On a remarkable formula of Ramanujan

TL;DR

This paper provides a simple proof of Ramanujan's Fourier transform formula for the Gamma function's modulus squared, extends it to the left half-plane, and calculates the jump across the imaginary axis.

Contribution

It introduces a straightforward proof of Ramanujan's formula, extends the result to the left half-plane, and computes the discontinuity across the imaginary axis.

Findings

Proof of Ramanujan's Fourier transform formula

Extension to the left half-plane via ODE solution

Calculation of the jump across the imaginary axis

Abstract

A simple proof of Ramanujan's formula for the Fourier transform of the square of the modulus of the Gamma function restricted to a vertical line in the right half-plane is given. The result is extended to vertical lines in the left half-plane by solving an inhomogeneous ODE. We then use it to calculate the jump across the imaginary axis.