Ramanujan's master theorem
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Ramanujan's master theorem
In mathematics, Ramanujan's master theorem (named after mathematician Srinivasa Ramanujan[1]) is a technique which 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
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.
Multidimensional version of this theorem also appear in quantum physics (through Feynman diagrams).[2]
A similar result was also obtained by J. W. L. Glaisher.[3]
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Alternative formalism
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 
.
Proof
The proof of Ramanujan's Master Theorem provided by G. H. Hardy[4] employs Cauchy's residue theorem as well as 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 Hurwitz zeta function:
by 
for 
. By means of Ramanujan master theorem and generating function of Bernoulli polynomials one will have following integral representation[5]:
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:
for valid for .
Special cases of 
and 
are
Mathematica 7 is unable to compute these examples.[6]
Evaluation of quartic integral
It is well known for the evaluation of
which is a well known quartic integral.[7]
References
- ^ 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.
External links
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