan\\tfrac{t}{a}\\Big) - \\sin(t\\log z)\\cos\\Big(s\\arctan\\tfrac{t}{a}\\Big)}{\\big(a^2 + t^2\\big)^{\\frac{s}{2}} \\sinh\\pi t }\\,dt", "\\Phi(\\omega, s, \\alpha) = \\sum_{n=0}^\\infty\\frac {\\omega^n} {(n+\\alpha)^s} = \\sum_{m=0}^{q-1} \\sum_{n=0}^\\infty \\frac {\\omega^{qn + m}}{(qn + m + \\alpha)^s} = \\sum_{m=0}^{q-1} \\omega^m q^{-s} \\zeta(s,\\frac{m + \\alpha}{q})", "\\,\\zeta(s,\\alpha)=L(0, \\alpha,s)=\\Phi(1,s,\\alpha)", "\\Phi(z,s,a)=z^{-a}\\left[\\Gamma(1-s)\\left(-\\log (z)\\right)^{s-1}+\\sum_{k=0}^\\infty \\zeta(s-k,a)\\frac{\\log^k (z)}{k!}\\right]", "\\Phi(e^{i\\varphi},s,a)=L\\big(\\tfrac{\\varphi}{2\\pi},a,s\\big)= \\frac{1}{a^s} + \\frac{1}{2\\Gamma(s)}\\int_{0}^{\\infty}\\frac{t^{s-1}e^{-at}\\big(e^{i\\varphi}-e^{-t}\\big)}{\\cosh{t}-\\cos{\\varphi}}\\,dt", "\\Phi(z,s,a)=\\frac{1}{\\Gamma(s)}\\int_0^\\infty\\frac{t^{s-1}e^{-at}}{1-ze^{-t}}\\,dt", "\\Phi(z,s,a)=\\frac{1}{2a^s}+\\frac{\\log^{s-1}(1/z)}{z^a}\\Gamma(1-s,a\\log(1/z))+\\frac{2}{a^{s-1}}\\int_0^\\infty\\frac{\\sin(s\\arctan(t)-ta\\log(z))}{(1+t^2)^{s/2}(e^{2\\pi at}-1)}\\,dt", "\\,\\zeta(s)=\\Phi (1,s,1)", "\\Phi(z,s,q)=\\frac{1}{1-z} \\sum_{n=0}^\\infty \\left(\\frac{-z}{1-z} \\right)^n\\sum_{k=0}^n (-1)^k \\binom{n}{k} (q+k)^{-s}", "\\Phi(z,s,a+x)=\\sum_{k=0}^\\infty \\Phi(z,s+k,a)(s)_{k}\\frac{(-x)^k}{k!};|x|<\\Re(a)", "\\Phi(z,s,a)=\\sum_{k=0}^n \\frac{z^k}{(a+k)^s}+z^n\\sum_{m=0}^\\infty (1-m-s)_{m}\\operatorname{Li}_{s+m}(z)\\frac{(a+n)^m}{m!};\\ a\\rightarrow-n", "\\Phi(z,s,a)=z^{-a}\\Gamma(1-s)\\sum_{k=-\\infty}^\\infty[2k\\pi i-\\log(z)]^{s-1}e^{2k\\pi ai}", "\\Phi(z,s,a) = \\frac{1}{1-z} \\frac{1}{a^{s}}    +    \\sum_{n=1}^{N-1} \\frac{(-1)^{n} \\mathrm{Li}_{-n}(z)}{n!} \\frac{(s)_{n}}{a^{n+s}}    +O(a^{-N-s})", "f(z,x,a) \\equiv \\frac{1-(z e^{-x})^{1-a}}{1-z e^{-x}}", "\\Phi(z,s,a) - \\frac{\\mathrm{Li}_{s}(z)}{z^{a}}=\\sum_{n=0}^{N-1}C_{n}(z,a) \\frac{(s)_{n}}{a^{n+s}}+O\\left( (\\Re a)^{1-N-s}+a z^{-\\Re a} \\right)" ],
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    "definition" : "series representation for the Lerch",
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    "definition" : "Dirichlet eta function",
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    "definition" : "Riemann zeta function",
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}

Specify your own input

Formula input
