mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, there are several functions known as Kummer's function. One is known as the confluent hypergeometric function of Kummer. Another one, defined below, is related to the

polylogarithm In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natur ...

. Both are named for

Ernst Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of h ...

. Kummer's function is defined by :

\Lambda_n(z)=\int_0^z \frac\;dt.

The duplication formula is :

\Lambda_n(z)+\Lambda_n(-z)= 2^\Lambda_n(-z^2)

. Compare this to the duplication formula for the polylogarithm: :

\operatorname_n(z)+\operatorname_n(-z)= 2^\operatorname_n(z^2).

An explicit link to the polylogarithm is given by :

\operatorname_n(z)=\operatorname_n(1)\;\;+\;\;
\sum_^ (-1)^ \;\frac  \;\operatorname_ (z) \;\;+\;\;
\frac \;\left \Lambda_n(-1) - \Lambda_n(-z) \right

References

*. Special functions {{mathanalysis-stub hu:Kummer-függvény