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.
The result is stated as follows:
If a complex-valued function has an expansion of the form
where is an analytic function, then the Mellin transform of is given by
where 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.
An alternative formulation of Ramanujan's master theorem is as follows:
which gets converted to the above form after substituting and using the functional equation for the gamma function.
The integral above is convergent for subject to growth conditions on .
A proof subject to "natural" assumptions (though not the weakest necessary conditions) to Ramanujan's master theorem was provided by G. H. Hardy(chapter XI) employing the residue theorem and the well-known Mellin inversion theorem.
Recall Euler's representation of the Gamma function
choosing , multiplying both sides by and then summing over to obtain:
Observing the sum in the RHS is a Taylor Series and writing in its series expansion:
Rewriting the LHS:
Then once again observing the sum over is a Taylor series:
Finally defining and letting we gain the master theorem:
Let be an analytic single-valued function defined on a half plane
for some . Suppose that, for some , satisfies the growth condition
for all . Let . This implies the series converges. Observing has poles at with residue , application of the residue theorem yields
for any . This integral converges absolutely and uniformly for for any . Applying the Mellin inversion theorem yields
The generating function of the Bernoulli polynomials is given by:
These polynomials are given in terms of the Hurwitz zeta function:
by for . Using the Ramanujan master theorem and the generating function of Bernoulli polynomials one has the following integral representation:
which is valid for .
Weierstrass's definition of the gamma function
is equivalent to expression
where is the Riemann zeta function.
Then applying Ramanujan master theorem we have:
valid for .
Special cases of and are
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
valid for .
Equivalently, if the spherical Bessel function is preferred, the formula becomes
valid for .
The solution is remarkable in that it is able to interpolate across the major identities for the gamma function. In particular, the choice of gives the square of the gamma function, gives the duplication formula, gives the reflection formula, and fixing to the evaluable or 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. 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.
This section identifies the integration formulas for integrand's with and without consecutive integer exponents and for single and double integrals. The integration formula for double integrals may be generalized to any multiple integral. In all cases, there is a parameter value or array of parameter values that solves one or more linear equations derived from the exponent terms of the integrand's series expansion.
This is the function series expansion, integral and integration formula for an integral whose integrand's series expansion contains consecutive integer exponents.
The parameter is a solution to this linear equation.
Applying the substitution generates the function series expansion, integral and integration formula for an integral whose integrand's series expansion may not contain consecutive integer exponents.
The parameter is a solution to this linear equation.
This is the function series expansion, integral and integration formula for a double integral whose integrand's series expansion contains consecutive integer exponents.
The parameters and are solutions to these linear equations.
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 is .
Applying the substitution generates the function series expansion, integral and integration formula for a double integral whose integrand's series expansion may not contain consecutive integer exponents. The integral and integration formula are The parameter matrix is a solution to this linear equation. .
In some cases, there may be more sums than variables. 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.
The free summation indices are elements of set . The matrix of free summation indices is and the coefficients of the free summation indices is matrix .
The remaining indices are set containing indices . Matrices and contain matrix elements that multiply or sum with the non-summation indices. The selected free summation indices must leave matrix non-singular. . This is the function's series expansion, integral and integration formula.
The parameters are linear functions of the parameters . .
<em>Bracket series</em> notations are notations that substitute for common power series notations (Table 1). Replacing power series notations with bracket series notations transforms the power series to a bracket series. A bracket series facilitates identifying the formula parameters needed for integration. It is also recommended to replace a sum raised to a power: with this bracket series expression:
This algorithm describes how to apply the integral formulas.
The bracket method will integrate this integral.
<ol> <li> Express the integrand as a power series. </li> <li>Transform the power series to a bracket series. </li> <li>Obtain the complexity index, formula parameters and series coefficient function.</li>
<li>Use table 2 to compute the integral.</li> </ol>
The bracket method will integrate this integral.
1. Express the integrand as a power series. Use the sum raised to a power formula.
2. Transform the power series to a bracket series.
3. Obtain the complexity index, formula parameters and series coefficient function.
4. Use table 2 to compute the integral