Methods of Mathematical Physics Partial Differential Equations

by Courant, Richard; Hilbert, David

• ISBN13: 9780471504399
• ISBN10: 0471504394
• Edition: 1st
• Format: Paperback
• Copyright: 1991-01-08
• Publisher: Wiley-VCH

Summary

Since the first volume of this work came out in Germany in 1937, this book, together with its first volume, has remained standard in the field. Courant and Hilbert's treatment restores the historically deep connections between physical intuition and mathematical development, providing the reader with a unified approach to mathematical physics. The present volume represents Richard Courant's final revision of 1961.

Author Biography

David Hilbert (1862-1943) received his PhD from the University of K+¦nigsberg, Prussia (now Kaliningrad, Russia) in 1884. He remained there until 1895, after which he was appointed Professor of Mathematics at the University of G+¦ttingen. He held this professorship for most of his life. Hilbert is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. His own discoveries alone would have given him that honour, yet it was his leadership in the field of mathematics throughout his later life that distinguishes him. Hilbert's name is given to Infinite-Dimensional space, called Hilbert space, used as a conception for the mathematical analysis of the kinetic gas theory and the theory of radiations.

Richard Courant (1888-1972) obtained his doctorate at the University of G+¦ttingen in 1910. Here, he became Hilbert-+s assistant. He returned to G+¦ttingen to continue his research after World War I, and founded and headed the university's Mathematical Institute. In 1933, Courant left Germany for England, from whence he went on to the United States after a year. In 1936, he became a professor at the New York University. Here, he headed the Department of Mathematics and was Director of the Institute of Mathematical Sciences - which was subsequently renamed the Courant Institute of Mathematical Sciences. Among other things, Courant is well remembered for his achievement regarding the finite element method, which he set on a solid mathematical basis and which is nowadays the most important way to solve partial differential equations numerically.

Table of Contents

I. Introductory Remarks	1	(61)
General Information about the Variety of Solutions	2	(10)
Examples	2	(6)
Differential Equations for Given Families of Functions	8	(4)
Systems of Differential Equations	12	(6)
The Question of Equivalence of a System of Differential Equations and a Single Differential Equation	12	(2)
Elimination from a Linear System with Constant Coefficients	14	(1)
Determined, Overdetermined, Underdetermined Systems	15	(3)
Methods of Integration for Special Differential Equations	18	(4)
Separation of Variables	18	(2)
Construction of Further Solutions by Superposition. Fundamental Solution of the Heat Equation. Poisson's Integral	20	(2)
Geometric Interpretation of a First Order Partial Differential Equation in Two Independent Variables. The Complete Integral	22	(6)
Geometric Interpretation of a First Order Partial Differential Equation	22	(2)
The Complete Integral	24	(1)
Singular Integrals	25	(2)
Examples	27	(1)
Theory of Linear and Quasi-Linear Differential Equations of First Order	28	(4)
Linear Differential Equations	28	(3)
Quasi-Linear Differential Equations	31	(1)
The Legendre Transformation	32	(7)
The Legendre Transformation for Functions of Two Variables	32	(2)
The Legendre Transformation for Functions of n Variables	34	(1)