Wikitext context	In [[mathematics]], the '''Lerch [[zeta function]]''', sometimes called the '''Hurwitz–Lerch zeta-function''', is a [[special function]] that generalizes the [[Hurwitz zeta function]] and the [[polylogarithm]]. It is named after the Czech mathematician [[Mathias Lerch]] [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lerch.html]. ==Definition== The Lerch zeta function is given by :<math>L(\lambda, \alpha, s) = \sum_{n=0}^\infty \frac { e^{2\pi i\lambda n}} {(n+\alpha)^s}.</math> A related function, the '''Lerch transcendent''', is given by :<math>\Phi(z, s, \alpha) = \sum_{n=0}^\infty \frac { z^n} {(n+\alpha)^s}.</math> The two are related, as :<math>\,\Phi(e^{2\pi i\lambda}, s,\alpha)=L(\lambda, \alpha,s).</math> ==Integral representations== An integral representation is given by :<math> \Phi(z,s,a)=\frac{1}{\Gamma(s)}\int_0^\infty \frac{t^{s-1}e^{-at}}{1-ze^{-t}}\,dt</math> for :<math>\Re(a)>0\wedge\Re(s)>0\wedge z<1\vee\Re(a)>0\wedge\Re(s)>1\wedge z=1. </math> A [[contour integral]] representation is given by :<math> \Phi(z,s,a)=-\frac{\Gamma(1-s)}{2\pi i}\int_0^{(+\infty)} \frac{(-t)^{s-1}e^{-at}}{1-ze^{-t}}\,dt</math> for :<math>\Re(a)>0\wedge\Re(s)<0\wedge z<1</math> where the contour must not enclose any of the points <math>t=\log(z)+2k\pi i,k\in Z.</math> A Hermite-like integral representation is given by :<math> \Phi(z,s,a)= \frac{1}{2a^s}+ \int_0^\infty \frac{z^t}{(a+t)^s}\,dt+ \frac{2}{a^{s-1}} \int_0^\infty \frac{\sin(s\arctan(t)-ta\log(z))}{(1+t^2)^{s/2}(e^{2\pi at}-1)}\,dt </math> for :<math>\Re(a)>0\wedge \|z\|<1 </math> and :<math> \Phi(z,s,a)=\frac{1}{2a^s}+ \frac{\log^{s-1}(1/z)}{z^a}\Gamma(1-s,a\log(1/z))+ \frac{2}{a^{s-1}} \int_0^\infty \frac{\sin(s\arctan(t)-ta\log(z))}{(1+t^2)^{s/2}(e^{2\pi at}-1)}\,dt </math> for :<math>\Re(a)>0. </math> Similar representations include :<math> \Phi(z,s,a)= \frac{1}{2a^s} + \int_{0}^{\infty}\frac{\cos(t\log z)\sin\Big(s\arctan\tfrac{t}{a}\Big) - \sin(t\log z)\cos\Big(s\arctan\tfrac{t}{a}\Big)}{\big(a^2 + t^2\big)^{\frac{s}{2}} \tanh\pi t }\,dt, </math> and :<math> \Phi(-z,s,a)= \frac{1}{2a^s} + \int_{0}^{\infty}\frac{\cos(t\log z)\sin\Big(s\arctan\tfrac{t}{a}\Big) - \sin(t\log z)\cos\Big(s\arctan\tfrac{t}{a}\Big)}{\big(a^2 + t^2\big)^{\frac{s}{2}} \sinh\pi t }\,dt, </math> holding for positive ''z'' (and more generally wherever the integrals converge). Furthermore, :<math> \Phi(e^{i\varphi},s,a)=L\big(\tfrac{\varphi}{2\pi},a,s\big)= \frac{1}{a^s} + \frac{1}{2\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-at}\big(e^{i\varphi}-e^{-t}\big)}{\cosh{t}-\cos{\varphi}}\,dt, </math> The last formula is also known as ''Lipschitz formula''. ==Special cases== The [[Hurwitz zeta function]] is a special case, given by :<math>\,\zeta(s,\alpha)=L(0, \alpha,s)=\Phi(1,s,\alpha).</math> The [[polylogarithm]] is a special case of the Lerch Zeta, given by :<math>\,\textrm{Li}_s(z)=z\Phi(z,s,1).</math> The [[Legendre chi function]] is a special case, given by :<math>\,\chi_n(z)=2^{-n}z \Phi (z^2,n,1/2).