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

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

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

\[\int _{ 0 }^{ \infty }{ { \left( x \right) }^{ s-1 } } f\left( x \right) dx=\Gamma (s)\phi (-s)\]

It was widely used by Ramanujan to calculate definite integrals and infinite series.

Multidimensional versions of this theorem also appear in quantum physics (through Feynman diagrams).

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

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

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 \[\zeta \left( 1-n,a \right) =-\frac { { B }_{ n }\left( a \right) }{ n } \quad for\quad n\ge 1\]

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

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

For all \(0<Re(s)<1\)

Weierstrass's definition of the Gamma function

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

is equivalent to expression

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

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

Then applying Ramanujan master theorem we have:

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

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

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

\[\int_0^\infty \frac{\gamma x+\log\Gamma(1+x)}{x^{5/2}} \, dx =\frac{2\pi}{3} \zeta\left( \frac{3}{2} \right) \]

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

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 well known quartic integral.