Lerch's Transcendent

Definition

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle { \LerchPhi@{z}{s}{a} = \sum_{n=0}^\infty \frac{z^n}{(a+n)^s} } }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle a \neq 0,-1,-2,\dots, |z| < 1}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > 1, |z| = 1}}

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \LerchPhi@{1}{s}{a} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > 1}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle a \neq 0,-1,-2,\dots}}

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Polylogarithm{s}@{z} = z \LerchPhi@{z}{s}{1} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > 1}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle |z| \leq 1}}

Properties

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \LerchPhi@{z}{s}{a} = z^m \LerchPhi@{z}{s}{a+m} + \sum_{n=0}^{m-1} \frac{z^n}{(a+n)^s} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle a \neq 0,-1,-2,\dots, |z| < 1}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > 1, |z| = 1}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle m = 1,2,3,\dots}}

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \LerchPhi@{z}{s}{a} = \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1} \expe^{-ax}}{1-z\expe^{-x}} \diff{x} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > 0}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{a} >0}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle z \in \Complex \setminus [1,\infty)}}

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \LerchPhi@{z}{s}{a} = \frac{1}{2} a^{-s} + \int_0^\infty \frac{z^x}{(a+x)^s} \diff{x} - 2 \int_0^\infty \frac{\sin@{x \ln@@{z} - s \atan@{x/a}}} {(a^2 + x^2)^{s/2} (\expe^{2 \cpi x} - 1)} \diff{x} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > 0}} if Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle |z| < 1}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > 1}} if Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle |z| = 1, \realpart{a} > 0}}

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Lerch's Transcendent

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Lerch's Transcendent