Hypergeometric functions Bessel functions

After Eulerian functions, which are universally used, it's undoubtedly the hypergeometric functions – the Gauss function and the confluent functions – that provide the simplest examples of the application of the fundamental processes of analysis. Indeed, the Gauss function, defined by an integer series, appears as a natural generalization of the geometric series, and is thus covered by the methods of analytic function theory. The same can be said of confluent functions, in particular Kummer's function, which generalizes the exponential function.

Although studied separately, Bessel functions are a notable special case of confluent hypergeometric functions, in that all their properties can be described from the latter.

What all these functions have in common is that they are each solutions of a second-order linear differential equation with polynomial coefficients: the Gauss equation, the Kummer equation and the Bessel equation. This fact is at the root of all the important properties of the functions considered. It is also responsible for the extraordinary development of the literature on hypergeometric functions, especially Bessel functions, since it is via the differential equation that these appear in many problems of mathematical physics (electrodynamics, vibration theory, heat theory), when the so-called separation of variables method is used to solve the equation in question.

In what follows, we deliberately take a basic approach, while endeavouring to be rigorous. Without claiming to be exhaustive – and by no means –, the subject matter covered provides an initial understanding of the problems, and enables more experienced readers to tackle the specialized works listed at the end of this article.