Application of the Legendre Transformation to Partial Differential Equations	35	(4)
The Existence Theorem of Cauchy and Kowalewsky	39	(23)
Introduction and Examples	39	(4)
Reduction to a System of Quasi-Linear Differential Equations	43	(4)
Determination of Derivatives Along the Initial Manifold	47	(1)
Existence Proof for Solutions of Analytic Differential Equations	48	(7)
Observation About Linear Differential Equations	54	(1)
Remark About Nonanlytic Differential Equations	54	(1)
Remarks on Critical Initial Data. Characteristics	55	(3)
Appendix 1 to Chapter I. Laplace's Differential Equations for the Support Function of a Minimal Surface	56	(2)
Appendix 2 to Chapter I. Systems of Differential Equations of First Order and Differential Equations of Higher Order	58	(1)
Plausibility Considerations	58	(1)
Conditions of Equivalence for Systems of Two First Order Partial Differential Equations and a Differential Equation of Second Order	58	(4)
II. General Theory of Partial Differential Equations of First Order	62	(92)
Geometric Theory of Quasi-Linear Differential Equations in Two Independent Variables	62	(7)
Characteristic Curves	62	(2)
Initial Value Problem	64	(2)
Examples	66	(3)
Quasi-Linear Differential Equations in n Independent Variables	69	(6)
General Differential Equations in Two Independent Variables	75	(9)
Characteristic Curves and Focal Curves. The Monge Cone	75	(4)
Solution of the Initial Value Problem	79	(3)
Characteristics as Branch Elements. Supplementary Remarks. Integral Conoid. Caustics	82	(2)
The Complete Integral	84	(2)
Focal Curves and the Monge Equation	86	(2)
Examples	88	(9)
The Differential Equation of Straight Light Rays, (grad u)2 = 1	88	(3)
The Equation F (uz, uy) = 0	91	(3)
Clairaut's Differential Equation	94	(1)
Differential Equation of Tubular Surfaces	95	(1)
Homogeneity Relation	96	(1)
General Differential Equation in n Independent Variables	97	(6)
Complete Integral and Hamilton-Jacobi Theory	103	(10)
Construction of Envelopes and Characteristic Curves	103	(3)
Canonical Forms of the Characteristic Differential Equations	106	(1)
Hamilton-Jacobi Theory	107	(2)
Example. The Two-Body Problem	109	(2)
Example. Geodesics on an Ellipsoid	111	(2)
Hamilton-Jacobi Theory and the Calculus of Variations	113	(14)
Euler's Differential Equations in Canonical Form	114	(1)
Geodetic Distance or Eiconal and Its Derivatives. Hamilton-Jacobi Partial Differential Equation	115	(4)
Homogeneous Integrands	119	(2)
Fields of Extremals. Hamilton-Jacobi Differential Equation	121	(3)
Cone of Rays. Huyghens' Construction	124	(1)
Hilbert's Invariant Integral for the Representation of the Eiconal	125	(2)
Theorem of Hamilton and Jacobi	127	(1)
Canonical Transformations and Applications	127	(5)
The Canonical Transformation	127	(2)
New Proof of the Hamilton-Jacobi Theorem	129	(1)
Variation of Constants (Canonical Perturbation Theory)	130	(2)
Appendix 1 to Chapter II	132	(1)
Further Discussion of Characteristic Manifolds	132	(7)
Remarks on Differentiation in n Dimensions	132	(3)
Initial Value Problem. Characteristic Manifolds	135	(4)
Systems of Quasi-Linear Differential Equations with the Same Principal Part. New Derivation of the Theory	139	(6)
Haar's Uniqueness Proof	145	(9)
Appendix 2 to Chapter II. Theory of Conservation Laws	147	(7)
III. Differential Equation of HIgher Order	154	(86)
Normal Forms for Linear and Quasi-Linear Differential Operators of Second Order in Two Independent Variables	154	(16)
Elliptic, Hyperbolic, and Parabolic Normal Forms. Mixed Types	155	(5)
Examples	160	(3)
