Definition:Lerch Transcendent

Definition

The Lerch transcendent $\map \Phi {z, s, a}$ is defined on $\set {z \in \C: 0 \le \cmod z < 1, s \in \C, a \ne 0, -1, -2, \cdots} \cup \set {z = 1, \map \Re s > 1, a \ne 0, -1, -2, \cdots}$ as the series:

$\ds \map \Phi {z, s, a} = \sum_{n \mathop = 0}^\infty \frac {z^n} {\paren {n + a}^s}$

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Also see

• The Polylogarithm, the unique analytic continuation of $\map \Phi {z, s, 0}$
• The Hurwitz zeta function, which is the special case of $\map \Phi {z, s, a}$ for $z = 1$.

• Results about the Lerch transcendent can be found here.

Source of Name

This entry was named for Mathias Lerch.

Sources

• Weisstein, Eric W. "Lerch Transcendent." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LerchTranscendent.html