There is nothing particularly special about a special function, except that some person in authority or textbook writer (not the same thing!) has decided to bestow the moniker. Special functions are sometimes called higher transcendental functions (higher than what?) | Chapter 6. Special Functions Introduction There is nothing particularly special about a special function except that some person in authority or textbook writer not the same thing has decided to bestow the moniker. Special functions are sometimes called higher transcendental functions higher than what orfunctions of mathematical physics but they occur in other fields also or functions that satisfy certain frequently occurring second-order differential equations but not all special functions do . One might simply call them useful functions and let it go at that it is surely only a matter of taste which functions we have chosen to include in this chapter. Good commercially available program libraries such as NAG or IMSL contain routines for a number of special functions. These routines are intended for users who will have no idea what goes on inside them. Such state of the art black boxes are often very messy things full of branches to completely different methods depending on the value of the calling arguments. Black boxes have or should have careful control of accuracy to some stated uniform precision in all regimes. We will not be quite so fastidious in our examples in part because we want to illustrate techniques from Chapter 5 and in part because we want you to understand what goes on in the routines presented. Some of our routines have an accuracy parameter that can be made as small as desired while others especially those involving polynomial fits give only a certain accuracy one that we believe serviceable typically six significant figures or more . We do not certify that the routines are perfect black boxes. We do hope that if you ever encounter trouble in a routine you will be able to diagnose and correct the problem on the basis of the information that we have given. In short the special function routines of this chapter are meant to be used we use them all the time but we also want you to be prepared to understand their inner workings. CITED .
