Theory of Incomplete Cylindrical Functions and their Applications

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.

I. Some Information from the Theory of Cylindrical Functions.- 1. The Differential Equation and Recursion Relationships for Cylindrical Functions.- 2. Generalized System of Functional Equations for Cylindrical Functions and the Inhomogenious Bessel Differential Equation.- 3. Integral Repesentations of Cylindrical Functions in the Poisson Form.- 4. Integral Representations of Cylindrical Functions of the Bessel-Schlaefli and Sonine Form.- II. General Theory of Incomplete Cylindrical Functions Expressed in Poisson Form.- 1. Definitions of Incomplete Cylindrical Functions as Poisson Integrals.- 2. Recurrence Relations for the Incomplete Cylindrical Functions.- 3. Some General Properties of Incomplete Cylindrical Functions.- 4. Differential Equations for Incomplete Cylindrical Functions.- 5. Incomplete Lipschitz-Hankel Integrals.- 6. The Relation between the Incomplete Lipschitz-Hankel Integrals and Incomplete Cylindrical Functions of Poisson Form.- 7. Series Representations for Incomplete Cylindrical Functions.- 8. Incomplete Beta and Gamma Functions and their Relation to Hypergeometric Functions.- 9. Asymptotic Series and Methods for their Construction.- 10. Asymptotic Expansion for Incomplete Cylindrical Functions of Poisson Form.- III. Incomplete Cylindrical Functions of Bessel Form.- 1. Definitions of Incomplete Cylindrical Functions of Bessel Form.- 2. Recursion Relations and Differential Equations for the Function ?v (w, z).- 3. Connection between Incomplete Cylindrical Functions of the Poisson and Bessel Forms.- 4. Incomplete Cylindrical Functions of Bessel Form with Half-Odd Indices.- 5. A Generating Function, Addition Formula, and Series for the Function ?v (w, z).- 6. Asymptotic Expansions for Incomplete Cylindrical Functions of the Bessel Form.- 7. Asymptotic Expansions for Incomplete Cylindrical Functions; the General Case.- 8. Incomplete Airy Integrals and their Generalizations.- 9. Asymptotic Expansions of Incomplete Cylindrical Functions of the Bessel Form for Large Indices.- IV. Incomplete Cylindrical Functions of Sonine-Schlaefli Form and the Incomplete Weber Integrals.- 1. The Concept of Incomplete Cylindrical Functions of Whittaker Form.- 2. Incomplete Cylindrical Functions of Sonine-Schlaefli Form and their Basic Properties.- 3. Incomplete Weber Integrals.- 4. The Connection between the Incomplete Weber Integral, Incomplete Cylindrical Functions, and Lipschitz-Hankel Integrals.- 5. The Connection between Incomplete Integrals of Weber, Lipschitz-Hankel and Incomplete Cylindrical Functions of Poisson Form.- 6. The Connection between Incomplete Weber Integrals and Lommel Functions of Two Variables.- V. Incomplete Cylindrical Functions of Real Arguments and their Relation to Certain Discontinuous Integrals.- 1. Incomplete Cylindrical Functions of Real Arguments.- 2. Unit Functions and their Basic Characteristics.- 3. New Integral Representations for Incomplete Cylindrical Functions of Real Arguments.- 4. Relation between Weber Integrals of Bessel Functions and Incomplete Cylindrical Functions.- 5. Discontinuous Integrals of Gallop and their Relation to Incomplete Cylindrical Functions.- 6. Sonine’s Discontinuous Integral and its Connection with Incomplete Bessel Functions.- VI. Integrals Involving Incomplete Cylindrical Functions.- 1. Improper Integral of Lipschitz-Hankel.- 2. Integrals of Weber Type Involving Incomplete Cylindrical Functions.- 3. Improper Integrals of Certain Incomplete Cylindrical Functions with Respect to Index.- 4. Some Discontinuous Integrals of Weher-Schafheitlin Type Involving Incomplete Cylindrical Functions.- 5. Improper Integrals of Hankel Type Involving Incomplete Cylindrical Functions.- 6. Definite Integrals Containing Incomplete Cylindrical Functions.- VII. Application of Incomplete Cylindrical Functions to Problems of Wave Propagation and Diffraction.- 1. Connections between Incomplete Cylindrical Functions and some Tabulated Special Integrals.- 2. Absorption of Radiation in the Earth’s Atmosphere.- 3. Radiation from a Vertical Dipole on the Earth’s Surface.- 4. The Problem of Diffraction by a Wedge.- 5. Diffraction of Waves by a Screen of Given Form.- 6. Some Problems in the Theory of Diffraction in Optical Apparatus.- VIII. Application of Incomplete Cylindrical Functions to some Problems of Solid State Theory and the Motion of Charged Particles in Electromagnetic Fields.- 1. Computation of Radiation and Absorption Fields in the Theory of Multiphonon Processes.- 2. Scattering of Light by Atoms with Interference of Excited States.- 3. Motion of Charged Particles in Constant Electric and Time-Varying Magnetic Fields.- 4. Motion of Charged Particles in Changing Electric and Magnetic Fields.- IX. Applications of Incomplete Cylindrical Functions to some Problems of Atomic and Nuclear Physics.- 1. Solution of a Particular Form of the Schrödinger Equation.- 2. Some Relations for Evaluating the Transparency Coefficients of Nuclei and the Average Loss of Impulse for Interaction of Particles with their Fields.- 3. Resonant Absorption of Radiation in Media of Finite Dimensions.- 4. Application of Incomplete Cylindrical Functions in the Study of a Heterogeneous Reactor with Small Number of Blocks.- 5. Application of Incomplete Cylindrical Functions to the Solution of the Diffusion Equation.- X. Other Applied Problems Leading to Incomplete Cylindrical Functions.- 1. Excitation of Betatron Oscillations.- 2. Transient Processes in Electrical and Microphonic Circuits.- 3. Influence of Optical Systems on the Amplitude of Transmitted Waves.- 4. Density Perturbation in a Gas due to a Rapidly Moving Body.- 5. Computation of the Sound Field in Closed Domains.- 6. Exchange Processes between Liquid and Solid Phases.- 7. Drift of Non-Fundamental Carriers in Semiconductors.- 8. Damping of Radiation in High Temperature Plasma in the Presence of Shielding.- 9. Other Special Integrals.- XI. Compendium of Tables and Computation Formulae for Evaluation of Incomplete Cylindrical Functions.- 1. Incomplete Bessel Functions, Struve Functions, Anger Functions, and Weber Functions.- 2. Incomplete Hankel Functions and MacDonald Functions.- 3. Incomplete Lipschitz-Hankel Integrals.- 4. Incomplete Weber Integrals.- 5. Tables of Incomplete Cylindrical Functions.- Author Index.