Lectures on Cauchy's Problem in Linear Partial Differential Equations

by Jacques Hadamard

Overview

Would well repay study by most theoretical physicists." — Physics Today
"An overwhelming influence on subsequent work on the wave equation." — Science Progress
"One of the classical treatises on hyperbolic equations." — Royal Naval Scientific Service
Delivered at Columbia University and the Universities of Rome and Zürich, these lectures represent a pioneering investigation. Jacques Hadamard based his research on prior studies by Riemann, Kirchhoff, and Volterra. He extended and improved Volterra's work, applying its theories relating to spherical and cylindrical waves to all normal hyperbolic equations instead of only to one. Topics include the general properties of Cauchy's problem, the fundamental formula and the elementary solution, equations with an odd number of independent variables, and equations with an even number of independent variables and the method of descent.

Product Details

ISBN-13:	9780486781488
Publisher:	Dover Publications
Publication date:	08/25/2014
Sold by:	Barnes & Noble
Format:	eBook
Pages:	320
File size:	15 MB
Note:	This product may take a few minutes to download.

Table of Contents

	Preface	iii
Book I.	General Properties of Cauchy's Problem
I.	Cauchy's Fundamental Theorem. Characteristics	3
II.	Discussion of Cauchy's Result	23
Book II.	The Fundamental Formula and the Elementary Solution
I.	Classic Cases and Results	47
II.	The Fundamental Formula	58
III.	The Elementary Solution	70
1.	General Remarks	70
2.	Solutions with an Algebroid Singularity	73
3.	The Case of the Characteristic Conoid	83
	Additional Note on the Equations of Geodesics	111
Book III.	The Equations with an Odd Number of Independent Variables
I.	Introduction of a New Kind of Improper Integral	117
1.	Discussion of Preceding Results	117
2.	The Finite Part of an Infinite Simple Integral	133
3.	The Case of Multiple Integrals	141
4.	Some Important Examples	150
II.	The Integration for an Odd Number of Independent Variables	159
III.	Synthesis of the Solution Obtained	181
IV.	Applications to Familiar Equations	207
Book IV.	The Equations with an Even Number of Independent Variables and the Method of Descent
I.	Integration of the Equation in 2m[subscript 1] Variables	215
1.	General Formulae	215
2.	Familiar Examples	236
3.	Application to a Discussion of Cauchy's Problem	247
II.	Other Applications of the Principle of Descent	262
1.	Descent from m Even to m Odd	262
2.	Properties of the Coefficients in the Elementary Solution	266
3.	Treatment of Non-Analytic Equations	277
	Index	313