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In mathematics, Ramanujan's master theorem (named after mathematician Srinivasa Ramanujan[1]) is a technique that provides an analytic expression for the Mellin transform of a function.
Page from Ramanujan's notebook stating his Master theorem.

The result is stated as follows:

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

$$f(x)=\sum_{k=0}^\infty \frac{\phi(k)}{k!}(-x)^k \!$$

then Mellin transform of $$f(x) \! is given by \( \int_0^\infty x^{s-1} f(x) \, dx = \Gamma(s)\phi(-s) \!$$

where $$\Gamma(s) \!$$ is the Gamma function.

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

Multidimensional version of this theorem also appear in quantum physics (through Feynman diagrams).[2]

A similar result was also obtained by J. W. L. Glaisher.[3]

Alternative formalism

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

$$\int_0^\infty x^{s-1} ({\lambda(0)-x\lambda(1)+x^{2}\lambda(2)-\cdots}) \, dx = \frac{\pi}{\sin(\pi s)}\lambda(-s)$$

which gets converted to original form after substituting $$\lambda(n) = \frac{\phi(n)}{\Gamma(1+n)} \!$$ and using functional equation for Gamma function.

The integral above is convergent for $$0< \operatorname{Re}(s)<1 \!.$$

Proof

The proof of Ramanujan's Master Theorem provided by G. H. Hardy[4] employs Cauchy's residue theorem as well as the well-known Mellin inversion theorem.
Application to Bernoulli polynomials

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

$$\frac{ze^{xz}}{e^z-1}=\sum_{k=0}^\infty B_k(x)\frac{z^k}{k!} \!$$

These polynomials are given in terms of 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\geq1 \!. By means of Ramanujan master theorem and generating function of Bernoulli polynomials one will have following integral representation:[5] \( \int_0^\infty x^{s-1} \left(\frac{e^{-ax}}{1-e^{-x}}-\frac{1}{x}\right) \, dx = \Gamma(s)\zeta(s,a) \!$$

valid for $$0<\operatorname{Re}(s)<1\!.$$
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 $$\zeta(k) \! i$$ s the Riemann zeta function.

Then applying Ramanujan master theorem we have:

$$\int_0^\infty x^{s-1} \frac{\gamma x+\log\Gamma(1+x)}{x^2} \, dx= \frac{\pi}{\sin(\pi s)}\frac{\zeta(2-s)}{2-s} \!$$

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

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

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

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

Mathematica 7 is unable to compute these examples.[6]
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.[7]

References

B. Berndt. Ramanujan’s Notebooks, Part I. Springer-Verlag, New York, 1985.
A generalized Ramanujan Master Theorem applied to the evaluation of Feynman diagrams by Iv´an Gonz´alez, V. H. Moll and Iv´an Schmidt
J. W. L. Glaisher. A new formula in definite integrals. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 48(315):53–55, Jul 1874.
G. H. Hardy. Ramanujan. Twelve Lectures on subjects suggested by his life and work. Chelsea Publishing Company, New York, N. Y., 3rd edition, 1978.
O. Espinosa and V. Moll. On some definite integrals involving the Hurwitz zeta function. Part 2. The Ramanujan Journal, 6:449–468, 2002.
Ramanujan's Master Theorem by Tewodros Amdeberhan, Ivan Gonzalez, Marshall Harrison, Victor H. Moll and Armin Straub, The Ramanujan Journal.

T. Amdeberhan and V. Moll. A formula for a quartic integral: a survey of old proofs and some new ones. The Ramanujan Journal, 18:91–102, 2009.