Handbook of continued fractions for special functions

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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 col
Annie Cut. Vigdis brevik petersen Brigitte verdonk. Haakon Waadeland Williams. jones Handbook of Continued fractions for Special Functions With contributions by Franky Backeljauw. Catherine Bonan-Hamada Verified numerical output Stefan Becuwe. Annie Cut S pringer Annie Cut University of Antwerp Department of Mathematics and Computer Science BE-2020 Antwerpen B elgin Vigdis brevik petersen Sor-Trondelag University College Faculty of Teacher and Interpreter Education No-7004 Trondheim Norway Brigitte verdonk University of Antwerp Department of Mathematics and Computer Science BE-2020 Antwerpen Belgium Haakon waadeland Norwegian University of Science and Technology Department of mathematical Sciences No-7491 Trondheim Norway William b. jones University of Colorado Department of mathematics Boulder Co 80309-0395 USA ISBN:978-1-4020-6948-2 e-ISBN:978-1-4020-69499 Library of congress control Number: 2007941383 c 2008 Springer Science+Business Media B V No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the publisher with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Printed on acid-free paper 987654321 Springer. com TABLE OF CONTENTS Preface Notation X111 0 General considerations 0. 1 Part one 0.2 Part two 0.3 Part three Part I: BASIC THEORY 1 Basics 1.1 Symbols and notation 1.2 Definitions 10 1.3 Recurrence relations 13 1.4 Equivalence transformations 15 1.5 Contractions and extensions 16 1.6 Continued fractions with prescribed approximants 18 1.7 Connection between continued fractions and series 19 Periodic and limit periodic continued fractions ...... 21 1.9 Tails of continued fractions 1.10 Continued fractions over normed fields 26 1.11 Generalisations of continued fractions 28 2 Continued fraction representation of functions .29 2.1 Symbols and notation ..29 2.2 Correspondence 30 2.3 Families of continued fractions 35 2.4 Correspondence of C-fractions 2.5 Correspondence of P-fractions 40 2.6 Correspondence of J-fractions and T-fractions ..41 2.7 Correspondence and three-term recurrences 42 3 Convergence criteria 45 3.1 Some classical theorems 45 3.2 Convergence sets and value sets 47 3.3 Parabola and oval theorems 49 TABLE OF CONTENTS 3.4 Correspondence and uniform convergence .52 3.5 Periodic and limit periodic continued fractions 53 3.6 Convergence and minimal solutions .56 4 Pade approximants 59 4.1 Definition and notation 59 4.2 Fundamental properties 60 4.3 Connection with regular C-fractions 64 4.4 Connection with P-fractions ..65 4.5 Extension of the Pade table ..67 4.6 Connection with M-fractions and the M-table 68 4.7 Convergence of Pade approximants 70 4.8 Formal orthogonality propert 72 5 moment theory and orthogonal functions 77 5.1 Moment theory .77 5.2 Stieltjes transforms 5.3 Construction of solutions 5.4 Orthogonal polynomials 91 5.5 Monic orthogonal polynomials on R and -fractions 92 5.6 Szego polynomials and ppc-fractions 5.7 Orthogonal Laurent polynomials and APT-fractions... 102 Part II: NUMERICS 6 Continued fraction construction 107 1 Regular C-fractions 107 6.2 C-fractions .113 6.3 S-fractions .114 6.4 P-fractions 114 6.5 J-fractions 120 6.6 M-fractions 122 6.7 Positive t-fractions .....,124 6.8 Thiele fractions 125 7 Truncation error bounds 129 7.1 Parabola theorems 129 7.2 The oval sequence theorem 131 7.3Th l sequence the 136 7.4 Specific a priori bounds for S-fractions 138 7.5 A posteriori truncation error bounds 140 TABLE OF CONTENTS 7. 6 Tails and truncation error bounds ...143 7. 7 Choice of modification 143 Continued fraction evaluation 149 The effect of finite precision arithmetic 149 8.2 Evaluation of approximants 152 3 The forward recurrence and minimal solutions 154 8. 4 Round-off error in the backward recurrence ......156 Part III: SPECIAL FUNCTIONS 9 On tables and graphs 163 9.1 Introduction 163 9.2 Comparative tables 163 9.3 Reliable graphs 168 10 Mathematical constants 175 10.1 Regular continued fractions 175 10.2 Archimedes?constant, symbol T 176 10.3 Euler's number, base of the natural logarithm .........178 10.4 Integer powers and roots of T and e 180 10.5 The natural logarithm, In(2) 181 10.6 Pythagoras'constant, the square root of two 183 10.7 The cube root of two 183 10.8 Eulers constant, symbol 185 10.9 Golden ratio, symbol g 185 10.10 The rabbit constant, symbol P ...186 10.11 Apery's constant, S 3) 188 10.12 Catalans constant, symbol C 10.13 Gompertz' constant, symbol G 10.14 Khinchin constant, symbol K 190 11 Elementary functions 193 11.1 The exponential functio ..193 11.2 The natural logarithm 196 11.3 Trigonometric functions 200 11. 