Ramanujan's master theorem explained

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 $$f(x)=\backslash sum\_^\backslash infty\; \backslash frac(-x)^k$$

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

$$\backslash int\_0^\backslash infty\; x^\; f(x)\; \backslash ,\; dx\; =\; \backslash Gamma(s)\backslash ,\backslash varphi(-s)$$

where $\backslash Gamma(s)$ 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:

$$\backslash int\_0^\backslash infty\; x^\backslash left(\backslash ,\backslash lambda(0)\; -\; x\backslash ,\backslash lambda(1)\; +\; x^2\backslash ,\backslash lambda(2)\; -\backslash ,\backslash cdots\backslash ,\backslash right)\; dx\; =\; \backslash frac\backslash ,\backslash lambda(-s)$$

which gets converted to the above form after substituting $\backslash lambda(n)\; \backslash equiv\; \backslash frac$ and using the functional equation for the gamma function.

The integral above is convergent for subject to growth conditions on $\backslash varphi$.

Proof

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

$$\backslash frac=\backslash sum\_^\backslash infty\; B\_k(x)\backslash ,\backslash frac$$

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

$$\backslash zeta(s,a)\; =\; \backslash sum\_^\backslash infty\; \backslash frac$$

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

$$\backslash int\_0^\backslash infty\; x^\backslash left(\backslash frac-\backslash frac\backslash right)\; dx\; =\; \backslash Gamma(s)\backslash ,\backslash zeta(s,a)\; \backslash !$$

which is valid for .

Application to the gamma function

Weierstrass's definition of the gamma function

$$\backslash Gamma(x)\; =\; \backslash frac\backslash ,\backslash prod\_^\backslash infty\; \backslash left(\backslash ,1\; +\; \backslash frac\backslash ,\backslash right)^\; e^\; \backslash !$$

is equivalent to expression

$$\backslash log\backslash Gamma(1+x)\; =\; -\backslash gamma\backslash ,x\; +\; \backslash sum\_^\backslash infty\; \backslash frac\backslash ,(-x)^k$$

where $\backslash zeta(k)$ is the Riemann zeta function.

Then applying Ramanujan master theorem we have:

$$\backslash int\_0^\backslash infty\; x^\; \backslash frac\; \backslash mathrm\; d\; x\; =\; \backslash frac\backslash frac\; \backslash !$$

valid for .

Special cases of $s\; =\; \backslash frac$ and $s\; =\; \backslash frac$ are

$$\backslash int\_0^\backslash infty\; \backslash frac\; \backslash ,\; \backslash mathrm\; d\; x\; =\; \backslash frac\backslash ,\backslash zeta\backslash left(\backslash frac\; \backslash right)$$

$$\backslash int\_0^\backslash infty\; \backslash frac\; \backslash ,\; \backslash mathrm\; d\; x\; =\; \backslash sqrt\; \backslash frac\; \backslash zeta\backslash left(\backslash frac\; 5\; 4\backslash right)$$

Application to Bessel functions

The Bessel function of the first kind has the power series$$J\_\backslash nu(z)=\backslash sum\_^\backslash infty\; \backslash frac\backslash bigg(\backslash frac\backslash bigg)^$$

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

$$\backslash frac\; \backslash int\_0^\backslash infty\; z^J\_\backslash nu(\backslash sqrt)\backslash ,dz\; =\; \backslash Gamma(s)\backslash Gamma(s-\backslash nu)$$

valid for .

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

$$\backslash frac\; \backslash int\_0^\backslash infty\; z^j\_\backslash nu(\backslash sqrt)\backslash ,dz\; =\; \backslash Gamma(s)\backslash Gamma\backslash bigg(\backslash frac+s-\backslash nu\backslash bigg)$$

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 $J\_0(\backslash sqrt\; z)$ gives the square of the gamma function, $j\_0(\backslash sqrt)$ gives the duplication formula, $z^J\_(\backslash sqrt\; z)$ gives the reflection formula, and fixing to the evaluable $s=\backslash frac$ or $s=1$ gives the gamma function by itself, up to reflection and scaling.

Bracket integration method

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.

Integration formulas

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 $n^$ or array of parameter values $N^$ that solves one or more linear equations derived from the exponent terms of the integrand's series expansion.

