Lerch transcendent

In mathematics, the Lerch zeta-function, sometimes called the Hurwitz–Lerch zeta-function, is a special function that generalizes the Hurwitz zeta-function and the polylogarithm. It is named after the Czech mathematician Mathias Lerch [1].

The Lerch zeta-function is given by

A related function, the Lerch transcendent, is given by

The two are related, as

An integral representation is given by

for

A contour integral representation is given by

for

where the contour must not enclose any of the points $t=\log(z)+2k\pi i,k\in Z.$ $t=\log(z)+2k\pi i,k\in Z.$

A Hermite-like integral representation is given by

for

and

for

The Hurwitz zeta-function is a special case, given by

The polylogarithm is a special case of the Lerch Zeta, given by

The Legendre chi function is a special case, given by