In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Lerch
zeta function
In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function
: \zeta(s) = \sum_^\infty \frac 1 .
Zeta functions include:
* Airy zeta function, related to the zeros of the Airy function
* ...
, sometimes called the Hurwitz–Lerch zeta function, is a
special function
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined by ...
that generalizes the
Hurwitz zeta function
In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables with and by
:\zeta(s,a) = \sum_^\infty \frac.
This series is absolutely convergent for the given values of and and can ...
and 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 ...
. It is named after Czech mathematician
Mathias Lerch
Mathias Lerch (''Matyáš Lerch'', ) (20 February 1860, Milínov – 3 August 1922, Sušice) was a Czech mathematician who published about 250 papers, largely on mathematical analysis and number theory. He studied in Prague and Berlin, and held t ...
, who published a paper about the function in 1887.
Definition
The Lerch zeta function is given by
:
A related function, the Lerch transcendent, is given by
:
The two are related, as
:
Integral representations
The Lerch transcendent has an integral representation:
:
The proof is based on using the integral definition of the
Gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
to write
:
and then interchanging the sum and integral. The resulting integral representation converges for