Normal Forms for Quasi-Linear Second Order Differential Equations in Two Variables	163	(4)
Example. Minimal Surfaces	167	(2)
Systems of Two Differential Equations of First Order	169	(1)
Classification in General and Characteristics	170	(10)
Notations	170	(1)
Systems of First Order with Two Independent Variables. Characteristics	171	(2)
Systems of First Order with n Independent Variables	173	(2)
Differential Equations of Higher Order. Hyperbolicity	175	(1)
Supplementary Remarks	176	(1)
Examples. Maxwell's and Dirac's Equations	176	(4)
Linear Differential Equations with Constant Coefficients	180	(17)
Normal Form and Classification for Equations of Second Order	181	(3)
Fundamental Solutions for Equations of Second Order	184	(3)
Plane Waves	187	(1)
Plane Waves Continued. Progressing Waves. Dispersion	188	(4)
Examples. Telegraph Equation. Undistorted Waves in Cables	192	(1)
Cylindrical and Spherical Waves	193	(4)
Initial Value Problems. Radiation Problems for the Wave Equation	197	(13)
Initial Value Problems for Heat Conduction. Transformation of the Theta Function	197	(4)
Initial Value Problems for the Wave Equation	201	(1)
Duhamel's Principle. Nonhomogeneous Equations. Retarded Potentials	202	(3)
Duhamel's Principle for Systems of First Order	204	(1)
Initial Value Problem for the Wave Equation in Two-Dimensional Space. Method of Descent	205	(1)
The Radiation Problem	206	(2)
Propagation Phenomena and Huyghens' Principle	208	(2)
Solution of Initial Value Problems by Fourier Integrals	210	(11)
Cauchy's Method of the Fourier Integral	210	(2)
Example	212	(3)
Justification of Cauchy's Method	215	(6)
Typical Problems in Differential Equations of Mathematical Physics	221	(11)
Introductory Remarks	221	(5)
Basic Principles	226	(4)
Remarks about ``Improperly Posed'' Problems	230	(1)
General Remarks About Linear Problems	231	(1)
Appendix 1 to Chapter III	232	(1)
Sobolev's Lemma	232	(2)
Adjoint Operators	234	(6)
Matrix Operators	234	(1)
Adjoint Differential Operators	235	(5)
Appendix 2 to Chapter III. The Uniqueness Theorem of Holmgren	237	(3)
IV. Potential Theory and Elliptic Differential Equations	240	(167)
Basic Notions	240	(21)
Equations of Laplace and Poisson, and Related Equations	240	(5)
Potentials of Mass Distributions	245	(7)
Green's Formulas and Applications	252	(6)
Derivatives of Potentials of Mass Distributions	258	(3)
Poisson's Integral and Applications	261	(14)
The Boundary Value Problem and Green's Function	261	(3)
Green's Function for the Circle and Sphere. Poisson's Integral for the Sphere and Half-Space	264	(4)
Consequences of Poisson's Formula	268	(7)
The Mean Value Theorem and Applications	275	(15)
The Homogeneous and Nonhomogeneous Mean Value Equation	275	(2)
The Converse of the Mean Value Theorems	277	(7)
Poisson's Equation for Potentials of Spatial Distributions	284	(2)
Mean Value Theorems for Other Elliptic Differential Equations	286	(4)
The Boundary Value Problem	290	(22)
Preliminaries. Continuous Dependence on the Boundary Values and on the Domain	290	(3)
Solution of the Boundary Value Problem by the Schwarz Alternating Procedure	293	(6)
The Method of Integral Equations for Plane Regions with Sufficiently Smooth Boundaries	299	(4)
Remarks on Boundary Values	303	(3)
Capacity and Assumption of Boundary Values	305	(1)
Perron's Method of Subharmonic Functions	306	(6)
The Reduced Wave Equations. Scattering	312	(8)
Background	312	(3)
Sommerfeld's Radiation Condition	315	(3)
Scattering	318	(2)
Boundary Value Problems for More General Elliptic Differential Equations. Uniqueness of the Solutions	320	(11)
