M. M. Agrest· M. S. Maksimov
Theory of Incomplete Cylindrical Funetions and their Applications
Translated from the Russian by H. E. Fettis J.W. Goresh D. A. Lee
With 20 Figures
Springer-Verlag Berlin Heidelberg NewYork 1971 Professor Matest M. Agrest Professor Michail S. :.vlaksimov
SuchumijUSSR SucbumijUSSR
Title of the Russian Original Edition: Teorija nepolnych zilindritscheskich funkzij i ejo prilosheniia Publisher: Atomizdat, Moscow/USSR, 1965
Translators:
Henry E. Fettis John W. Goresh t David A. Lee J\Iathematician Aerospace Engineer ::\1 a thematician Applied l\{athcmatics Hypcrsonic· Research Applied Mathcmatics Research Laboratory Facility Research Laboratory
Aerospace Research Laboratories Wright Patterson Air Force Base, Ohio. USA_
Geschäftsführcnde Herausgeber:
Professor Dr. B. Eckmann
Eidgenössische Technische Hochschule Zürich
Professor Dr. B. L. van der \Vaerden Mathematisches Iustitut der Universität Zürich
AMS Subject Classifications (1970) Primary 3302 - 33A40 - 33A 70 - G5AOõ G5D 20 Secondary 78A3õ - 78A45 - 81AG3 - 81A69
ISBN -13: 978-3-642-65023-9 e-ISBN -13:978-3-642-65021-5 DOI: 10.1007/978-.3-642-65021-5
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, rc-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copy• right Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag, Berlin' Heidelberg 1971. Softeover reprint of the hardeover 1st edition 1971
Library 01 Congress Catalog Card Number 78·139673 M. M. Agrest· M. S. Maksimov
Theory of Incomplete Cylindrical Funetions and their Applications
Translated from the Russian by H. E. Fettis J.W. Goresh D. A. Lee
With 20 Figures
Springer-Verlag NewYork Heidelberg Berlin 1971 Professor Matest M. Agrest Professor Michail S. :.\JIaksimov
SuchumijUSSR SuchumijUSSK
Title of the Russian Original Edition: Teorija nepolnych zilindritscheskich funkzij i ejo priloshenija Publisher: Atomizdat, MoscowjUSSR, 1965
Transla tors :
Henry E. Fettis John W. Goresh t David A. Lee Mathcmatician Aerospace Engineer :\lathematician Applied :\fathcmatics Hypersonic Research Applied !\lathcmatics Research Laboratory Facility Research Lauoratory
Aeraspaee Research Laboratüries Wright Patterson Air Force Base, Ohio, USA
Geschäftsführende Herausgeber:
Professor Dr. B. Eckmann
Eidgenössische Technisehe Hochschule Zürich
Professor Dr. B. L. van der 'Vaerdcn Mathematisches Institut der Universität Zürich
AMS Subj ect Classifications (1970) Primary 3302 - 33A40 - 33A70 - GöAOö !3öD20 Secondary 78A3ö - 78A4ö - 81AG3 - 81AG9
ISBN 0-387-0.1111-2 Springer-Verlag New York-Heidelberg-Berlin ISBN 3-ö40-0ö111-2 Springer-Verlag Berlin-Heidelberg-New York
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerncd, specificaIIy those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine Dr similaI' me ans, and storage in data banks. Under § 54 of the German Copy• right Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag, Berlin· Heidclberg 1971. Printed in Germany. Library 01 Caugress Catalog Card Number 78-139673 Preface to the English Edition In preparing the English edition of this unique work, every effort has been made to obtain an easily read and lueid exposition of the material. This has frequently been done at the expense of a literal translation of the original text and it is felt that such liberties as have been taken with the author's language are justified in the interest of ease in readingo None of us pretends to be an authority in the Russian language, and we trust that the original intent of the authors has not been lost. The equations, whieh were for the most part taken verbatim from the original work, were eheeked only eursorily; obvious and previously noted errors have been eorreeted. Fortunately, the Russian and English mathematieal notations are generally in good agreement. An exeeption is the shortened abbreviations for the hyperbolie functions (e.g. sh for sinh), and the symbol Jm rather that Im to denote the imaginary part. As near as possible, these diserepaneies have been correeted. In preparing the Bibliography, works having an English equivalent have been translated into the English title, but in the text the referenee to the Russian work was retained, as it was impraetieal to attempt to find in eaeh ease the eorresponding eitation in the English edition. Authors' names and titles associated with purely Russian works have been transliterated as nearly as possible to the English equivalent, along with the equivalent English title of the work cited. The numerieal tables appear exaetly as they did in the original work, and no attempt was made to eheek their aeeuraey. This may be done at a later date. The translators' interest in this subjeet goes baek many years. One of them first eneountered these funetions in eonneetion with a problem in unsteady aerodynamies, and later used it as a thesis subjeet. AGREST and MAKSIMOV have extended the theory mueh further than could ori• ginally have been expected, and have included a wealth of appliea• tions in almost all fields of mathematical physics.
