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

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

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.