Zernike circle polynomials and infinite integrals involving the product of Bessel functions

TL;DR

This paper derives explicit formulas for infinite integrals involving Bessel functions related to Zernike circle polynomials, enabling precise calculations of various coefficients and responses in optical and acoustical applications.

Contribution

It explicitly evaluates integrals involving Bessel functions for Zernike polynomials, advancing analytical methods in optical and acoustical modeling.

Findings

Explicit formulas for expansion coefficients of scaled-and-shifted circle polynomials.

Closed-form expressions for correlation coefficients of circle polynomials.

Analytical solutions for Fourier coefficients and transient acoustical responses.

Abstract

Several quantities related to the Zernike circle polynomials admit an expression as an infinite integral involving the product of two or three Bessel functions. In this paper these integrals are identified and evaluated explicitly for the cases of (a) the expansion coefficients of scaled-and-shifted circle polynomials, (b) the expansion coefficients of the correlation of two circle polynomials, (c) the Fourier coefficients occurring in the cosine representation of the circle polynomials, (d) the transient response of a baffled-piston acoustical radiator due to a non-uniform velocity profile on the piston.