</math> The [[Riemann zeta function]] is given by :<math>\,\zeta(s)=\Phi (1,s,1).</math> The [[Dirichlet eta function]] is given by :<math>\,\eta(s)=\Phi (-1,s,1).</math> ==Identities== For λ rational, the summand is a [[root of unity]], and thus <math>L(\lambda, \alpha, s)</math> may be expressed as a finite sum over the Hurwitz zeta-function. Suppose <math>\lambda = \frac{p}{q}</math> with <math>p, q \in \mathbb{Z}</math> and <math>q > 0</math>. Then <math>z = \omega = e^{2 \pi i \frac{p}{q}}</math> and <math>\omega^q = 1</math>. :<math>\Phi(\omega, s, \alpha) = \sum_{n=0}^\infty \frac {\omega^n} {(n+\alpha)^s} = \sum_{m=0}^{q-1} \sum_{n=0}^\infty \frac {\omega^{qn + m}}{(qn + m + \alpha)^s} = \sum_{m=0}^{q-1} \omega^m q^{-s} \zeta(s,\frac{m + \alpha}{q})</math> Various identities include: :<math>\Phi(z,s,a)=z^n \Phi(z,s,a+n) + \sum_{k=0}^{n-1} \frac {z^k}{(k+a)^s}</math> and :<math>\Phi(z,s-1,a)=\left(a+z\frac{\partial}{\partial z}\right) \Phi(z,s,a)</math> and :<math>\Phi(z,s+1,a)=-\,\frac{1}{s}\frac{\partial}{\partial a} \Phi(z,s,a).</math> ==Series representations== A series representation for the Lerch transcendent is given by :<math>\Phi(z,s,q)=\frac{1}{1-z} \sum_{n=0}^\infty \left(\frac{-z}{1-z} \right)^n \sum_{k=0}^n (-1)^k \binom{n}{k} (q+k)^{-s}.</math> (Note that <math>\tbinom{n}{k}</math> is a [[binomial coefficient]].) The series is valid for all ''s'', and for complex ''z'' with Re(''z'')<1/2. Note a general resemblance to a similar series representation for the Hurwitz zeta function. <ref name="benorin"> {{cite web \| url=https://www.physicsforums.com/insights/the-analytic-continuation-of-the-lerch-and-the-zeta-functions/ \| title=The Analytic Continuation of the Lerch Transcendent and the Riemann Zeta Function \| access-date=28 April 2020 }} </ref> A [[Taylor series]] in the first parameter was given by [[Arthur Erdélyi\|Erdélyi]]. It may be written as the following series, which is valid for :<math>\|\log(z)\|<2 \pi;s\neq 1,2,3,\dots; a\neq 0,-1,-2,\dots</math> :<math> \Phi(z,s,a)=z^{-a}\left[\Gamma(1-s)\left(-\log (z)\right)^{s-1} +\sum_{k=0}^\infty \zeta(s-k,a)\frac{\log^k (z)}{k!}\right] </math> {{cite journal\|author=B. R. Johnson\|title=Generalized Lerch zeta-function\|journal=Pacific J. Math.\|volume=53\|number=1\|date=1974\|pages=189–193\|doi=10.2140/pjm.1974.53.189\|doi-access=free}} If n is a positive integer, then :<math> \Phi(z,n,a)=z^{-a}\left\{ \sum_{{k=0}\atop k\neq n-1}^ \infty \zeta(n-k,a)\frac{\log^k (z)}{k!} +\left[\psi(n)-\psi(a)-\log(-\log(z))\right]\frac{\log^{n-1}(z)}{(n-1)!}\right\}, </math> where <math>\psi(n)</math> is the [[digamma function]]. A [[Taylor series]] in the third variable is given by :<math> \Phi(z,s,a+x)=\sum_{k=0}^\infty \Phi(z,s+k,a)(s)_{k}\frac{(-x)^k}{k!