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The functions discussed in this chapter form the most unique part of ’s mathematical capabilities. This explains the relatively large size of this chapter as well as the relatively large number of exercises. can only be considered a complete system for doing various parts of mathematics and its applications on a computer if we include all of its graphical, pattern recognition, programming, numerical, and symbolic capabilities. On the other hand, capabilities such as factorization of polynomials, differentiation, integration, solution of equations and differential equations, computation of sums, products, and limit values can hardly be duplicated by other conventional programming systems that can handle graphics and numerical computations. So the commands for the operations Factor, D, Integrate, FourierTransform, Solve, Reduce, Root, CylindricalDecomposition, Eliminate, Resultant, GroebnerBasis, DSolve, Limit, and Series are in some sense the most important functions in and the content of this chapter. As already discussed in the beginning (in Chapter 2 of the Programming volume []), every object in is an expression. This uniformity of the underlying data structures allows symbolic expressions to be integrated uniformly into the whole system and to work naturally together with the graphical, pattern recognition, programming, and numerical functions of .

In this chapter, we discuss the classical orthogonal polynomials. Our main purpose is to provide formulas that uniquely define the polynomials (including normalization factors) and to demonstrate some of ’s integration, differentiation, and series expansion capabilities. Further, we will make use of graphics to visualize the orthogonal polynomials in a variety of ways. In this chapter, we sometimes use the word “check” not in the sense of a mathematical proof, but rather in the sense of checking formulas by means of special cases and computational examples. (For the use of as a theorem prover, see [], [].) Occasionally we will encounter special functions (especially the Gamma function and the Gauss hypergeometric function) in this chapter when carrying out integrals involving orthogonal polynomials. We will discuss these functions in the next chapter.

This chapter presents some of the special functions that are among the most important for technical and scientific applications. We do not attempt to discuss all special functions that are built-in , not all of their important properties and applications, nor do we look at all of them graphically. Moreover, in contrast to the previous chapter, we do not discuss the corresponding differential equations, series expansions, or differentiation and integration formulas. Instead, our aim is to introduce the nomenclature, to illustrate the power of as a tool for dealing with special functions, and to show how they can be used to provide effective solutions to a variety of problems. So some famous functions (like the Riemann Zeta function) will not be discussed in the following sections because it would be outside of the scope of the , but we will nevertheless refer to some of the not-discussed functions in the exercises of this chapter. For a quite complete treatment, including series, product, and integral representations, as well as visualizations of all special functions of , see the comprehensive site http://functions.wolfram.com (and the forthcoming sites http://dlmf.nist.gov [], [] and http://algo.inria.fr/esf []).