Linear Differential Equations	320	(2)
Nonlinear Equations	322	(2)
Rellich's Theorem on the Monge-Ampere Differential Equation	324	(2)
The Maximum Principle and Applications	326	(5)
A Priori Estimates of Schauder and Their Applications	331	(19)
Schauder's Estimates	332	(4)
Solution of the Boundary Value Problem	336	(5)
Strong Barrier Functions and Applications	341	(2)
Some Properties of Solutions of L[u] = f	343	(3)
Further Results on Elliptic Equations; Behavior at the Boundary	346	(4)
Solution of the Beltrami Equations	350	(7)
The Boundary Value Problem for a Special Quasi-Linear Equation. Fixed Point Method of Leray and Schauder	357	(5)
Solution of Elliptic Differential Euquations by Means of Integral Equations	362	(13)
Construction of Particular Solutions. Fundamental Solutions. Parametrix	363	(4)
Further Remarks	367	(1)
Appendix to Chapter IV. Nonlinear Equations	367	(1)
Perturbation Theory	368	(1)
The Equation Δ = f(x, u)	369	(6)
Supplement to Chapter IV. Function Theoretic Aspects of the Theory of Elliptic Partial Differential Equations	374	(1)
Definition of Pseudoanalytic Functions	375	(2)
An Integral Equation	377	(1)
Similarity Principle	378	(4)
Applications of the Similarity Principle	382	(2)
Formal Powers	384	(2)
Differentiation and Integration of Pseudoanalytic Functions	386	(3)
Example. Equations of Mixed Type	389	(2)
General Definition of Pseudoanalytic Functions	391	(1)
Quasiconformality and a General Representation Theorem	392	(3)
A Nonlinear Boundary Value Problem	395	(4)
An Extension of Riemann's Mapping Theorem	399	(1)
Two Theorems on Minimal Surfaces	400	(1)
Equations with Analytic Coefficients	400	(1)
Proof of Privaloff's Theorem	401	(2)
Proof of the Schauder Fixed Point Theorem	403	(4)
V. Hyperbolic Differential Equations in Two Independent Variables	407	(144)
Introduction	407	(1)
Characteristics for Differential Equations Mainly of Second Order	408	(16)
Basic Notions. Quasi-Linear Equations	408	(6)
Characteristics on Integral Surfaces	414	(2)
Characteristics as Curves of Discontinuity. Wave Fronts. Propagation of Discontinuities	416	(2)
General Differential Equations of Second Order	418	(3)
Differential Equations of Higher Order	421	(2)
Invariance of Characteristics under Point Transformations	423	(1)
Reduction to Quasi-Linear Systems of First Order	423	(1)
Characteristic Normal Forms for Hyperbolic Systems of First Order	424	(5)
Linear, Semilinear and Quasi-Linear Systems	424	(3)
The Case k = 2. Linearization by the Hodograph Transformation	427	(2)
Applications to Dynamics of Compressible Fluids	429	(9)
One-Dimensional Isentropic Flow	429	(3)
Spherically Symmetric Flow	432	(1)
Steady Irrotational Flow	432	(2)
Systems of Three Equations for Nonisentropic Flow	434	(2)
Linearized Equations	436	(2)
Uniqueness. Domain of Dependence	438	(11)
Domains of Dependence, Influence, and Determinacy	438	(2)
Uniqueness Proofs for Linear Differential Equations of Second Order	440	(5)
General Uniqueness Theorem for Linear Systems of First Order	445	(3)
Uniqueness for Quasi-Linear Systems	448	(1)
Energy Inequalities	449	(1)
Riemann's Representation of Solutions	449	(12)
The Initial Value Problem	450	(1)
Riemann's Function	450	(4)
Symmetry of Riemann's Function	454	(1)
Riemann's Function and Radiation from a Point. Generalization to Higher Order Problems	455	(2)
Examples	457	(4)
Solution of Hyperbolic Linear and Semilinear Initial Value Problems by Iteration	461	(15)
Construction of the Solution for a Second Order Equation	462	(2)
