Handbook of Continued Fractions for Special Functions

Annie A.M. Cuyt, Vigdis Petersen, Brigitte Verdonk, Haakon Waadeland, William B. Jones, F. Backeljauw, C. Bonan-Hamada9781402069482, 1402069480

Special functions are pervasive in all fields of science and industry. The most well-known application areas are in physics, engineering, chemistry, computer science and statistics. Because of their importance, several books and websites (see for instance http: functions.wolfram.com) and a large collection of papers have been devoted to these functions. Of the standard work on the subject, the Handbook of mathematical functions with formulas, graphs and mathematical tables edited by Milton Abramowitz and Irene Stegun, the American National Institute of Standards claims to have sold over 700 000 copies!

But so far no project has been devoted to the systematic study of continued fraction representations for these functions. This handbook is the result of such an endeavour. We emphasise that only 10% of the continued fractions contained in this book, can also be found in the Abramowitz and Stegun project or at the Wolfram website!

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Table of contents :
Cover……Page 1
Handbook of Continued Fractions for Special Functions……Page 2
Copyright……Page 4
Table of Contents……Page 5
Preface……Page 10
Notation……Page 12
0 General considerations……Page 16
Part I Basic Theory……Page 22
1 Basics……Page 23
2 Continued fraction representation of functions……Page 43
3 Convergence criteria……Page 58
4 Padé approximants……Page 72
5 Moment theory and orthogonal functions……Page 89
Part II Numerics……Page 117
6 Continued fraction construction……Page 118
7 Truncation error bounds……Page 140
8 Continued fraction evaluation……Page 159
Part III Special Functions……Page 170
9 On tables and graphs……Page 171
10 Mathematical constants……Page 182
11 Elementary functions……Page 199
12 Gamma function and related functions……Page 227
13 Error function and related integrals……Page 258
14 Exponential integrals and related functions……Page 279
15 Hypergeometric functions……Page 295
16 Confluent hypergeometric functions……Page 322
17 Bessel functions……Page 345
18 Probability functions……Page 372
19 Basic hypergeometric functions……Page 391
Bibliography……Page 401
Index……Page 420