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 ${\displaystyle f(x)}$ has an expansion of the form

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

then the Mellin transform of ${\displaystyle f(x)}$ is given by

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

where ${\displaystyle \Gamma (s)}$ 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 Glaisher.^[3]

Alternative formalism[]

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

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

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

The integral above is convergent for ${\displaystyle 0<{\mathcal {Re}}(s)<1}$ subject to growth conditions on ${\displaystyle \varphi }$.^[4]

Proof[]

A proof subject to "natural" assumptions (though not the weakest necessary conditions) to Ramanujan's Master theorem was provided by G. H. Hardy^[5] employing the residue theorem and the well-known Mellin inversion theorem.

Application to Bernoulli polynomials[]

The generating function of the Bernoulli polynomials ${\displaystyle B_{k}(x)}$ is given by:

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

These polynomials are given in terms of the Hurwitz zeta function:

${\displaystyle \zeta (s,a)=\sum _{n=0}^{\infty }{\frac {1}{\,(n+a)^{s}\,}}}$

by ${\displaystyle ~\zeta (1-n,a)=-{\frac {B_{n}(a)}{n}}~}$ for ${\displaystyle ~n\geq 1~}$. Using the Ramanujan master theorem and the generating function of Bernoulli polynomials one has the following integral representation:^[6]

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

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

Application to the Gamma function[]

Weierstrass's definition of the Gamma function

${\displaystyle \Gamma (x)={\frac {\,e^{-\gamma \,x\,}}{x}}\,\prod _{n=1}^{\infty }\left(\,1+{\frac {x}{n}}\,\right)^{-1}e^{x/n}\!}$

is equivalent to expression

${\displaystyle \log \Gamma (1+x)=-\gamma \,x+\sum _{k=2}^{\infty }{\frac {\,\zeta (k)\,}{k}}\,(-x)^{k}}$

where ${\displaystyle \zeta (k)}$ is the Riemann zeta function.

Then applying Ramanujan master theorem we have:

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

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

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

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

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

Application to Bessel functions[]

The Bessel function of the first kind has the power series

${\displaystyle J_{\nu }(z)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{\Gamma (k+\nu +1)k!}}{\bigg (}{\frac {z}{2}}{\bigg )}^{2k+\nu }}$

By Ramanujan's master theorem, together with some identities for the gamma function and rearranging, we can evaluate the integral

${\displaystyle {\frac {2^{\nu -2s}\pi }{\sin {(\pi (s-\nu ))}}}\int _{0}^{\infty }z^{s-1-\nu /2}J_{\nu }({\sqrt {z}})\,dz=\Gamma (s)\Gamma (s-\nu )}$

valid for ${\displaystyle 0<2{\mathcal {Re}}(s)<{\mathcal {Re}}(\nu )+{\tfrac {3}{2}}}$.

Equivalently, if the spherical Bessel function ${\displaystyle j_{\nu }(z)}$ is preferred, the formula becomes

${\displaystyle {\frac {2^{\nu -2s}{\sqrt {\pi }}(1-2s+2\nu )}{\cos {(\pi (s-\nu ))}}}\int _{0}^{\infty }z^{s-1-\nu /2}j_{\nu }({\sqrt {z}})\,dz=\Gamma (s)\Gamma {\bigg (}{\frac {1}{2}}+s-\nu {\bigg )}}$

valid for ${\displaystyle 0<2{\mathcal {Re}}(s)<{\mathcal {Re}}(\nu )+2}$.

The solution is remarkable in that it is able to interpolate across the major identities for the gamma function. In particular, the choice of ${\displaystyle J_{0}({\sqrt {z}})}$ gives the square of the gamma function, ${\displaystyle j_{0}({\sqrt {z}})}$ gives the duplication formula, ${\displaystyle z^{-1/2}J_{1}({\sqrt {z}})}$ gives the reflection formula, and fixing to the evaluable ${\displaystyle s={\tfrac {1}{2}}}$ or ${\displaystyle s=1}$ gives the gamma function by itself, up to reflection and scaling.

References[]

External links[]

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