};\|x\|<\Re(a), </math> where <math>(s)_{k}</math> is the [[Pochhammer symbol]]. Series at ''a'' = -''n'' is given by :<math> \Phi(z,s,a)=\sum_{k=0}^n \frac{z^k}{(a+k)^s} +z^n\sum_{m=0}^\infty (1-m-s)_{m}\operatorname{Li}_{s+m}(z)\frac{(a+n)^m}{m!};\ a\rightarrow-n </math> A special case for ''n'' = 0 has the following series :<math> \Phi(z,s,a)=\frac{1}{a^s} +\sum_{m=0}^\infty (1-m-s)_m \operatorname{Li}_{s+m}(z)\frac{a^m}{m!}; \|a\|<1, </math> where <math>\operatorname{Li}_s(z)</math> is the [[polylogarithm]]. An [[asymptotic series]] for <math>s\rightarrow-\infty</math> :<math> \Phi(z,s,a)=z^{-a}\Gamma(1-s)\sum_{k=-\infty}^\infty [2k\pi i-\log(z)]^{s-1}e^{2k\pi ai} </math> for <math>\|a\|<1;\Re(s)<0 ;z\notin (-\infty,0) </math> and :<math> \Phi(-z,s,a)=z^{-a}\Gamma(1-s)\sum_{k=-\infty}^\infty [(2k+1)\pi i-\log(z)]^{s-1}e^{(2k+1)\pi ai} </math> for <math>\|a\|<1;\Re(s)<0 ;z\notin (0,\infty). </math> An asymptotic series in the [[incomplete gamma function]] :<math> \Phi(z,s,a)=\frac{1}{2a^s}+ \frac{1}{z^a}\sum_{k=1}^\infty \frac{e^{-2\pi i(k-1)a}\Gamma(1-s,a(-2\pi i(k-1)-\log(z)))} {(-2\pi i(k-1)-\log(z))^{1-s}}+ \frac{e^{2\pi ika}\Gamma(1-s,a(2\pi ik-\log(z)))}{(2\pi ik-\log(z))^{1-s}} </math> for <math>\|a\|<1;\Re(s)<0.</math> == Asymptotic expansion == The polylogarithm function <math>\mathrm{Li}_n(z)</math> is defined as :<math> \mathrm{Li}_0(z)=\frac{z}{1-z}, \qquad \mathrm{Li}_{-n}(z)=z \frac{d}{dz} \mathrm{Li}_{1-n}(z). </math> Let :<math> \Omega_{a} \equiv \begin{cases} \mathbb{C}\setminus[1,\infty) & \text{if } \Re a > 0, \\ {z \in \mathbb{C}, \|z\|<1} & \text{if } \Re a \le 0. \end{cases} </math> For <math>\|\mathrm{Arg}(a)\|<\pi, s \in \mathbb{C}</math> and <math>z \in \Omega_{a}</math>, an asymptotic expansion of <math>\Phi(z,s,a)</math> for large <math>a</math> and fixed <math>s</math> and <math>z</math> is given by :<math> \Phi(z,s,a) = \frac{1}{1-z} \frac{1}{a^{s}} + \sum_{n=1}^{N-1} \frac{(-1)^{n} \mathrm{Li}_{-n}(z)}{n!} \frac{(s)_{n}}{a^{n+s}} +O(a^{-N-s}) </math> for <math>N \in \mathbb{N}</math>, where <math>(s)_n = s (s+1)\cdots (s+n-1)</math> is the [[Falling and rising factorials\|Pochhammer symbol]].<ref>{{cite journal \|last1=Ferreira \|first1=Chelo \|last2=López \|first2=José L. \|title=Asymptotic expansions of the Hurwitz–Lerch zeta function \|journal=Journal of Mathematical Analysis and Applications \|date=October 2004 \|volume=298 \|issue=1 \|pages=210–224 \|doi=10.1016/j.jmaa.2004.05.040}}</ref> Let :<math> f(z,x,a) \equiv \frac{1-(z e^{-x})^{1-a}}{1-z e^{-x}}. </math> Let <math>C_{n}(z,a)</math> be its Taylor coefficients at <math>x=0</math>. Then for fixed <math>N \in \mathbb{N}, \Re a > 1</math> and <math>\Re s > 0</math>, :<math> \Phi(z,s,a) - \frac{\mathrm{Li}_{s}(z)}{z^{a}} = \sum_{n=0}^{N-1} C_{n}(z,a) \frac{(s)_{n}}{a^{n+s}} + O\left( (\Re a)^{1-N-s}+a z^{-\Re a} \right), </math> as <math>\Re a \to \infty</math>.