Gustav Doetsch Introduction to the Theory and Application of the Laplace Transformation With 51 Figures and a Table of Laplace Transforms Translation by Walter Nader Springer-Verlag Berlin Heidelberg New York 1974
Gustav Doetsch Professor Emeritus of Mathematics, University of Freiburg Walter Nader Assistant Professor, University of Alberta, Canada Translation of the German Original Edition: Einfuhrung in. Theorie und Anwendung der Laplace-Transformation Zweite, neubearbeitete und erweiterte Auflage (Mathematische Reibe, Band 24, Sarnmlung LMW) Birkhauser Verlag Basel, 1970 AMS SUbject Classification (1970): 44AlO, 42A68, 34A10, 34A25, 35A22, 30A84 ISBN -13: 978-3-642-65692-7 e-isbn -13: 978-3-642-65690-3 DOl: 10.1007/978-3-642-65690-3 ISBN -13: 978-3-642-65692-7 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, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 54 of the German Copyright 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 1974. Softcover reprint of the hardcover 1st edition 1974. Library of Congress Catalog Card N'umber 73-10661.
Preface In anglo-american literature there exist numerous books, devoted to the application of the Laplace transformation in technical domains such as electrotechnics, mechanics etc. Chiefly, they treat problems which, in mathematical language, are governed by ordinary and partial differential equations, in various physically dressed forms. The theoretical foundations of the Laplace transformation are presented usually only in a simplified manner, presuming special properties with respect to the transformed functions, which allow easy proofs. By contrast, the present book intends principally to develop those parts of the theory of the Laplace transformation, which are needed by mathematicians, physicists a,nd engineers in their daily routine work, but in complete generality and with detailed, exact proofs. The applications to other mathematical domains and to technical problems are inserted, when the theory is adequately developed to present the tools necessary for their treatment. Since the book proceeds, not in a rigorously systematic manner, but rather from easier to more difficult topics, it is suited to be read from the beginning as a textbook, when one wishes to familiarize oneself for the first time with the Laplace transformation. For those who are interested only in particular details, all results are specified in "Theorems" with explicitly formulated assumptions and assertions. Chapters 1-14 treat the question of convergence and the mapping properties of the Laplace transformation. The interpretation of the transformation as the mapping of one function space to another (original and image functions) constitutes the dominating idea of all subsequent considerations. Chapters 14-22 immediately take advantage of the mapping properties for the solution of ordinary differential equations and of systems of such equations. In this part, especially important for practical applications, the concepts and the special cases occurring in technical literature are considered in detail. Up to this point no complex function theory is required. Chapters 23-31 enter the more difficult parts of the theory. They are devoted to the complex inversion integral and its various evaluations (by deformation of the path of integration and by series developments) and to the Parseval equation. In these considerations the Fourier transformation is used as an auxiliary tool. Its principal properties are explained for this purpose. Also the question of the representability of a function as a Laplace transform is answered here. Chapters 32-37 deal with a topic which is of special interest for both theory and application and which is commonly neglected in other books: that is the deduction of asymptotic expansions for the image function from properties of the original function, and conversely the passage from the image function to asymptotic expansions of the original function. In the latter case the inversion integral with angular path plays a decisive role; therefore its properties are developed, for the first time in the literature, in full detail. In technical problems this part provides the basis for the investigation of the behaviour of physical systems for large values of the-time variable. Chapter 38 presents the ordinary differential equation with polynomial coefficients. Here, again, the inversion integral with angular path is used for the contruction of the classical solution.
IV Preface As examples of boundary value problems in partial differential equations, Chapter 39 treats the equation of heat conduction and the telegraph equation. The results of Chapters 35-37 are used to deduce the stationary state of the solutions, which is of special interest for engineering. In Chapter 40 the linear integral equations of convolution type are solved. As an application the integral and the derivative of non-integral order in the intervall (0,00) are defined. Not only to procure a broader basis for the theory, but also to solve certain problems in practical engineering in a satisfactory manner, it is necessary to amplify the space of functions by the modern concept of distribution. The Laplace transformation can be defined for distributions in different ways. The usual method defines the Fourier transformation for distributions and on this basis the Laplace transformation. This method requires the limitation to "tempered" distributions and involves certain difficulties with regard to the definition of the convolution and the validity of the "convolution theorem". Here, however, a direct definition of the Laplace transformation is introduced, which is limited to distributions "of finite order". With this definition the mentioned difficulties do not appear; moreoverithas the advantage that in this partial distribution space, a necessary und sufficient condition for the representability of an analytic function as a Laplace transform can be formulated, which is only sufficient but not necessary in the range of the previous mentioned definition. The theory of distributions does not only make possible the legitimate treatment of such physical phenomena as the "impulse", but also the solution of a problem that has caused many discussions in the technical literature. When the initial value problem for a system of simultaneous differential equations is posed in the sense of classical mathematics, the initial values are understood as limits in the origin form the right. In general, this problem can be solved only if the initial values comply with certain "conditions of compatibility", a requirement which is fulfilled seldom in practice. Since, however, also in such a case the