Integral and Series Representations of the Dirac Delta Function

TL;DR

This paper provides rigorous mathematical justifications for various integral and series representations of the Dirac delta function, commonly used in physics, using asymptotic analysis of special functions.

Contribution

It offers new rigorous proofs for existing integral and series representations of the Dirac delta function involving special functions.

Findings

Validates integral representations involving Airy and Coulomb functions

Establishes series representations with Laguerre polynomials and spherical harmonics

Uses asymptotic analysis to justify these representations

Abstract

Mathematical justifications are given for several integral and series representations of the Dirac delta function which appear in the physics literature. These include integrals of products of Airy functions, and of Coulomb wave functions; they also include series of products of Laguerre polynomials and of spherical harmonics. The methods used are essentially based on the asymptotic behavior of these special functions.