Notations and Results for Linear and Semilinear Systems of First Order	464	(2)
Construction of the Solution	466	(4)
Remarks. Dependence of Solutions on Parameters	470	(1)
Mixed Initial and Boundary Value Problems	471	(5)
Cauchy's Problem for Quasi-Linear Systems	476	(2)
Cauchy's Problem for Single Hyperbolic Differential Equations of Higher Order	478	(8)
Reduction to a Characteristic System of First Order	479	(1)
Characteristic Representation of L[u]	480	(2)
Solution of Cauchy's Problem	482	(1)
Other Variants for the Solution. A Theorem	483	(2)
P. Ungar
Remarks	485	(1)
Discontinuities of Solutions. Shocks	486	(4)
Generalized Solutions. Weak Solutions	486	(2)
Discontinuities for Quasi-Linear Systems Expressing Conservation Laws. Shocks	488	(2)
Appendix 1 to Chapter V. Application of Characteristics as Coordinates	490	(1)
Additional Remarks on General Nonlinear Equations of Second Order	490	(5)
The Quasi-Linear Differential Equation	491	(3)
The General Nonlinear Equation	494	(1)
The Exceptional Character of the Monge-Ampere Equation	495	(4)
Transition from the Hyperbolic to the Elliptic Case Through Complex Domains	499	(2)
The Analyticity of the Solutions in the Elliptic Case	501	(4)
Function-Theoretic Remark	501	(1)
Analyticity of the Solutions of Δ = f (x, y, u, p, q)	502	(3)
Remark on the General Differential Equation F (x, y, u, p, q, r, s, t) = 0	505	(1)
Use of Complex Quantities for the Continuation of Solutions	505	(3)
Appendix 2 to Chapter V. Transient Problems and Heaviside Operational Calculus	507	(1)
Solution of Transient Problems by Integral Representation	508	(9)
Explicit Example. The Wave Equation	508	(3)
General Formulation of the Problem	511	(1)
The Integral of Duhamel	512	(3)
Method of Superposition of Exponential Solutions	515	(2)
The Heaviside Method of Operators	517	(18)
The Simplest Operators	518	(3)
Examples of Operators and Applications	521	(4)
Applications to Heat Conduction	525	(2)
Wave Equation	527	(1)
Justification of the Operational Calculus. Interpretation of Further Operators	528	(7)
General Theory of Transient Problems	535	(16)
The Laplace Transformation	535	(4)
Solution of Transient Problems by the Laplace Transformation	539	(5)
Example. The Wave and Telegraph Equations	544	(7)
VI. Hyperbolic Differential Equations in More Than Two Independent Variables	551	(1)
Introduction	551	(1)
PART I. Uniqueness, Construction, and Geometry of Solutions	552	(124)
Differential Equations of Second Order. Geometry of Characteristics	552	(17)
Quasi-Linear Differential Equations of Second Order	552	(4)
Linear Differential Equations	556	(2)
Rays or Bicharacteristics	558	(2)
Characteristics as Wave Fronts	560	(1)
Invariance of Characteristics	561	(2)
Ray Cone, Normal Cone, and Ray Conoid	563	(1)
Connection with a Riemann Metric	564	(2)
Reciprocal Transformations	566	(2)
Huyghens' Construction of Wave Fronts	568	(1)
Space-Like Surfaces. Time-Like Directions	569	(1)
Second Order Equations. The Role of Characteristics	569	(8)
Discontinuities of Second Order	570	(1)
The Differential Equation along a Characteristic Surface	571	(2)
Propagation of discontinuities along Rays	573	(1)
Illustration. Solution of Cauchy's Problem for the Wave Equation in Three Space Dimensions	574	(3)
Geometry of Characteristics for Higher Order Operators	577	(20)
Notation	577	(2)
Characteristic Surfaces, Forms, and Matrices	579	(2)
Interpretation of the Characteristic Condition in Time and Space. Normal Cone and Normal Surface. Characteristic Nullvectors and Eigenvalues	581	(2)
