The perturbed Bessel equation, I. A Duality Theorem

TL;DR

This paper introduces a duality theorem for the perturbed Bessel equation by transforming it into a monodromic functional equation and extending classical formulas to broader function classes.

Contribution

It develops a duality theorem linking perturbed Bessel equations to monodromic functional equations, expanding Goursat's formula to larger function classes.

Findings

Established a duality theorem for perturbed Bessel equations.

Extended Goursat's hypergeometric function formula.

Linked differential equations with monodromic functional equations.

Abstract

The Euler-Gauss linear transformation formula for the hypergeometric function was extended by Goursat for the case of logarithmic singularities. By replacing the perturbed Bessel differential equation by a monodromic functional equation, and studying this equation separately from the differential equation by an appropriate Laplace-Borel technique, we associate with the latter equation another monodromic relation in the dual complex plane. This enables us to prove a duality theorem and to extend Goursat's formula to much larger classes of functions.