## 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. Second edition has been updated, restructured and modernised to answer specific difficulties typically found by students encountering delta functions for the first timeAttention is focused on techniques, with a liberal selection of worked examples and exercisesReadily accessible to a broad audience of scientists, mathematicians and engineers and can be used as a working manual in its own right |

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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 pre-delta function pseudo-function 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 well-defined write