Consecutive integer exponents, 1 variable

This is the function series expansion, integral and integration formula for an integral whose integrand's series expansion contains consecutive integer exponents. The parameter

n ^{\ast} is a solution to this linear equation.$$n^+c=0,\; \backslash \; n^=\; -c$$

General exponents, 1 variable

Applying the substitution $y=x^$ 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 $n^$ is a solution to this linear equation.$$a\; \backslash \; n^+c=0,\; \backslash \; n^=-a^\; c$$

Consecutive integer exponents, double integral

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 $n^\_$ and $n^\_$ are solutions to these linear equations.$$n^\_+c\_=0,\; \backslash \; n^\_+c\_=0,\; \backslash \; n^\_=-c\_,\; \backslash \; n^\_=-c\_$$

General exponents, double integral

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 $A$ is $\backslash det|A|$. Applying the substitution $$y\_=x^\_\; x^\_,\; \backslash quad\; y\_=x^\_\; x^\_$$ 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 $N^$ is a solution to this linear equation.$$AN^+C=0,\; \backslash \; N^=-A^\; C$$.

Positive complexity index

In some cases, there may be more sums then 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 number of brackets is the number of linear equations associated with an integral. This term reflects the common practice of bracketing each linear equation.
• The complexity index is the number of integrand sums minus the number of brackets (linear equations). Each series expansion of the integrand contributes one sum.
• The summation indices (variables) are the indices that index terms in a series expansion. In the example, there are 3 summation indices $n\_,\; n\_$ and $n\_$ because the integrand is a product of 3 series expansions.
• The free summation indices (variables) are the summation indices that remain after completing all integrations. Integration reduces the number of sums in the integrand by replacing the series expansions (sums) with an integration formula. Therefore, there are fewer summation indices after integration. The number of chosen free summation indices equals the complexity index.

Integrals with a positive complexity index

The free summation indices $\backslash bar\_,\backslash ldots,\; \backslash bar\_$ are elements of set $F$. The matrix of free summation indices is $\backslash bar$ and the coefficients of the free summation indices is matrix $\backslash bar$.The remaining indices are set $B$ containing indices $n\_,\backslash ldots,\; n\_$. Matrices $A,\; C$ and $N^$ contain matrix elements that multiply or sum with the non-summation indices. The selected free summation indices must leave matrix $A$ non-singular. . This is the function's series expansion, integral and integration formula. The parameters $n^\_,\; \backslash ldots,\; n^\_$ are linear functions of the parameters $\backslash bar^\_,\; \backslash ldots,\; \backslash bar^\_$. $$A\; \backslash \; N^+\backslash bar\; \backslash \; \backslash bar+C=0,\; \backslash \; N^=\; -A^(\backslash bar\; \backslash \; \backslash bar+C)$$.

Bracket series

Table 1. Bracket series notations

Notation type	Power series notation	Bracket series notation
Indicator	$$\backslash frac$$	$$\backslash phi\_$$
Multi-indicator	$$\backslash prod\_^N\; \backslash left(\backslash frac\; \backslash right)$$	$$\backslash phi\_$$
Bracket	$$\backslash int^\_\; dx\; \backslash \; x^$$	$$\backslash langle\; a\_n\_+\; \backslash ldots\; +\; a\_n\_+c\; \backslash rangle$$

Bracket series 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: $$\backslash frac$$with this bracket series expression:$$\backslash sum^\_\; \backslash ldots\; \backslash sum^\_\; \backslash \; \backslash phi\_\; \backslash \; x^\_\; \backslash dots\; x^\_\backslash frac.$$

Algorithm

This algorithm describes how to apply the integral formulas.

Table 2. Integral formulas

Complexity index	Integral formula
Zero, single integral	$$a^\; \backslash \; \backslash Gamma(-n^)\backslash ,\backslash varphi(n^)$$
Zero, multiple integral	$$\backslash det$$	A	^ \ \Gamma(-n^_) \ldots \Gamma(-n^_) \ \varphi(n^_, \ldots, n^_)
Positive	$$\backslash det$$	A	^ \sum^\infty_ \frac \ldots \frac \ \Gamma(-n^_) \ldots \Gamma(-n^_) \ \varphi(n^_, \ldots, n^_, \bar_, \dots, \bar_)

Input Integral expression

Output Integral value or integral cannot be assigned a value

1. Express the integrand as a power series.

1. Transform the integrand's power series to a bracket series.

1. Obtain the complexity index, formula parameters and series coefficient function.

1. 1. Complexity index is the number of integrand sums minus number of brackets.

