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Special functions and generalized functions

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posted on 2014-12-15, 10:40 authored by Fatma. Al-Sirehy
In 1950, Laurent Schwartz marked a convenient starting point for the theory of generalized functions as a subject in its own right. He developed and unified much of the earlier work by Hadamard, Bochner, Sobolev and others. Since then an enormous literature dealing with both theory and applications has grown up, and the subject has undergone extensive further development. The original Schwartz treatment defined a distribution as a linear continuous functional on a space of test functions.;This thesis can be considered a part of the development going in that direction. It is partly an extension of earlier contributions by Fisher, Kuribayashi, Itano and others.;After introducing the background and basic definitions in Chapter One, we developed some basic results concerning the cosine integral Ci(lambda x) and its associated functions Ci+(lambda x) and Ci-(lambdax) as well as the neutrix convolution products of the cosine integral.;Chapter Three is devoted to similar results concerning the sine integral Si(lambdax).;In Chapter Four, we generalize some earlier results by Fisher and Kuribayashi concerning the product of the two dimensions xl+ and x-l-r+ . Moreover, other results are obtained concerning the neutrix product of |x|lambda-r lnp |x| and sgn x| x|lambda-r. Other theorems are proved about the matrix product of some other distributions such as xl+ ln x+ and x-l-r- .;Chapter Five contains new results about the composition of distributions. It involves the applications of the neutrix limit to establish such relationships between different distributions.


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University of Leicester

Qualification level

  • Doctoral

Qualification name

  • PhD



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