Lerch transcendent
In mathematics, the Lerch transcendent, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about a similar function in 1887. The Lerch transcendent, is given by:
It only converges for any real number, where, or, and.
Special cases
The Lerch transcendent is related to and generalizes various special functions.The Lerch zeta function is given by:
The Hurwitz zeta function is the special case
The polylogarithm is another special case:
The Riemann zeta function is a special case of both of the above:
The Dirichlet eta function:
The Dirichlet beta function:
The Legendre chi function:
The inverse tangent integral:
The polygamma functions for positive integers n:
The Clausen function:
Integral representations
The Lerch transcendent has an integral representation:The proof is based on using the integral definition of the gamma function to write
and then interchanging the sum and integral. The resulting integral representation converges for Re > 0, and Re > 0. This analytically continues to z outside the unit disk. The integral formula also holds if z = 1, Re > 1, and Re > 0; see Hurwitz zeta function.
A contour integral representation is given by
where C is a Hankel contour counterclockwise around the positive real axis, not enclosing any of the points which are poles of the integrand. The integral assumes Re > 0.
Other integral representations
A Hermite-like integral representation is given byfor
and
for
Similar representations include
and
holding for positive z. Furthermore,
The last formula is also known as Lipschitz formula.
Identities
For λ rational, the summand is a root of unity, and thus may be expressed as a finite sum over the Hurwitz zeta function. Suppose with and. Then and.Various identities include:
and
and
Series representations
A series representation for the Lerch transcendent is given byThe series is valid for all s, and for complex z with Re<1/2. Note a general resemblance to a similar series representation for the Hurwitz zeta function.
A Taylor series in the first parameter was given by Arthur Erdélyi. It may be written as the following series, which is valid for
If n is a positive integer, then
where is the digamma function.
A Taylor series in the third variable is given by
where is the Pochhammer symbol.
Series at a = −n is given by
A special case for n = 0 has the following series
where is the polylogarithm.
An asymptotic series for
for
and
for
An asymptotic series in the incomplete gamma function
for
The representation as a generalized hypergeometric function is
Asymptotic expansion
The polylogarithm function is defined asLet
For and, an asymptotic expansion of for large and fixed and is given by
for, where is the Pochhammer symbol.
Let
Let be its Taylor coefficients at. Then for fixed and,
as.