Talk:Ramanujan's master theorem

Add a bottom section.
Hello, I would like to add a bottom section to enhance the Wikipedia page. The topic added is not large enough to merit a separate page. The additional content will enhance the quality of the Wikipedia page. The references overlap with the existing references. I added 2 additional references, added labels to the references, and capitalized Ramanujan's Master Theorem as it is a proper noun. Thanks.

Proposal
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 $ f(x) $ has an expansion of the form


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

then the Mellin transform of $f(x)$ is given by


 * $$ \int_0^\infty x^{s-1} f(x) dx = \Gamma(s)\,\varphi(-s) $$

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).

A similar result was also obtained by Glaisher.

Alternative formalism
An alternative formulation of Ramanujan's Master Theorem is as follows:


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

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

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

Proof
A proof subject to "natural" assumptions (though not the weakest necessary conditions) to Ramanujan's Master theorem was provided by G. H. Hardy employing the 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:


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

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


 * $$ \zeta(s,a) = \sum_{n=0}^\infty \frac{1}{\,(n+a)^s\,} $$

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


 * $$ \int_0^\infty x^{s-1}\left(\frac{e^{-ax}}{\,1 - e^{-x}\,}-\frac{1}{x}\right) dx = \Gamma(s)\,\zeta(s,a) \!$$

which is valid for $ 0 < \mathcal{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:


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

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

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


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


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

Application to Bessel functions
The Bessel function of the first kind has the power series
 * $$ J_\nu(z)=\sum_{k=0}^\infty \frac{(-1)^k}{\Gamma(k+\nu+1)k!}\bigg(\frac{z}{2}\bigg)^{2k+\nu} $$

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


 * $$ \frac{2^{\nu-2s}\pi}{\sin{(\pi(s-\nu))}} \int_0^\infty z^{s-1-\nu/2}J_\nu(\sqrt{z})\,dz = \Gamma(s)\Gamma(s-\nu)$$

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

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


 * $$ \frac{2^{\nu-2s}\sqrt{\pi}(1-2s+2\nu)}{\cos{(\pi(s-\nu))}} \int_0^\infty z^{s-1-\nu/2}j_\nu(\sqrt{z})\,dz = \Gamma(s)\Gamma\bigg(\frac{1}{2}+s-\nu\bigg)$$

valid for $ 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 $J_{0}(\sqrt{z}) $ gives the square of the gamma function, $j_{0}(\sqrt{z})$  gives the duplication formula, $z^{-1/2}J_{1}(\sqrt{z})$  gives the reflection formula, and fixing to the evaluable $s=\tfrac{1}{2}$  or $s=1$  gives the gamma function by itself, up to reflection and scaling.

Bracket Integration Method
The Bracket Integration Method applies Ramanujan's Master Theorem to a broad range of integrals. The Bracket Integration Method generates an integral of a series expansion, introduces simplifying notations, solves linear equations, and completes the integration using formulas arising from Ramanujan's Master Theorem.

Generate an integral of a series expansion
This method transforms the integral to an integral of a series expansion involving M variables, $x_1, \ldots x_M$, and S summation parameters, $n_1, \dots n_S$. A multivariate integral may assume this form.

Apply special notations

 * The bracket ($< \dots >$ ), indicator ($\phi$ ), and monomial power notations replace terms in the series expansion.
 * Application of these notations transforms the integral to a bracket series containing B brackets.
 * Each bracket series has an index defined as index=number of sums - number of brackets.
 * Among all bracket series representations of an integral, the representation with a minimal index is preferred.

Solve linear equations

 * The array of coefficients $a_{jk}$ must have maximum rank,  linearly independent leading columns to solve the following set of  linear equations.
 * If the index is non-negative, solve this equation set for each $n^{\ast}_{j}$ . The terms $n^{\ast}_{j}$  may be  linear functions of $\{n_{B+1} \dots n_{S} \}$.
 * If the index is zero, equation ($$) simplifies to solving this equation set for each $n^{\ast}_{j}$
 * If the index is negative, the integral cannot be determined.

Apply formulas

 * If the index is non-negative, the formula for the integral is this form.
 * These rules apply.
 * A series is generated for each choice of free summation parameters, $\{ n_{B}+1, \dots N_{S} \}$.
 * Series converging in a common region are added.
 * If a choice generates a divergent series or null series (a series with zero valued terms), the series is rejected.
 * A bracket series of negative index is assigned no value.
 * If all series are rejected, then the method cannot be applied.
 * If the index is zero, the formula $$ simplifies to this formula and no sum occurs.

Mathematical Basis
.
 * Apply this variable transformation to the general integral form ($$).
 * This is the transformed integral ($$) and the result from applying Ramanujan's Master Theorem ($$).
 * The number of brackets (B) equals the number of integrals (M) ($$). In addition to generating the algorithm's formulas ($$,$$), the variable transformation also generates the algorithm's linear equations ($$,$$).

Example

 * The Bracket Integration Method is applied to this integral.
 * $$ \int_{0}^{\infty} x^{3/2} \cdot e^{-x^3/2} \ dx $$


 * Generate the integral of a series expansion ($$).
 * $$ \int_{0}^{\infty} \sum_{n=0}^{\infty} 2^{-n} \cdot \frac{(-1)^{n}}{n!}  \cdot  x^{(3 \cdot n+5/2)-1} \ dx $$


 * Apply special notations ($$, $$).
 * $$ \sum_{n=0}^{\infty} 2^{-n}  \cdot \phi(n) \cdot  <3 \cdot n+ \frac{5}{2}>   $$


 * Solve the linear equation ($$).
 * $$ 3 \cdot n^{\ast}+ \frac{5}{2}=0, \ n^{\ast}= \frac{-5}{6} $$


 * Apply the formula ($$).
 * $$ \frac {2^{\frac{5}{6}} \cdot \Gamma(\frac{5}{6})}{3} $$

Revision
This is a proposed revision. Although current content is correct, a rewrite makes the content easier to understand for the reader. Additional content and references were added. If any concerns, please send me a message, and let us discuss. ThanksTMM53 (talk) 05:59, 21 May 2024 (UTC)