<ref>{{cite journal \|last1=Cai \|first1=Xing Shi \|last2=López \|first2=José L. \|title=A note on the asymptotic expansion of the Lerch's transcendent \|journal=Integral Transforms and Special Functions \|date=10 June 2019 \|volume=30 \|issue=10 \|pages=844–855 \|doi=10.1080/10652469.2019.1627530\|arxiv=1806.01122 \|s2cid=119619877 }}</ref> ==Software== The Lerch transcendent is implemented as LerchPhi in [http://www.maplesoft.com/support/help/Maple/view.aspx?path=LerchPhi Maple] and [https://functions.wolfram.com/ZetaFunctionsandPolylogarithms/LerchPhi/ Mathematica], and as lerchphi in [http://mpmath.org/doc/current/functions/zeta.html#lerch-transcendent mpmath] and [https://docs.sympy.org/latest/modules/functions/special.html#sympy.functions.special.zeta_functions.lerchphi SymPy]. ==References== {{Reflist}} * {{dlmf \| id= 25.14 \| first= T. M. \| last= Apostol \| title= Lerch's Transcendent}}. * {{citation \| first1= H. \| last1= Bateman \| author1-link= Harry Bateman \| first2= A. \| last2= Erdélyi \| author2-link= Arthur Erdélyi \| title= Higher Transcendental Functions, Vol. I \| year= 1953 \| location= New York \| publisher= McGraw-Hill \| url=http://apps.nrbook.com/bateman/Vol1.pdf}}. (See § 1.11, "The function Ψ(''z'',''s'',''v'')", p. 27) * {{cite book \|author-first1=Izrail Solomonovich \|author-last1=Gradshteyn \|author-link1=Izrail Solomonovich Gradshteyn \|author-first2=Iosif Moiseevich \|author-last2=Ryzhik \|author-link2=Iosif Moiseevich Ryzhik \|author-first3=Yuri Veniaminovich \|author-last3=Geronimus \|author-link3=Yuri Veniaminovich Geronimus \|author-first4=Michail Yulyevich \|author-last4=Tseytlin \|author-link4=Michail Yulyevich Tseytlin \|author-first5=Alan \|author-last5=Jeffrey \|editor-first1=Daniel \|editor-last1=Zwillinger \|editor-first2=Victor Hugo \|editor-last2=Moll \|translator=Scripta Technica, Inc. \|title=Table of Integrals, Series, and Products \|publisher=Academic Press \|date=2015 \|orig-year=October 2014 \|edition=8 \|language=English \|isbn=978-0-12-384933-5 \|lccn=2014010276 <!-- \|url=https://books.google.com/books?id=NjnLAwAAQBAJ \|access-date=2016-02-21-->\|title-link=Gradshteyn and Ryzhik \|chapter=9.55.