In mathematics, Ramanujan's master theorem (named after mathematician Srinivasa Ramanujan) is a technique that provides an analytic expression for the Mellin transform of a function.
The result is stated as follows:
Assume function has an expansion of the form
for some function (say analytic or integrable) , then the Mellin transform of is given by
where denotes the gamma function. It was widely used by Ramanujan to calculate definite integrals and infinite series.
Multi-dimensional versions of this theorem also appear in quantum physics (through Feynman diagrams).
An alternative formulation of Ramanujan's master theorem is as follows:
for all .
This gets converted to the result expressed before when a substitution of is made and the functional equation for the gamma function is further used.
Note: Some authors denote and .
The generating function of the Bernoulli polynomials is given by
These polynomials are given in terms of Hurwitz zeta function:
By means of Ramanujan master theorem and generating function of Bernoulli polynomials, one will have the following integral representation:
for all .
It is well known for the evaluation of
which is a quartic integral.