Ebook: Introduction to the Theory of Distributions

Every physicist and mathematician uses distributions (sometimes called generalized functions), albeit often unknowingly. From its origins in the Dirac delta `function', distribution theory continues to influence many research areas from quantum mechanics to partial differential equations, but has also grown into an important field in its own right. For anyone interested in learning about the field, this is clearly the first port of call. It presents a balanced introduction to the subject on a level suitable for anyone with a basic grounding in analysis (no knowledge of functional analysis is required).

The book begins by defining the two building blocks of the theory---test functions and distributions. It then quickly expands, filling in the important details of differentiation, multiplication, tensor products and convolution. All of this is written with sufficient mathematical rigor, but never too much that it interferes with the basic understanding of the subject, and is supported throughout by useful exercises. The book then builds up the theory of Fourier and Laplace transforms of distributions, which has important applications in the study of linear partial differential equations. The second edition contains an indispensable new chapter on the calculus of wavefront sets, which, among its uses, allows the propagation of singularities of solutions to partial differential equations to be properly treated. All in all, while the book is not for the common man, and does require a certain level of mathematical maturity, it does present an excellent introduction to an important, and often poorly understood, area of mathematics.