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Kummer's function

Mathematical function

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Source updated September 10, 2026

In mathematics, Kummer's function, named after Ernst Kummer, is a complex function related to the polylogarithm. It is defined by

$\Lambda _{n}(z)=\int _{0}^{z}{\frac {\log ^{n-1}|t|}{1+t}}\;dt.$

Its duplication formula is

$\Lambda _{n}(z)+\Lambda _{n}(-z)=2^{1-n}\Lambda _{n}(-z^{2})$ .

Compare this to the duplication formula for the polylogarithm:

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

An explicit link to the polylogarithm is given by

$\operatorname {Li} _{n}(z)=\operatorname {Li} _{n}(1)\;\;+\;\;\sum _{k=1}^{n-1}(-1)^{k-1}\;{\frac {\log ^{k}|z|}{k!}}\;\operatorname {Li} _{n-k}(z)\;\;+\;\;{\frac {(-1)^{n-1}}{(n-1)!}}\;\left[\Lambda _{n}(-1)-\Lambda _{n}(-z)\right].$

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Sources and credits

This article is adapted from the Wikipedia article “Kummer's function”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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