Construction of Characteristic Surfaces or Fronts. Rays, Ray Cone, Ray Conoid	583	(2)
Wave Fronts and Huyghens' Construction. Ray Surface and Normal Surfaces	585	(4)
Example	588	(1)
Invariance Properties	589	(1)
Hyperbolicity. Space-Like Manifolds, Time-Like Directions	589	(4)
Symmetric Hyperbolic Operators	593	(1)
Symmetric Hyperbolic Equations of Higher Order	594	(1)
Multiple Characteristic Sheets and Reducibility	595	(2)
Lemma on Bicharacteristic Directors	597	(21)
Examples. Hydrodynamics, Crystal Optics, Magnetohydrodynamics	599	(1)
Introduction	599	(1)
The Differential Equation System of Hydrodynamics	600	(2)
Crystal Optics	602	(3)
The Shapes of the Normal and Ray Surfaces	605	(4)
Cauchy's Problem for Crystal Optics	609	(3)
Magnetohydrodynamics	612	(6)
Propagation of Discontinuities and Cauchy's Problem	618	(18)
Introduction	618	(1)
Discontinuities of First Derivatives for Systems of First Order. Transport Equation	618	(2)
Discontinuities of Initial Values. Introduction of Ideal Functions. Progressing Waves	620	(4)
Propagation of Discontinuities for Systems of First Order	624	(2)
Characteristics with Constant Multiplicity	626	(3)
Examples for Propagation of Discontinuities Along Manifolds of More Than One Dimension. Conical Refraction	627	(2)
Resolution of Initial Discontinuities and Solution of Cauchy's Problem	629	(3)
Characteristic Surfaces as Wave Fronts	631	(1)
Solution of Cauchy's Problem by Convergent Wave Expansions	632	(1)
Systems of Second and Higher Order	633	(2)
Supplementary Remarks. Weak Solutions. Shocks	635	(1)
Oscillatory Initial Values. Asymptotic Expansion of the Solution. Transition to Geometrical Optics	636	(6)
Preliminary Remarks. Progressing Waves of Higher Order	636	(1)
Construction of Asymptotic Solutions	637	(3)
Geometrical Optics	640	(2)
Examples of Uniqueness Theorems and Domain of Dependence for Initial Value Problems	642	(7)
The Wave Equation	643	(3)
The Differential Equation (Darboux Equation)	646	(1)
Maxwell's Equations in Vacuum	647	(2)
Domains of Dependence for Hyperbolic Problems	649	(3)
Introduction	649	(1)
Description of the Domain of Dependence	650	(2)
Energy Integrals and Uniqueness for Linear Symmetric Hyperbolic Systems of First Order	652	(9)
Energy Integrals and Uniqueness for the Cauchy Problem	652	(2)
Energy Integrals of First and Higher Order	654	(2)
Energy Inequalities for Mixed Initial and Boundary Value Problems	656	(3)
Energy Integrals for Single Second Order Equations	659	(2)
Energy Estimates for Equations of Higher Order	661	(7)
Introduction	661	(1)
Energy Identities and Inequalities for Solutions of Higher Order Hyperbolic Operators. Method of Leray and Garding	662	(4)
Other Methods	666	(2)
The Existence Theorem	668	(8)
Introduction	668	(1)
The Existence Theorem	669	(2)
Remarks on Persistence of Properties of Initial Values and on Corresponding Semigroups. Huyghens' Minor Principle	671	(2)
Focussing. Example of Nonpersistence of Differentiability	673	(2)
Remarks about Quasi-Linear Systems	675	(1)
Remarks about Problems of Higher Order or Nonsymmetric Systems	675	(1)
PART II. Representation of Solutions	676	(123)
Introduction	676	(5)
Outline. Notations	676	(1)
Some Integral Formulas. Decomposition of Functions into Plane Waves	677	(4)
Equations of Second Order with Constant Coefficients	681	(18)
Cauchy's Problem	681	(2)
Construction of the Solution for the Wave Equation	683	(3)
Method of Descent	686	(2)
Further Discussion of the Solution. Huyghens' Principle	688	(3)
The Nonhomogeneous Equation. Duhamel's Integral	691	(1)
