Delta Functions: An Introduction to Generalised FunctionsDelta Functions has now been updated, restructured and modernised into a second edition, to answer specific difficulties typically found by students encountering delta functions for the first time. In particular, the treatment of the Laplace transform has been revised with this in mind. The chapter on Schwartz distributions has been considerably extended and the book is supplemented by a fuller review of Nonstandard Analysis and a survey of alternative infinitesimal treatments of generalised functions. Dealing with a difficult subject in a simple and straightforward way, the text is readily accessible to a broad audience of scientists, mathematicians and engineers. It can be used as a working manual in its own right, and serves as a preparation for the study of more advanced treatises. Little more than a standard background in calculus is assumed, and attention is focused on techniques, with a liberal selection of worked examples and exercises.

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Contents
Table of Contents  1 
The Dirac Delta Function  26 
Properties of the Delta Function  47 
Timeinvariant Linear Systems  76 
The Laplace Transform  99 
Fourier Series and Transforms  128 
Other Generalised Functions  154 
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Common terms and phrases
absolutely integrable algebraic apply arbitrary Chapter closed interval continuous function continuously differentiable function convergence convolution corresponding definition denote Dirac delta function elementary calculus equation equivalence class everywhere example exists expressions F(iu finite interval finite number follows formal Fourier series Fourier transform func function defined function F generalised functions given Hence hyperfinite hypernatural number hyperreal number impulse response impulse response function infinitesimal input internal function inverse Laplace transform least upper bound Lebesgue integral linear functional linear space mathematical Mean Value Theorem neighbourhood Nonstandard Analysis nonstandard extension notation ordinary functions origin output particular pointwise positive integer predelta function pseudofunction r+oo real number system relatively bounded result Riemann integral sampling property Schwartz distributions sense sequence signal Similarly simple standard function Stieltjes integral subset Suppose symbol T)dT theory tion uniform norm unit step function vanishes identically vector welldefined write