Ramanujan's master theorem

In mathematics, Ramanujan's master theorem (named after Srinivasa Ramanujan^[1]) 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 {\phi (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)\phi (-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).^[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 the above form after substituting $\lambda (n)={\frac {\phi (n)}{\Gamma (1+n)}}\!$ and using the functional equation for the 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 the Cauchy's 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 {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 $n\geq 1\!$ . 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 $\zeta (k)\!$ 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}}}\,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 54}\right)

Mathematica 7 was unable to compute these examples.^[6]

References

↑ Berndt, B. (1985). Ramanujan’s Notebooks, Part I. New York: Springer-Verlag.
↑ González, Iván; Moll, V. H.; Schmidt, Iván. "A generalized Ramanujan Master Theorem applied to the evaluation of Feynman diagrams" (PDF).
↑ Glaisher, J. W. L. (1874). "A new formula in definite integrals". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 48 (315): 53–55.
↑ Hardy, G. H. (1978). Ramanujan. Twelve Lectures on subjects suggested by his life and work (3rd ed.). New York: Chelsea. ISBN 0-8284-0136-5.
↑ Espinosa, O.; Moll, V. (2002). "On some definite integrals involving the Hurwitz zeta function. Part 2". The Ramanujan Journal. 6 (4): 449–468. doi:10.1023/A:1021171500736.
↑ Amdeberhan, Tewodros; Gonzalez, Ivan; Harrison, Marshall; Moll, Victor H.; Straub, Armin (2012). "Ramanujan's Master Theorem". The Ramanujan Journal. 29 (1–3): 103–120. doi:10.1007/s11139-011-9333-y.

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