4 Inverse trigonometric functions 11.5 Hyperbolic functions 210 11.6 Inverse hyperbolic functions 213 11.7 The power function 217 TABLE OF CONTENTS 12 Gamma function and related functions 221 12.1 Gamma function 221 12.2 Binet function 224 12.3 Polygamma functions 229 12. 4 Trigamma function 232 12.5 Tetragamma function 235 12.6 Incomplete gamma functions 238 13 Error function and related integrals 253 13. 1 Error function and Dawson's integral ..253 13.2 Complementary and complex error function ∴......261 13.3 Repeated integrals 268 13.4 Fresnel integrals 269 14 Exponential integrals and related functions 14.1 Exponential integrals 275 14.2 Related functi 285 15 Hypergeometric functions 291 15.1 Definition and basic properties 291 15.2 Stieltjes transform 295 15.3 Continued fraction representations 295 15.4 Pade approximants .....309 15.5 Monotonicity properties .313 15.6 Hypergeometric series p Fq 315 16 Confluent hypergeometric functions 319 16. 1 Kummer functions 319 16.2 Confluent hypergeometric series 2 Fo 330 16.3 Confluent hypergeometric limit function 33 16.4 Whittaker functions .334 16.5 Parabolic cylinder functions 337 17 Bessel functions 343 17.1 Bessel functions 343 17.2 Modified bessel functions 356 18 Probability functions 371 18.1 Definitions and elementary properties ..371 18.2 Normal and log-normal distributions ..373 18.3 Repeated integrals .377 18.4 Gamma and chi-square distribution ..378 TABLE OF CONTENTS 18.5 Beta. F- and student's t-distributions 382 19 Basic hypergeometric functions 391 19.1 Definition and basic properties .....391 19.2 Continued fraction representations .395 19.3 Higher order basic hypergeometric functions 399 Bibliography 401 Index 421 PREFACE The idea to write a Handbook of Continued fractions for Special functions originated more than 15 years ago, but the project only got started end of 2001 when a pair of Belgian and a pair of norwegian authors agreed to join forces with the initiator W. B Jones. The book splits naturally into three parts: Part i discussing the concept, correspondence and conver- gence of continued fractions as well as the relation to Pade approximants and orthogonal polynomials, Part II on the numerical computation of the continued fraction elements and approximants, the truncation and round- off error bounds and finally Part Ill on the families of special functions for which we present continued fraction representations 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 functions. wolfram. com)and a large col- lection of papers have been devoted to these functions. of the standard work on the subject, the Handbook of mathematical functions with for mulas, graphs and mathematical tables edited by Milton Abramowitz and Irene stegun, the American National Institute of Standards and Technol- ogy claims to have sold over 700 000 copies(over 150 000 directly and more than fourfold that number through commercial publishers)! 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 The fact that the belgian and norwegian authors could collaborate in pairs at their respective home institutes in Antwerp(Belgium) and Trondheim (Norway) offered clear advantages. Nevertheless, most progress with the manuscript was booked during the so-called handbook workshops which were organised at regular intervals, three to four times a year, by the first four authors A. Cuyt, V.B. Petersen, B. Verdonk and H. Waadeland. They got together a staggering 16 times, at different host institutes, for a total of 28 weeks to compose. streamline and discuss the contents of the different chapters The Belgian and Norwegian pair were also welcomed for two or more weeks at the MFO(Oberwolfach, Germany), CWI(Amsterdam, The Nether lds), University of La Laguna(Tenerife, Spain), the University of Stel lenbosch(South-Africa) and last, but certainly not least, the University of

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