}} * {{citation \| first1= Jesus \| last1= Guillera \| first2= Jonathan \| last2= Sondow \| arxiv= math.NT/0506319 \| mr = 2429900 \| title= Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent \| journal= The Ramanujan Journal \| volume= 16 \| year= 2008 \| pages= 247–270 \| issue= 3 \| doi= 10.1007/s11139-007-9102-0\| s2cid= 119131640 }}. (Includes various basic identities in the introduction.) * {{citation \| first= M. \| last= Jackson \| title= On Lerch's transcendent and the basic bilateral hypergeometric series <sub>2</sub>''ψ''<sub>2</sub> \| year= 1950 \| journal= J. London Math. Soc. \| volume= 25 \| issue= 3 \| pages= 189–196 \| doi= 10.1112/jlms/s1-25.3.189 \| mr= 0036882}}. * {{citation \| first1= F. \| last1= Johansson \| first2= Ia. \| last2= Blagouchine \| arxiv= 1804.01679 \| mr = 3925487 \| title= Computing Stieltjes constants using complex integration \| journal= Mathematics of Computation \| volume= 88 \| year= 2019 \| pages= 1829–1850 \| issue= 318 \| doi= 10.1090/mcom/3401\| s2cid= 4619883 }}. * {{citation \| first1= Antanas \| last1= Laurinčikas \| first2= Ramūnas \| last2= Garunkštis \| title= The Lerch zeta-function \| publisher= Kluwer Academic Publishers \| location= Dordrecht \| year= 2002 \| isbn= 978-1-4020-1014-9 \| mr= 1979048}}. * {{citation \| first= Mathias \| last= Lerch \| authorlink= Mathias Lerch \| title= Note sur la fonction <math>\scriptstyle{\mathfrak K}(w,x,s) = \sum_{k=0}^\infty {e^{2k\pi ix} \over (w+k)^s}</math> \| language= French \| year= 1887 \| journal= Acta Mathematica \| volume= 11 \| issue= 1–4 \| pages= 19–24 \| doi= 10.1007/BF02612318 \| mr= 1554747 \| jfm= 19.0438.01\| s2cid= 121885446 \| url= https://zenodo.org/record/1681743 \| doi-access= free }}. ==External links== * {{citation \| first1= Sergej V. \| last1= Aksenov \| first2= Ulrich D. \| last2= Jentschura \| year=2002 \| url= http://aksenov.freeshell.org/lerchphi.html \| title= C and Mathematica Programs for Calculation of Lerch's Transcendent}}. * Ramunas Garunkstis, ''[http://www.mif.vu.lt/~garunkstis Home Page]'' (2005) ''(Provides numerous references and preprints.)'' * Ramunas Garunkstis, ''[http://www.mif.vu.lt/~garunkstis/preprintai/approx.pdf Approximation of the Lerch Zeta Function]'' (PDF) * S. Kanemitsu, Y. Tanigawa and H. Tsukada, ''[http://www.iisc.ernet.in/nias/HRJ/vol27/Ktt.pdf A generalization of Bochner's formula]{{dead link\|date=December 2017 \|bot=InternetArchiveBot \|fix-attempted=yes }}'', (undated, 2005 or earlier) * {{MathWorld \| urlname= LerchTranscendent \| title= Lerch Transcendent}} * {{cite web \| url= http://dlmf.nist.gov/25.14 \|title= §25.14, Lerch's Transcendent \|date= 2010 \|work= NIST Digital Library of Mathematical Functions \|publisher= National Institute of Standards and Technology \|accessdate= 28 January 2012 }} [[Category:Zeta and L-functions]]
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