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.
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] An alternative formulation of Ramanujan's master theorem is as follows:
[4] A proof subject to "natural" assumptions (though not the weakest necessary conditions) to Ramanujan's master theorem was provided by G. H. Hardy[5](chapter XI) employing the residue theorem and the well-known Mellin inversion theorem.
The generating function of the Bernoulli polynomials
These polynomials are given in terms of the Hurwitz zeta function:
Using the Ramanujan master theorem and the generating function of Bernoulli polynomials one has the following integral representation:[6]
Then applying Ramanujan master theorem we have:
The Bessel function of the first kind has the power series
By Ramanujan's master theorem, together with some identities for the gamma function and rearranging, we can evaluate the integral
Equivalently, if the spherical Bessel function
The solution is remarkable in that it is able to interpolate across the major identities for the gamma function.
gives the gamma function by itself, up to reflection and scaling.
The bracket integration method (method of brackets) applies Ramanujan's master theorem to a broad range of integrals.
[7] The bracket integration method generates the integrand's series expansion, creates a bracket series, identifies the series coefficient and formula parameters and computes the integral.
[8] This section identifies the integration formulas for integrand's with and without consecutive integer exponents and for single and double integrals.
that solves one or more linear equations derived from the exponent terms of the integrand's series expansion.
This section describes the integration formula for a double integral whose integrand's series expansion may not contain consecutive integer exponents.
Matrices contain the parameters needed to express the exponents in a series expansion of the integrand, and the determinant of invertible matrix
generates the function series expansion, integral and integration formula for a double integral whose integrand's series expansion may not contain consecutive integer exponents.
For example, if the integrand is a product of 3 functions of a common single variable, and each function is converted to a series expansion sum, the integrand is now a product of 3 sums, each sum corresponding to a distinct series expansion.
and the coefficients of the free summation indices is matrix
contain matrix elements that multiply or sum with the non-summation indices.
The selected free summation indices must leave matrix
A bracket series facilitates identifying the formula parameters needed for integration.
It is also recommended to replace a sum raised to a power:[19]
This algorithm describes how to apply the integral formulas.
Obtain the complexity index, formula parameters and series coefficient function.
