Ramanujan 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 a function.

The result is stated as follows:

Assume function $f(x)$ has an expansion of the form

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

for some function (say analytic or integrable) $\phi(k)$, then the Mellin transform of $f(x)$ is given by

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

where $\Gamma(s)$ denotes the gamma function. It was widely used by Ramanujan to calculate definite integrals and infinite series.

Multi-dimensional versions of this theorem also appear in quantum physics (through Feynman diagrams).

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

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

for all $0 < \mathfrak{R} (s) < 1$.

This gets converted to the result expressed before when a substitution of $\lambda(n) = \dfrac{\phi(n)}{\Gamma(n+1)}$ is made and the functional equation for the gamma function is further used.

Note: Some authors denote $\phi(n) \to \lambda(n)$ and $\lambda(n) \to \phi(n)$.

The generating function of the Bernoulli polynomials $B_k(x)$ is given by

$\frac { z{ e }^{ xz } }{ { e }^{ z }-1 } =\sum _{ k=0 }^{ \infty }{ { B }_{ k }\left( x \right) \frac { { \left( z \right) }^{ k } }{ k! } }.$

These polynomials are given in terms of Hurwitz zeta function:

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

by $\zeta \left( 1-n,a \right) =-\frac { { B }_{ n }\left( a \right) }{ n } \quad\text{ for } n\ge 1.$

By means of Ramanujan master theorem and generating function of Bernoulli polynomials, one will have the following integral representation:

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

for all $0<\mathfrak{R}(s)<1$.

Weierstrass's definition of the gamma function is

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

where $\gamma$ denotes the Euler-Mascheroni constant.

This is equivalent to the expression

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

where $\zeta(k)$ denotes the Riemann zeta function.

Then, applying Ramanujan master theorem, we have

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

for all $0<\mathfrak{R}(s)<1$.

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

$\begin{aligned} \int_0^\infty \frac{\gamma x+\log\Gamma(1+x)}{x^{5/2}} \, dx &=\frac{2\pi}{3} \zeta\left( \frac{3}{2} \right) \\\\ \int_0^\infty \frac{\gamma x+\log\Gamma(1+x)}{x^{9/4}} \,dx &= \sqrt{2} \frac{4\pi}{5} \zeta\left(\frac 5 4\right). \end{aligned}$

It is well known for the evaluation of

$F(a,m)=\int_0^\infty \frac{dx}{(x^4+2ax^2+1)^{m+1}},$

which is a quartic integral.