Handbook of Continued Fractions for Special Functions

by Cuyt, Annie; Petersen, Vigdis Brevik; Verdonk, Brigitte; Waadeland, Haakon; Jones, William B.

• ISBN13: 9781402069482
• ISBN10: 1402069480
• Format: Hardcover
• Copyright: 2008-06-03
• Publisher: Springer Nature

Summary

"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 and a large collection of papers have been devoted to these functions. 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 special functions websites!"--BOOK JACKET.

Table of Contents

Preface	p. xi
Notation	p. xiii
General considerations	p. 1
Part one	p. 1
Part two	p. 2
Part three	p. 2
Basic Theory
Basics	p. 9
Symbols and notation	p. 9
Definitions	p. 10
Recurrence relations	p. 13
Equivalence transformations	p. 15
Contractions and extensions	p. 16
Continued fractions with prescribed approximants	p. 18
Connection between continued fractions and series	p. 19
Periodic and limit periodic continued fractions	p. 21
Tails of continued fractions	p. 23
Continued fractions over normed fields	p. 26
Generalisations of continued fractions	p. 28
Continued fraction representation of functions	p. 29
Symbols and notation	p. 29
Correspondence	p. 30
Families of continued fractions	p. 35
Correspondence of C-fractions	p. 39
Correspondence of P-fractions	p. 40
Correspondence of J-fractions and T-fractions	p. 41
Correspondence and three-term recurrences	p. 42
Convergence criteria	p. 45
Some classical theorems	p. 45
Convergence sets and value sets	p. 47
Parabola and oval theorems	p. 49
Correspondence and uniform convergence	p. 52
Periodic and limit periodic continued fractions	p. 53
Convergence and minimal solutions	p. 56
Pade approximants	p. 59
Definition and notation	p. 59
Fundamental properties	p. 60
Connection with regular C-fractions	p. 64
Connection with P-fractions	p. 65
Extension of the Pade table	p. 67
Connection with M-fractions and the M-table	p. 68
Convergence of Pade approximants	p. 70
Formal orthogonality property	p. 72
Moment theory and orthogonal functions	p. 77
Moment theory	p. 77
Stieltjes transforms	p. 85
Construction of solutions	p. 90
Orthogonal polynomials	p. 91
Monic orthogonal polynomials on R and J-fractions	p. 92
Szego polynomials and PPC-fractions	p. 100
Orthogonal Laurent polynomials and APT-fractions	p. 102
Numerics
Continued fraction construction	p. 107
Regular C-fractions	p. 107
C-fractions	p. 113
S-fractions	p. 114
P-fractions	p. 11
J-fractions	p. 120
M-fractions	p. 122
Positive T-fractions	p. 124
Thiele fractions	p. 125
Truncation error bounds	p. 129
Parabola theorems	p. 129
The oval sequence theorem	p. 131
The interval sequence theorem	p. 136
Specific a priori bounds for S-fractions	p. 138
A posteriori truncation error bounds	p. 140
Tails and truncation error bounds	p. 143
Choice of modification	p. 143
Continued fraction evaluation	p. 149
The effect of finite precision arithmetic	p. 149
Evaluation of approximants	p. 152
The forward recurrence and minimal solutions	p. 154
Round-off error in the backward recurrence	p. 156
Special Functions
On tables and graphs	p. 163
Introduction	p. 163
Comparative tables	p. 163
Reliable graphs	p. 168
Mathematical constants	p. 175
Regular continued fractions	p. 175
Archimedes' constant, symbol [pi]	p. 176
Euler's number, base of the natural logarithm	p. 178
Integer powers and roots of [pi] and e	p. 180
The natural logarithm, ln(2)	p. 181
Pythagoras' constant, the square root of two	p. 183
The cube root of two	p. 183
Euler's constant, symbol [gamma]	p. 185
Golden ratio, symbol [phi]	p. 185
The rabbit constant, symbol [rho]	p. 186
Apery's constant, [zeta](3)	p. 188
Catalan's constant, symbol C	p. 189
Gompertz' constant, symbol G	p. 190
Khinchin's constant, symbol K	p. 190
Elementary functions	p. 193
The exponential function	p. 193
The natural logarithm	p. 196
Trigonometric functions	p. 200
Inverse trigonometric functions	p. 204
Hyperbolic functions	p. 210
Inverse hyperbolic functions	p. 213
The power function	p. 217
Gamma function and related functions	p. 221
Gamma function	p. 221
Binet function	p. 224
Polygamma functions	p. 229
Trigamma function	p. 232
Tetragamma function	p. 235
Incomplete gamma functions	p. 238
Error function and related integrals	p. 253
Error function and Dawson's integral	p. 253
Complementary and complex error function	p. 261
Repeated integrals	p. 268
Fresnel integrals	p. 269
Exponential integrals and related functions	p. 275
Exponential integrals	p. 275
Related functions	p. 285
Hypergeometric functions	p. 291
Definition and basic properties	p. 291
Stieltjes transform	p. 295
Continued fraction representations	p. 295
Pade approximants	p. 309
Monotonicity properties	p. 313
Hypergeometric series [subscript p]F[subscript q]	p. 315
Confluent hypergeometric functions	p. 319
Kummer functions	p. 319
Confluent hypergeometric series [subscript 2]F[subscript 0]	p. 330
Confluent hypergeometric limit function	p. 333
Whittaker functions	p. 334
Parabolic cylinder functions	p. 337
Bessel functions	p. 343
Bessel functions	p. 343
Modified Bessel functions	p. 356
Probability functions	p. 371
Definitions and elementary properties	p. 371
Normal and log-normal distributions	p. 373
Repeated integrals	p. 377
Gamma and chi-square distribution	p. 378
Beta, F- and Student's t-distributions	p. 382
Basic hypergeometric functions	p. 391
Definition and basic properties	p. 391
Continued fraction representations	p. 395
Higher order basic hypergeometric functions	p. 399
Bibliography	p. 401
Index	p. 421
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