Ramanujan's master theorem
The result is stated as follows:
Assume function has an expansion of the form
then Mellin transform of is given by
where is the Gamma function.
It was widely used by Ramanujan to calculate definite integrals and infinite series.
An alternative formulation of Ramanujan's master theorem is as follows:
which gets converted to original form after substituting and using functional equation for Gamma function.
The integral above is convergent for .
Application to Bernoulli polynomials
The generating function of the Bernoulli polynomials is given by:
These polynomials are given in terms of Hurwitz zeta function:
by for . By means of Ramanujan master theorem and generating function of Bernoulli polynomials one will have following integral representation:
valid for .
Application to the Gamma function
Weierstrass's definition of the Gamma function
is equivalent to expression
where is the Riemann zeta function.
Then applying Ramanujan master theorem we have:
valid for .
Special cases of and are
Evaluation of quartic integral
It is well known for the evaluation of
which is a well known quartic integral.
- B. Berndt. Ramanujan’s Notebooks, Part I. Springer-Verlag, New York, 1985.
- A generalized Ramanujan Master Theorem applied to the evaluation of Feynman diagrams by Iv´an Gonz´alez, V. H. Moll and Iv´an Schmidt
- J. W. L. Glaisher. A new formula in definite integrals. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 48(315):53–55, Jul 1874.
- G. H. Hardy. Ramanujan. Twelve Lectures on subjects suggested by his life and work. Chelsea Publishing Company, New York, N. Y., 3rd edition, 1978.
- O. Espinosa and V. Moll. On some definite integrals involving the Hurwitz zeta function. Part 2. The Ramanujan Journal, 6:449–468, 2002.
- Ramanujan's Master Theorem by Tewodros Amdeberhan, Ivan Gonzalez, Marshall Harrison, Victor H. Moll and Armin Straub, The Ramanujan Journal.
- T. Amdeberhan and V. Moll. A formula for a quartic integral: a survey of old proofs and some new ones. The Ramanujan Journal, 18:91–102, 2009.