Cauchy's Problem for the General Linear Equation of Second Order	692	(3)
The Radiation Problem	695	(4)
Method of Spherical Means. The Wave Equation and the Darboux Equation	699	(7)
Darboux's Differential Equation for Mean Values	699	(1)
Connection with the Wave Equation	700	(3)
The Radiation Problem of the Wave Equation	703	(1)
Generalized Progressing Spherical Waves	704	(2)
The Initial Value Problem for Elastic Waves Solved by Spherical Means	706	(5)
Method of Plane Mean Values. Application to General Hyperbolic Equations with Constant Coefficients	711	(7)
General Method	711	(4)
Application to the Solution of the Wave Equation	715	(3)
Application to the Equations of Crystal Optics and Other Equations of Fourth Order	718	(9)
Solution of Cauchy's Problem	718	(5)
Further Discussion of the Solution. Domain of Dependence. Gaps	723	(4)
The Solution of Cauchy's Problem as Linear Functional of the Data. Fundamental Solutions	727	(17)
Description. Notations	727	(3)
Construction of the Radiation Function by Decomposition of the Delta Function	730	(3)
Regularity of the Radiation Matrix	733	(3)
The Generalized Huyghens Principle	735	(1)
Example. Special Linear Systems with Constant Coefficients. Theorem on Gaps	736	(1)
Example. The Wave Equation	737	(3)
Example. Hadamard's Theory for Single Equations of Second Order	740	(4)
Further Examples. Two Independent Variables. Remarks	744	(1)
Ultrahyperbolic Differential Equations and General Differential Equations of Second Order with Constant Coefficients	744	(10)
The General Mean Value Theorem of Asgeirsson	744	(4)
Another Proof of the Mean Value Theorem	748	(1)
Application to the Wave Equation	749	(1)
Solutions of the Characteristic Initial Value Problem for the Wave Equation	749	(3)
Other Applications. The Mean Value Theorem for Confocal Ellipsoids	752	(2)
Initial Value Problems for Non-Space-Like Initial Manifolds	754	(6)
Functions Determined by Mean Values over Spheres with Centers in a Plane	754	(2)
Applications to the Initial Value Problem	756	(4)
Remarks About Progressing Waves, Transmission of Signals and Huyghens' Principle	760	(6)
Distortion-Free Progressing Waves	760	(3)
Spherical Waves	763	(1)
Radiation and Huyghens' Principle	764	(2)
Appendix to Chapter VI. Ideal Functions or Distributions	766	(1)
Underlying Definitions and Concepts	766	(8)
Introduction	766	(1)
Ideal Elements	767	(1)
Notations and Definitions	768	(1)
Iterated Integration	769	(1)
Linear Functionals and Operators. Bilinear Form	769	(2)
Continuity of Functionals. Support of Test Functions	771	(1)
Lemma About r-Continuity	772	(1)
Some Auxiliary Functions	773	(1)
Examples	774	(1)
Ideal Functions	774	(14)
Introduction	774	(1)
Definition by Linear Differential Operators	775	(2)
Definition by Weak Limits	777	(1)
Definition by Linear Functionals	778	(1)
Equivalence. Representation of Functionals	779	(2)
Some Conclusions	781	(1)
Example. The Delta-Function	781	(2)
Identification of Ideal with Ordinary Functions	783	(2)
Definite Integrals. Finite Parts	785	(3)
Calculus with Ideal Functions	788	(4)
Linear Processes	789	(1)
Change of Independent Variables	789	(1)
Examples. Transformations of the Delta-Function	790	(1)
Multiplication and Convolution of Ideal Functions	791	(1)
Additional Remarks. Modifications of the Theory	792	(7)
Introduction	792	(1)
Different Spaces of Test Functions. The Space. Fourier Transforms	793	(2)
Periodic Functions	795	(1)
Ideal Functions and Hilbert Spaces. Negative Norms. Strong Definitions	796	(1)
Remark on Other Classes of Ideal Functions	797	(2)
Bibliography	799	(20)
Index	819