Ohio, May 1971 Aerospace Research Laboratories Wright Patterson Air Force Base H. E. FETTIS· J. W. GORESH t . D. A. LEE Contents
List of Symbols IX
Introduction. . 1
Chapter I. Some Informatian from the Theory of Cylindrieal Functions 5 1. The Differential Equation and Reeursion Rclationships for Cylin- drieal Functions ...... 5 2. Generalized System of Funetional Equations for Cylindrieal Functions and the Inhomogenious Bessel Differential Equation ...... 8 3. Integral Repesentations of Cylindrical Functions in the Poissan Form 11 4. Integral Representations of Cylindrieal Functions of the Bessel• Sehlaefli and Sonine Form...... 17
Chapter II. General Theory of Ineomplete Cylindrieal Functions Expressed in Poissan Form ...... 21 1. Definitions of Ineomplete Cylindrieal Functions as Poissan Integrals 22 2. Reeurrenee Relations for the Ineomplete Cylindrieal Funetions 27 3. Some General Properties of Ineomplete Cylindrieal Functions 32 4. Differential Equations for Incomplete Cylindrieal Functions 35 5. Ineomplete Lipsehitz-Hankel Integrals ...... 41 6. The Relation between the Ineomplete Lipsehitz-Hanke1 Integrals and Ineomplete Cylindrieal Functions of Poissan Form ...... 48 7. Series Representations for Ineomplete Cylindrieal Functions . . . . 52 8. Ineomplete Beta and Gamma Functions and their Relation to Hyper- geometrie Functions ...... 57 9. Asymptotic Series and Methods for their Construetion. . . . . 61 10. Asymptotie Expansion for Ineomplete Cylindrieal Functions of Poissan Form ...... 66
Chapter III. Ineomplete Cylindrieal Functions of Bessel Form 76 1. Definitions of Ineomplete Cylindrieal Functions of Bessel Form 76 2. Reeursion Relations and Differential Equations for the Function e.(w, z) ...... 79 3. Connection between Ineomplete Cylindrieal Functions of the Poissan and Bessel Forms ...... 82 4. Ineomplete Cylindrieal Functions of Bessel Form with Half-Odd Indiees ...... 85 5. A Generating Funetion, Addition Formula, and Series for the Function e.(w, z) ...... 89 Contents VII
6. Asymptotic Expansions for Incomplete Cylindrical Functions of the Bessel Form ...... 95 7. Asymptotic Expansions for Incomplete Cylindrical Functions; the Gen- eral Case ...... 103 8. Incomplete Airy Integrals and their Generalizations 108 9. Asymptotic Expansions of Incomplete Cylindrical Functions of the Bessel Form for Large Indices • ...... 112
Chapter IV. Incomplete Cylindrical Functions of Sonine-Schlaefli Form and the Incomplete Weber Integrals ...... 116 1. The Concept of Incomplete Cylindrical Functions of Whittaker Form 116 2. Incomplete Cylindrical Functions of Sonine-Schlaefli Form and their Basic Properties ...... 117 3. Incomplete Weber Integrals ...... 121 4. The Connection between the Incomplete Weber Integral, Ineomplete Cylindrieal Functions, and Lipschitz-Hankel Integrals ...... 129 5. The Conneetion between Ineomplete Integrals of Weber, Lipschitz• Hanke! and Incomplete Cylindrical Functions of Poisson Form. . . 135 6. The Connection between Incomplete Weber Integrals and Lommel Functions of Two Variables . . . . . 138
Chapter V. Incomplete Cylindrieal Functions of Real Arguments and their Relation to Certain Discontinuous Intcgrals ...... 141 1. Incomplete Cylindrical Functions of Real Arguments 141 2. Unit Functions and their Basic Characteristics . . . 143 3. New Integral Representations for Ineomplete Cylindrical Functions of Real Arguments ...... 146 4. Relation between Weber Integrals of Bessel Functions and Incomplete Cylindrieal Functions ...... 152 5. Discontinuous Integrals of Gallop and their Relation to Ineomplete Cylindrieal Functions ...... 155 6. Sonine's Discontinuous Integral and its Connection with Incomplete Bessel Functions . . . . • . . • ...... 160
