# Ramanujan's master theorem

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 ${\displaystyle f(x)\!}$ has an expansion of the form

${\displaystyle f(x)=\sum _{k=0}^{\infty }{\frac {\phi (k)}{k!}}(-x)^{k}\!}$

then the Mellin transform of ${\displaystyle f(x)\!}$ is given by

${\displaystyle \int _{0}^{\infty }x^{s-1}f(x)\,dx=\Gamma (s)\phi (-s)\!}$

where ${\displaystyle \Gamma (s)\!}$ 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 J. W. L. Glaisher.[3]

## Alternative formalism

An alternative formulation of Ramanujan's master theorem is as follows:

${\displaystyle \int _{0}^{\infty }x^{s-1}({\lambda (0)-x\lambda (1)+x^{2}\lambda (2)-\cdots })\,dx={\frac {\pi }{\sin(\pi s)}}\lambda (-s)}$

which gets converted to the above form after substituting ${\displaystyle \lambda (n)={\frac {\phi (n)}{\Gamma (1+n)}}\!}$ and using the functional equation for the gamma function.

The integral above is convergent for ${\displaystyle 0<\operatorname {Re} (s)<1}$ subject to growth conditions on ${\displaystyle \phi }$.[4]

## Proof

Ramanujan's Master Theorem is false in general (without special assumptions). A proof subject to "natural" assumptions (though not the weakest necessary conditions) was provided by G. H. Hardy[5] employing the residue theorem and the well-known Mellin inversion theorem.

## Application to Bernoulli polynomials

The generating function of the Bernoulli polynomials ${\displaystyle B_{k}(x)\!}$ is given by:

${\displaystyle {\frac {ze^{xz}}{e^{z}-1}}=\sum _{k=0}^{\infty }B_{k}(x){\frac {z^{k}}{k!}}\!}$

These polynomials are given in terms of the Hurwitz zeta function:

${\displaystyle \zeta (s,a)=\sum _{n=0}^{\infty }{\frac {1}{(n+a)^{s}}}\!}$

by ${\displaystyle \zeta (1-n,a)=-{\frac {B_{n}(a)}{n}}\!}$ for ${\displaystyle n\geq 1\!}$. Using the Ramanujan master theorem and the generating function of Bernoulli polynomials one has the following integral representation:[6]

${\displaystyle \int _{0}^{\infty }x^{s-1}\left({\frac {e^{-ax}}{1-e^{-x}}}-{\frac {1}{x}}\right)\,dx=\Gamma (s)\zeta (s,a)\!}$

valid for ${\displaystyle 0<\operatorname {Re} (s)<1\!}$.

## Application to the Gamma function

Weierstrass's definition of the Gamma function

${\displaystyle \Gamma (x)={\frac {e^{-\gamma x}}{x}}\prod _{n=1}^{\infty }\left(1+{\frac {x}{n}}\right)^{-1}e^{x/n}\!}$

is equivalent to expression

${\displaystyle \log \Gamma (1+x)=-\gamma x+\sum _{k=2}^{\infty }{\frac {\zeta (k)}{k}}(-x)^{k}\!}$

where ${\displaystyle \zeta (k)\!}$ is the Riemann zeta function.

Then applying Ramanujan master theorem we have:

${\displaystyle \int _{0}^{\infty }x^{s-1}{\frac {\gamma x+\log \Gamma (1+x)}{x^{2}}}\,dx={\frac {\pi }{\sin(\pi s)}}{\frac {\zeta (2-s)}{2-s}}\!}$

valid for ${\displaystyle 0.

Special cases of ${\displaystyle s={\frac {1}{2}}\!}$ and ${\displaystyle s={\frac {3}{4}}\!}$ are

${\displaystyle \int _{0}^{\infty }{\frac {\gamma x+\log \Gamma (1+x)}{x^{5/2}}}\,dx={\frac {2\pi }{3}}\zeta \left({\frac {3}{2}}\right)}$
${\displaystyle \int _{0}^{\infty }{\frac {\gamma x+\log \Gamma (1+x)}{x^{9/4}}}\,dx={\sqrt {2}}{\frac {4\pi }{5}}\zeta \left({\frac {5}{4}}\right)}$

Mathematica 7 was unable to compute these examples.[7]

## References

1. ^ Berndt, B. (1985). Ramanujan’s Notebooks, Part I. New York: Springer-Verlag.
2. ^ González, Iván; Moll, V. H.; Schmidt, Iván. "A generalized Ramanujan Master Theorem applied to the evaluation of Feynman diagrams" (PDF).
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.
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. doi:10.1007/s11139-011-9333-y.
5. ^ 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.
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:. doi:10.1023/A:1021171500736.
7. ^ 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.