Chapter 2 - Mathematical Preliminaries

Abstract

In this chapter mathematical preliminaries for various standard functions used throughout the book are presented, such as the gamma and beta functions, the hypergeometric function, and the associated Legendre functions. The essential modelling content of the book is the use of the continuum assumption for atomic surface densities and the replacement of double summations with double surface integrals involving atomic potential functions. For certain surfaces these integrals can often be evaluated in terms of well-known analytical functions, but generally these integrations are highly nontrivial. The mathematical perspective here is that many of the integrals can be identified from integral representations of the various hypergeometric functions including Appell’s hypergeometric function. These formulae are important since they often lead to evaluating the integral in terms of other special functions, such as Legendre polynomials, and through algebraic packages such as MAPLE and MATLAB the hypergeometric functions can be readily evaluated numerically. This chapter ensures that the reader is fully conversant with the major properties of all the special functions needed to undertake this process including the Dirac delta function, the Heaviside unit function, the gamma and beta functions, and the hypergeometric functions.