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Essential Mathematical Methods for the Physical Sciences by K. F. Riley, M. P. Hobson

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9

Special functions

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 ...

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