In mathematics, Ramanujan's master theorem (named after Srinivasa Ramanujan) is a technique that provides an analytic expression for the Mellin transform of an analytic function.
Page from Ramanujan's notebook stating his Master theorem.
The result is stated as follows:
If a complex-valued function has an expansion of the form
then the Mellin transform of is given by
where is the gamma function.
It was widely used by Ramanujan to calculate definite integrals and infinite series.
Higher-dimensional versions of this theorem also appear in quantum physics (through Feynman diagrams).
A similar result was also obtained by J. W. L. Glaisher.
An alternative formulation of Ramanujan's master theorem is as follows:
which gets converted to the above form after substituting and using the functional equation for the gamma function.
The integral above is convergent for subject to growth conditions on .
A proof subject to "natural" assumptions (though not the weakest necessary conditions) to Ramanujan's Master theorem was provided by G. H. Hardy employing the residue theorem and the well-known Mellin inversion theorem.
Application to Bernoulli polynomials
The generating function of the Bernoulli polynomials is given by:
These polynomials are given in terms of the Hurwitz zeta function:
by for .
Using the Ramanujan master theorem and the generating function of Bernoulli polynomials one has the 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
- ^ Berndt, B. (1985). Ramanujan’s Notebooks, Part I. New York: Springer-Verlag.
- ^ González, Iván; Moll, V. H.; Schmidt, Iván. "A generalized Ramanujan Master Theorem applied to the evaluation of Feynman diagrams".
- ^ Glaisher, J. W. L. (1874). "A new formula in definite integrals". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 48 (315): 53–55.
- ^ Amdeberhan, Tewodros; Gonzalez, Ivan; Harrison, Marshall; Moll, Victor H.; Straub, Armin (2012). "Ramanujan's Master Theorem". The Ramanujan Journal. 29 (1–3): 103–120. doi:10.1007/s11139-011-9333-y.
- ^ Hardy, G. H. (1978). Ramanujan. Twelve Lectures on subjects suggested by his life and work (3rd ed.). New York: Chelsea. ISBN 0-8284-0136-5.
- ^ Espinosa, O.; Moll, V. (2002). "On some definite integrals involving the Hurwitz zeta function. Part 2". The Ramanujan Journal. 6 (4): 449–468. arXiv:math/0107082. doi:10.1023/A:1021171500736.