Essential Mathematical Methods for the Physical Sciences

In the previous two chapters, we introduced the most important second-order linear ODEs in physics and engineering, listing their regular and irregular singular points in Table 7.1 and their Sturm–Liouville forms in Table 8.1. These equations occur with such frequency that solutions to them, which obey particular commonly occurring boundary conditions, have been extensively studied and given special names.

In this chapter, we discuss these so-called “special functions” and their properties. Inevitably, for each set of functions in turn, the discussion has to cover the differential equation they satisfy, their polynomial or power series form with some particular examples, their orthogonality and normalization properties, and their recurrence relations. In addition, as first introduced in this chapter, most sets possess a Rodrigues' formula and a generating function.

Although each of these aspects needs to be treated in sufficient detail for the enquiring reader to be satisfied about the validity of the results stated, their serial presentation, for one set of functions after another, tends to become rather overwhelming. Consequently it is suggested that once the reader has become familiar with the general nature of each of the aspects, by studying, say, Sections 9.1 to 9.3 on Legendre functions, associated Legendre functions and spherical harmonics, he or she may treat other sets of functions more lightly, turning in the first instance to the summary beginning on p. 377, and only referring to the detailed derivations, proofs and worked examples in Sections 9.4 to 9.9 when specific needs arise.