On the Integral of the Product of Three Bessel Functions over an Infinite Domain

Abstract

Fourier-space representation of the partial differential equations describing nonlinear dynamics of continuous media in cylindrical geometry can be achieved using Chandrasekhar–Kendall (C–K) functions defined over infinite domain as an orthogonal basis for solenoidal vector fields and their generating function and its gradient as orthogonal bases for scalar and irrotational vector fields, respectively. All differential and integral operations involved in translating the partial differential equations into transform space are then carried out on the basis functions, leaving a set of time evolution equations, which describe the rate of change of the spectral coefficient of an evolving mode in terms of an aggregate effect of pairs of interacting modes computed as an integral over a product of spectral coefficients of two physical quantities along with a kernel, which involves the following integral: