## Lectures on Cauchy's Problem in Linear Partial Differential EquationsWould 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 ServiceDelivered 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. 1923 ed. |

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### Contents

CAUCHYS FUNDAMENTAL THEOREM CHARACTER | 3 |

DISCUSSION OF CAUCHYS RESULT | 23 |

CLASSIC CASES AND RESULTS | 47 |

THE FUNDAMENTAL FORMULA | 58 |

THE ELEMENTARY SOLUTION | 70 |

INTRODUCTION OF A NEW KIND OF IMPROPER | 117 |

THE INTEGRATION FOR AN ODD NUMBER OF INDE | 159 |

### Common terms and phrases

adjoint admit aforesaid analytic function approach zero arbitrary assumed Book boundary calculation Calculus of Variations Cauchy Cauchy's data Cauchy's problem characteristic cone characteristic conoid coefficients considered const constant contains convergence coordinates corresponding curvilinear coordinates cylinder Darboux deduced defined denoting determinate different from zero double integral duly inclined edition element elementary solution equal exist expansion expression factor finite formula generatrix geodesic gives half conoid holomorphic function hyperbolic improper integral independent variables infinite infinitesimal inside instance integral equation integrand integration with respect introduced Jacobian latter method neighbourhood notation obtained operations ordinary ordinary differential equations parameters partial derivatives partial differential equation plane polynomial punctual transformation quadratic form quantity replaced result right-hand side satisfy simple integral singular space surface tangent term theorem theory tion transversal Unabridged republication upper limit vanish vertex Volterra's