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Advanced Mathematics for Applications by Andrea Prosperetti

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20

Theory of Distributions

20.1 Introduction

The immense usefulness of the “δ-function”, amply documented in this book, joint to its “non-existence” in the conventional sense in which a function is understood, would be motivation enough to develop a framework generalizing in a mathematically satisfactory way the concept of function. But a long list of additional justifications for such an extension is readily developed:

  1. A naive limiting process suggests that the d’Alembert general solution of the wave equation in the form f (xct) + g(x + ct) (p. 8) should also make sense when f and g are not twice differentiable functions. It should be possible to decide whether a proposed non-differentiable function is indeed a solution of the equation ...

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