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

From: Martin Muldoon
muldoon@mathstat.yorku.ca

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.

**Contents**:

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)

Bibliography

Notations and Symbols

Index

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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