1. 1. Parameters $n^$ or array $N^$ are solutions to linear equations $an^+c=0$ (zero complexity index, single integral), $AN^+C=0$ (zero complexity index, single integral) or $AN^+\backslash bar\; \backslash bar+C=0$ (positive complexity index).

1. 1. Identify parameter $a$ or (zero complexity index, single integral) or compute $\backslash det\; |A|$ (all other cases) from the associated linear equations.

1. 1. Identify the series coefficient function $\backslash varphi$ of the bracket series.

1. If the complexity index is negative, return integral cannot be assigned a value.

1. If the complexity index is zero, select the formula from table 2 for zero complexity index, single or multiple integral, compute the integral value with this formula, and return this integral value.

1. If the complexity index is positive, select the formula from table 2 for positive complexity index, and compute the integral value as a series expansion with this formula for all possible choices of the free summation indices. Select the lowest complexity index, convergent series expansion, adding series that converge in the same region.

1. 1. If all series expansions are divergent series or null series (all series terms zero), then return integral cannot be assigned a value.

1. 1. If the series expansion is non-null and non-divergent, return this series expansion as the integral value.

Examples

Zero complexity index

The bracket method will integrate this integral. $$\backslash int\_0^\backslash infty\; x^\; \backslash \; e^\; \backslash \; dx$$

1. Express the integrand as a power series.$$\backslash int\_0^\backslash infty\; \backslash sum\_^\backslash infty\; 2^\; \backslash \; \backslash frac\; \backslash \; x^\; \backslash \; dx$$
2. Transform the power series to a bracket series.$$\backslash sum\_^\backslash infty\; 2^\; \backslash \; \backslash phi(n)\; \backslash cdot\; \backslash left\; \backslash langle\; 3\; \backslash \; n+\; \backslash frac\; \backslash right\; \backslash rangle$$
3. Obtain the complexity index, formula parameters and series coefficient function.

Complexity index is zero.

$3\; \backslash \; n^+\; 5/2=0$

$n^=\; -5/6,\; \backslash \; a=3$

$\backslash varphi(n)=2^$.

4. Use table 2 to compute the integral.

$$\backslash int\_0^\backslash infty\; x^\; \backslash cdot\; e^\; \backslash \; dx$$$$=a^\; \backslash \; \backslash Gamma(-n^)\; \backslash \; \backslash varphi(n^)$$ $$=\; \backslash frac$$

Positive complexity index

The bracket method will integrate this integral.$$\backslash int^\_\; \backslash frac\; \backslash \; dx$$ 1. Express the integrand as a power series. Use the sum raised to a power formula.$$\backslash int^\_\; \backslash sum\_\; \backslash \; \backslash frac\; \backslash phi\_\; 1^\; x^\; \backslash langle\; n\_+n\_+n\_+1/2\; \backslash rangle\; \backslash \; dx$$2. Transform the power series to a bracket series. $$\backslash int^\_\; \backslash sum\_\; \backslash frac\; \backslash phi\_\; \backslash langle\; 5\; \backslash \; n\_+\; 3\; \backslash \; n\_+1\; \backslash rangle\; \backslash langle\; n\_+n\_+n\_+1/2\; \backslash rangle$$3. Obtain the complexity index, formula parameters and series coefficient function.

Complexity index is 1 as 3 sums and 2 brackets.

Select $n\_$ as the free index, $\backslash bar\_$. The linear equations, solutions, determinant and series coefficient are$$5n^\_+\; 3\; \backslash bar\_+1=0,\; \backslash \; n^\_+n^\_+\backslash bar\_+1/2=0$$ $$AN^\; +\; \backslash bar\; \backslash bar\; +\; C\; =0$$$$\backslash det\; |A|=5$$ $$n^\_=\; -\backslash frac\; \backslash bar\_-\backslash frac,\; \backslash \; n^\_=\; -\backslash frac\; \backslash bar\_-\backslash frac.$$ $$\backslash varphi(n^\_,n^\_,\; \backslash bar\_)=\; \backslash frac=\backslash frac$$ 4. Use table 2 to compute the integral

References

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• Book: Berndt, B. . Ramanujan's Notebooks, Part I . Springer-Verlag . New York . 1985 .
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• Gonzalez . Ivan . Jiu . Lin . Moll . Victor H. . An extension of the method of brackets. Part 2 . Open Mathematics . 2020 . 18 . 1 . 983–995 . 10.1515/math-2020-0062 . en . 2391-5455. free . 1707.08942 .
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