corresponding physical process ensues, someone mathematical description must exist. Such a description is possible, when 1. the functions are replaced by distributions, which are defined always over the whole axis and not only over the right half-axis, which is the domain of the initial value problem, and 2. the given initial values are understood as limits from the left; they originate, then, from the values of the unknowns in the left half-axis, Le. from the past of the system, which agrees exactly with the physical intuition. Instead of a general system, whose treatment would turn out very tedious, the system of first order for two unknown functions is solved completely with all details as a pattern, whereby already all essential steps are encountered. This work represents the translation of a book which appeared in first edition in 1958 and in amplified second edition in 1970 in "BirkhauserVerlag" (Basel und Stuttgart). The translation, which is based on the second edition, was prepared by Professor Dr. Walter Nader (University of Alberta, Edmonton, Canada) with extraordinary care. In innumerable epistolary discussions with the author, the translator has attempted to render the assertions of the German text in an adequate English structure. For his indefatigable endeavour I wish to express to Prof. Nader my warmest thanks. Riedbergstrasse 8 D-7800 Freiburg L Br. Western Germany GUSTAV DOETSCH
Table of Contents 1. Introduction of the Laplace Integral from Physical and Mathematical Points of View... 2. Examples of Laplace Integrals. Precise Definition of Integration 3. The Half-Plane of Convergence..... 4. The Laplace Integral as a Transformation 5. The Unique Inverse of the Laplace Transformation 6. The Laplace Transforn;t as an Analytic Function. 7. The Mapping of a Linear Substitution of the Variable. 8. The Mapping of Integration.'. 9. The Mapping of Differentiation. 10. The Mapping of the Convolution 11. Applications of the Convolution Theorem: Integral Relations. 12. The Laplace Transformation of Distributions...... 13. The Laplace Transforms of Several Special Distributions. 14. Rules of Mapping for the J..l-Transformation of Distributions 15. The Initial Value Problem of Ordinary Differential Equations with Constant Coefficients........... The Differential Equation of the First Order Partial Fraction Expansion of a Rational Function The Differential Equation of Order n 1. The homogeneous differential equation with arbitrary initial values 2. The inhomogeneous differential equation with vanishing initial values The Transfer Function...................... 16. The Ordinary Differential Equation, specifying Initial Values for Derivatives of Arbitrary Order, and Boundary Values........... 17. The Solutions of the Differential Equation for Specific Excitations 1. The Step Response............ 2. Sinusoidal Excitations. The Frequency Response 18. The Ordinary Linear Differential Equation in the Space of Distributions. The Impulse Response Response to the Excitation tj(.n) The Response to Excitation by a Pseudofunction A New Interpretation of the Concept Initial Value 19. The Normal System of Simultaneous Differential Equations 1 7 13 19 20 26 30 36 39 44 55 58 61 64 69 70 76 78 80 82 84 86 92 93 94 103 104 105 106 107 109
VI Table of Contents 1. The Normal Homogeneous System. for Arbitrary Initial Values 111 2. The Normal Inhomogeneous System with Vanishing Initial Values. 113 20. The Anomalous System of Simultaneous Differential Equations, with Initial Conditions which can be fulfilled........ 115 21. The Normal System in the Space of Distributions 125 22. The Anomalous System with Arbitrary Initial Values, in the Space of Distributions.................. 131 23. The Behaviour of the Laplace Transform near Infinity 139 24. The Complex Inversion Formula for the Absolutely Converging Laplace Transformation. The Fourier Transformation......... 148 25. Deformation of the Path of Integration of the Complex Inversion Integral.. 161 26. The Evaluation of the Complex Inversion Integral by Means of the Calculus of Residues............................ 169 27. The Complex Inversion Formule for the Simply Converging Laplace Transformation............................. 178 28. Sufficient Conditions for the Representability as a Laplace Transform of a Function............................. 184 29. A Condition, Necessary and Sufficient, for the Representability as a Laplace Transform of a Distribution..................... 189 30. Determination of the Original Function by Means of Series Expansion of the Image Function.......................... 192 31. The Parseval Formula of the Fourier Transformation and of the Laplace Transformation. The Image of the Product.......... 201 32. The Concepts: Asymptotic Representation, Asymptotic Expansion 218 33. Asymptotic Behaviour of the Image Function near Infinity 221 Asymptotic Expansion of Image Functions......... 228 34. Asymptotic Behaviour of the Image Function near a Singular Point on the Line of Convergence........................ 231 35. The Asymptotic Behaviour of the Original Function near Infinity, when the Image Function has Singularities of Unique Character......... 234 36. The Region of Convergence of the Complex Inversion Integral with Angular Path. The Holomorphy of the Represented Function.......... 239 37. The Asymptotic Behaviour of an Original Function near Infinity, when its Image Function is Many-Valued at the Singular Point with Largest Real Part............................... 250 38. Ordinary Differential Equations with Polynomial Coefficients. Solution by Means of the Laplace Transformation and by Means of Integrals with Angular Path of Integration......................... 262 The Differential Equation of the Bessel Functions............. 263 The General Linear Homogeneous Differential Equation with Linear Coefficients 270 39. Partial Differential Equations....... 278 1. The Equation of Diffusion or Heat ConductIon 279 The Case of Infinite Length 282 The Case of Finite Length 288
Table of Contents 1. Thermal Conductor with Vanishing Initial Temperature 2. Thermal Conductor with Vanishing End Temperatures Asymptotic Expansion of the Solution 2. The Telegraph Equation...... Asymptotic Expansion of the Solution 40. Integral Equations........ 1. The Linear Integral Equation of the Second Kind, of the Convolution Type. 304 2. The Linear Integral Equation of the First Kind, of the Convolution Type. 308 The Abel Integral Equation................. 309 VII 288 289 291 294 297 APPENDIX: Some Concepts and Theorems from the Theory of Distributions 313 Table of Laplace Transforms 317 Operations....... Functions and Distributions INDEX..... 304 317 318 322