SIAM AG on Orthogonal Polynomials and Special Functions


Extract from OP-SF NET

Topic #10 ---------------- OP-SF NET ----------------- March 14, 1996

From: Martin Muldoon
Subject: New book on special functions

Nico M. Temme, Special Functions: an Introduction to the Classical Special Functions of Mathematical Physics, Wiley, New York-Chichester-Brisbane-Toronto-Singapore, 1996, xii + 374 pp, ISBN 0-471-11313-1.

1. Bernoulli, Euler and Stirling Numbers
2. Useful Methods and Techniques (Theorems from Analysis, Asymptotic Expansions of Integrals)
3. The Gamma Function
4. Differential Equations (Separating the Wave Equation, DE's in the Complex Plane, Sturm's Comparison Theorem, Integrals as Solutions, Liouville Transformation)
5. Hypergeometric Functions (includes very brief introduction to q-functions)
6. Orthogonal Polynomials
7. Confluent Hypergeometric Functions (includes many special cases)
8. Legendre Functions
9. Bessel Functions
10. Separating the Wave Equation
11. Special Statistical Distribution Functions (Error functions, Incomplete Gammma Functions, Incomplete Beta Functions, Non-Central Chi-Squared Distribution, Incomplete Bessel Function)
12. Elliptic Integrals and Elliptic Functions
13. Numerical Aspects of Special Functions (mainly recurrence relations)
Notations and Symbols

The author mentions that part of the material was collected from well-known books such as those by Hochstadt, Lebedev, Olver, Rainville, Szego, and Whittaker & Watson as well as lecture notes by Lauwerier and Boersma. But there is much recent material, especially in the areas of asymptotic expansions and numerical aspects. About half of the approximately 200 references are dated 1975 and later.

The author states that the book "has been written with students of mathematics, physics and engineering in mind, and also researchers in these areas who meet special functions in their work, and for whom the results are too scattered in the general literature." Complex analysis (especially contour integration) would appear to be the main prerequisite. The book is clearly written, with good motivating examples and exercises. For example, the first chapter opens with a remarkable example of Borwein, Borwein and Dilcher (Amer. Math. Monthly 96 (1989), 681-687 which explains, using Euler numbers and Boole's summation formula, why the sum of 50 000 terms of the very slowly converging alternating series for Pi/4, though it gives an answer correct to only six digits, yet has nearly all its first 50 digits correct.

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