A study of bessel functions and regular coulomb wave functions

Abstract

This thesis presents a comprehensive study of the Bessel function and the regular Coulomb wave function, with a focus on their analytic properties, the distribution of their zeros, and the interrelations explored in the literature. The work begins with a detailed derivation of Bessel’s differential equation and systematically explores its solutions, Jν(z) and Yν(z), including their series representations, recurrence relations, and linear independence. Rigorous proofs are provided for the reality, simplicity, and interlacing properties of the zeros of Bessel functions and their derivatives, employing advanced tools such as the Weierstrass factorization theorem, Mittag-Leffler’s theorem, and Laguerre’s separation theorem. The generating function for Bessel functions is derived, and its implications for solution structures are discussed. The study extends to the regular Coulomb wave function FL(η, ρ), defining it via confluent hypergeometric functions and demonstrating its role as a one-parameter generalization of the Bessel function. The thesis investigates the reality and distribution of zeros of the Coulomb wave function, leveraging determinantal criteria (Grommer and Chebotarev theorems) and moment Hankel matrices. Special attention is given to recursive computations of sums over zeros and their connection to Rayleigh functions. The results yield explicit criteria for the number and nature (real or complex) of zeros.