Ramanujan's master theorem

In mathematics, Ramanujan's master theorem (named after Srinivasa Ramanujan) is a technique that provides an analytic expression for the Mellin transform of an analytic function.

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

$f(x)=\sum _{k=0}^{\infty }{\frac {\,\varphi (k)\,}{k!}}(-x)^{k}$

then the Mellin transform of $f(x)$ is given by

$\int _{0}^{\infty }x^{s-1}f(x)dx=\Gamma (s)\,\varphi (-s)$

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 Glaisher.

Alternative formalism

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

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

which gets converted to the above form after substituting $\lambda (n)\equiv {\frac {\varphi (n)}{\,\Gamma (1+n)\,}}$ and using the functional equation for the gamma function.

The integral above is convergent for $0<{\mathcal {Re}}(s)<1$ subject to growth conditions on $\varphi$.

Proof

A proof subject to "natural" assumptions (though not the weakest necessary conditions) to Ramanujan's Master theorem was provided by G. H. Hardy employing the 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:

${\frac {z\,e^{x\,z}}{\,e^{z}-1\,}}=\sum _{k=0}^{\infty }B_{k}(x)\,{\frac {z^{k}}{k!}}$

These polynomials are given in terms of the 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 $~n\geq 1~$. Using the Ramanujan master theorem and the generating function of Bernoulli polynomials one has the following integral representation:

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

which is valid for $~0<{\mathcal {Re}}(s)<1~$.

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:

$\int _{0}^{\infty }x^{s-1}{\frac {\,\gamma \,x+\log \Gamma (1+x)\,}{x^{2}}}\operatorname {d} x={\frac {\pi }{\sin(\pi s)}}{\frac {\zeta (2-s)}{2-s}}\!$

valid for $0<{\mathcal {Re}}(s)<1$.

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

$\int _{0}^{\infty }{\frac {\,\gamma x+\log \Gamma (1+x)\,}{x^{5/2}}}\,\operatorname {d} x={\frac {2\pi }{3}}\,\zeta \left({\frac {3}{2}}\right)$

$\int _{0}^{\infty }{\frac {\,\gamma \,x+\log \Gamma (1+x)\,}{x^{9/4}}}\,\operatorname {d} x={\sqrt {2}}{\frac {4\pi }{5}}\zeta \left({\frac {5}{4}}\right)$

External links

• "Ramanujan's Master Theorem". mathworld.wolfram.com.
• "rmt" (PDF). ArminStraub. publications.