Ramanujan's master theorem

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In mathematics, Ramanujan's master theorem (named after Srinivasa Ramanujan[1]) 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).[2]

A similar result was also obtained by Glaisher.[3]

Alternative formalism[edit]

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 .[4]

Proof[edit]

A proof subject to "natural" assumptions (though not the weakest necessary conditions) to Ramanujan's Master theorem was provided by G. H. Hardy[5] employing the residue theorem and the well-known Mellin inversion theorem.

Application to Bernoulli polynomials[edit]

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:[6]

which is valid for .

Application to the Gamma function[edit]

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

Application to Bessel functions[edit]

The Bessel function of the first kind has the power series

By Ramanujan's master theorem, together with some identities for the gamma function and rearranging, we can evaluate the integral

valid for .

Equivalently, if the spherical Bessel function is preferred, the formula becomes

valid for .

The solution is remarkable in that it is able to interpolate across the major identities for the gamma function. In particular, the choice of gives the square of the gamma function, gives the duplication formula, gives the reflection formula, and fixing to the evaluable or gives the gamma function by itself, up to reflection and scaling.

References[edit]

  1. ^ Berndt, B. (1985). Ramanujan's Notebooks, Part I. New York: Springer-Verlag.
  2. ^ González, Iván; Moll, V.H.; Schmidt, Iván (2011). "A generalized Ramanujan Master Theorem applied to the evaluation of Feynman diagrams". arXiv:1103.0588 [math-ph].
  3. ^ 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. doi:10.1080/14786447408641072.
  4. ^ Amdeberhan, Tewodros; Gonzalez, Ivan; Harrison, Marshall; Moll, Victor H.; Straub, Armin (2012). "Ramanujan's Master Theorem". The Ramanujan Journal. 29 (1–3): 103–120. CiteSeerX 10.1.1.232.8448. doi:10.1007/s11139-011-9333-y. S2CID 8886049.
  5. ^ Hardy, G.H. (1978). Ramanujan: Twelve lectures on subjects suggested by his life and work (3rd ed.). New York, NY: Chelsea. ISBN 978-0-8284-0136-4.
  6. ^ 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. S2CID 970603.

External links[edit]