Ramanujan's master theorem

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.

File:Ramanujanmasterth.jpg

Page from Ramanujan's notebook stating his Master theorem.

The result is stated as follows:

If a complex-valued function $f(x)\!$ has an expansion of the form

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

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

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

where $\Gamma (s)\!$ 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).

A similar result was also obtained by J. W. L. Glaisher.

Alternative formalism

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

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

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

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

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 employing the residue theorem and the well-known Mellin inversion theorem.

Application to Bernoulli polynomials

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

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

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

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

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

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

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

Application to the Gamma function

Weierstrass's definition of the Gamma function

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

is equivalent to expression

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

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

Then applying Ramanujan master theorem we have:

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

valid for $0<Re(s)<1\!$.

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

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

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

Mathematica 7 was unable to compute these examples.

External links

• http://mathworld.wolfram.com/RamanujansMasterTheorem.html
• http://arminstraub.com/files/publications/rmt.pdf