Ramanujan's Master Theorem and two formulas for zero-order Hankel transform

TL;DR

This paper derives two formulas for the zero-order Hankel transform of even functions using Ramanujan's Master Theorem, illustrating their validity with examples and computing a new parametric integral of J_0(x).

Contribution

It introduces novel formulas for Hankel transforms based on Ramanujan's Master Theorem, expanding analytical tools for integral transforms.

Findings

Derived two new formulas for Hankel transforms of order zero.

Validated formulas with multiple examples.

Computed a previously unknown integral involving J_0(x).

Abstract

Using Ramanujan's Master Theorem, two formulas are derived which define the Hankel transforms of order zero with even functions by inverse Mellin transforms, provided these functions and their derivatives obey special conditions. Their validity is illustrated by a number of examples. With a help of one of these formulas, one as yet unknown parametric improper integral of the Bessel function J_0(x) is calculated.