Chapter VI. Integrals Involving Incomplete Cylindrical Functions 166 1. Improper Integral of Lipschitz-Hankel ...... 166 2. Integrals of Weber Type Involving lneomplete Cylindrieal Functions 168 3. Improper Integrals of Certain Incomplete Cylindrical Functions with Respeet to Index...... 172 4. Some Discontinuous Integrals of Weber-Sehafheitlin Type Involving Ineomplete Cylindrical Functions...... 176 5. Improper Integrals of Hankel Type Involving Incomplete Cylindrieal Functions ...... 180 6. Definite Integrals Containing Incomplete Cylindrical Functions 184
Chapter VII. Application of Incomplete Cylindrieal Functions to Problems of Wave Propagation and Diffraetion...... 190 1. Connections between Ineomplete Cylindrical Functions and some Tabulated Special Integrals ...... 191 2. Absorption of Radiation in the Earth's Atmosphere. . . 197 3. Radiation from a Vertical Dipole on the Earth's Suriaee . 200 VIII Contents
4. The Problem of Diffraetion by a \Vedge ...... 205 5. Ditfraetion of \Vaves by a Sereen of Gi\'en Form ...... 213 6. Some Problems in the Theory of Diffraetion in Optieal Apparatus 219
Chapter VIII. Application of Ineomplete Cylindrieal Functions to so me Prob- lems of Solid State Theoryand the Motion of Charged Particles in Electro• magnetic Fields ...... 228 1. Computation of Radiation and Absorption Fields in the Theory of Multiphonon Processes ...... 228 2. Scattering of Light by Atoms with Interference of Excited States. . 232 3. Motion of Charged Partic!es in Constant Eleetrie and Time-Varying Magnetie Fields ...... 237 4. Motion of Charged Particles in Changing Electrie and Magnetie Fields 243
Chapter IX. Applieations of Incomplete Cylindrieal Funetions to some Prob• lems of Atomic and Nuclear Physies ...... 249 1. Solution of a Partieular Form of the Sehrödinger Equation 249 2. Some Relations for E\'aluating the Transpareney Coeffieients of Nuc!ei and the Average Loss of Impulse for Interaetion of Particles with their Fields ...... 253 3. Resonant Absorption of Radiation in Media of Finite Dimensions . 256 4. Applieation of Incomplete Cylindrical Funetions in the Study of a Heterogeneous Reaetor with Small Number of Bloeks ...... 262 5. Application of Ineomplete Cylindrieal Funetions to the Salutian of the Diffusion Equation ...... 267
Chapter X. Othcr Applied Problems Leading to Incomplete Cylindrical Fune- tions ...... 274 1. Exeitation of Betatron Oseillations ...... 274 2. Transient Processes in Electrical and ::\Iicrophonic Circuits . . . . . 274 3. Influenee of Optieal Systems on the Amplitude of Transmitted Waves 275 4. Density Perturbation in a Gas due to a Rapidly Moving Body 275 5. Computation of the Sound Field in Closed Domains. . 279 6. Exchange Processes between Liquid and Solid Phases . . . . 280 7. Drift of Non-Fundamental Carriers in Semieonduetors 282 8. Damping of Radiation in High Temperature Plasma in the Pres ene e of Shielding . . . . . 283 9. Other Speeial Integrals 285
Chapter XI. Compendium of Tables and Computation Formulae for Evalua• tion of Incomplete Cylindrieal Funetions ...... 288 1. Ineomplete Bessel Funetions, Struve Funetions, Anger Funetions, and Weber Funetions ...... 288 2. Incomplete Hankel Funetions and MaeDonald Functions 291 3. Ineomplete Lipsehitz-Hankel Integrals 292 4. Incomplete Weber Integrals ...... 294 5. Tables of Ineomplete Cylindrical Functions 297
Bibliography 324 Author Index 327 Subject Index 329 List of Symbols v.. Bessel Differential Operator j .. (z) Bessel Function of First Kind N.(::) Bessel Function of Second I A. = 2.r(v + ~ )r(1/ 2 ) 5.(z) Semi-Cylindrical Function d {} =z-az Re Real Part of Im Imaginary Part of r(z) Gamma Function tSt'j (e, z), tS:;') (e, z) Incomplete Hanke! Function Et (w, z) Incomplete Cylindrical Function (Poisson Form) j. (w, z) Incomplete Bessel Function (Poisson Form) H.(w, z) Incomplete Struve Function I.(w, z) Modified Incomplete Bessel Function L .. (w,z) Modified Incomplete Struve Function Ft (w, z), F;- (w, z) Alternative Forms of Above ](. (w, z) Incomplete Hankel Function (Alternative Form) $. (w, z) Incomplete MacDonald Function "" (IXl)m (IX2)m ••• (IXp)m zm pFq (CX1, IX2, ... , cxp; Yl, y2, ... , ')'q; z) = L; ( ) ( ) .~:~:-(--) m=O yl m Y2 m Yq m m!