This comprehensive, detailed reference to Mathematica provides the reader with both a working knowledge of Mathematica in general and a detailed knowledge of key aspects of Mathematica needed to create the fastest, shortest and most elegant implementations possible to solve problems from the natural sciences. The GuideBook gives the user a deeper understanding of Mathematica by instructive implementations, explanations, and examples from a range of disciplines at varying levels of complexity. The GuideBook is published in three volumes -- 1 -Programming, 2-Graphics, and 3-Mathematics -- each with a CD. The 3 volumes total 3000 pages and contain more than 15,000 Mathematica inputs, over 1500 graphics, 4000+ references, and more than 500 exercises. The Mathematica GuideBook is an indispensable resource for students, researchers and professionals in mathematics, the sciences and engineering and will thus find a natural place on the bookshelf next to The Mathematica Book. This third volume,3-Mathematics (chapters 10-14) covers Mathematica's mathematical capabilities. Chapter 10 deals with numerical calculations and is important for all Mathematica users, covering calculations involving real and complex numbers,then main numerical functions which are implemented in Mathematica - interpolation, Fourier transforms, solution of equations, minimization of functions, and the solution of differential equations. Chapters 11 and 12 deal with symbolic calculations using Mathematica - the real heart of the programming system, with concentration on topics that are important in classical analysis. Chapters 13 and 14 discuss classical orthogonal polynomials and the special functions of mathematical physics, focusing on the use of special functions in a number of physics- related applications in the text as well as in the exercises.