# 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).

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## Alternative formalism

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

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

## Application to Bernoulli polynomials

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

## Application to the Gamma function

Weierstrass's definition of the Gamma function

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

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

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

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

## Evaluation of quartic integral

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.

**Cite as:**Ramanujan Master Theorem.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/ramanujan-mastered-theorem/