Ramanujan's master theorem

In mathematics, Ramanujan's master theorem (named after mathematician 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 $0< \operatorname{Re}(s)<1 \!$.

Proof

The proof of Ramanujan's Master Theorem provided by G. H. Hardy employs the Cauchy's 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 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 $n\geq1 \!$. By means of Ramanujan master theorem and generating function of Bernoulli polynomials one will have 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 is unable to compute these examples.

External links

1. http://mathworld.wolfram.com/RamanujansMasterTheorem.html
2. http://www.youtube.com/watch?v=gLp5OsfUlNE
3. http://arminstraub.com/files/publications/rmt.pdf