Introduction to Hyperfunctions and Their Integral Transforms: An Applied and Computational Approach
This textbook presents an introduction to the subject of generalized functions and their integral transforms by an approach based on the theory of functions of one complex variable. It includes many concrete examples.
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Chapter 1 Introduction to Hyperfunctions
Chapter 2 Analytic Properties
Chapter 3 Laplace Transforms
Chapter 4 Fourier Transforms
Chapter 5 Hilbert Transforms
Chapter 6 Mellin Transforms
Chapter 7 Hankel Transforms
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Introduction to Hyperfunctions and Their Integral Transforms: An Applied and ...
Limited preview - 2010
analytic continuation arbitrary assume change of variables closed contour compact complex plane Consider contour integral converges uniformly correspondence deﬁned defining function F(z Deﬁnition denoted differential Dirac impulse domain entire function equivalent defining function established Example exists F_(z F+(z fc=i ﬁnite Fourier series Fourier transform given hyperfunction Hankel transform Hilbert transform holomorphic function holomorphic hyperfunction hyperfunction f(x hyperfunctions deﬁned image function integral equation integrand interval Let f(x Let us compute log(z loop lower component lower half-plane lower hyperfunctions Mellin transform obtain ordinary function parameter periodic hyperfunction polynomial Problem Proof Proposition real analytic function real axis respect right-hand side right-sided sense of hyperfunctions sgn(x shows Similarly sin(a7r singular solution strip of convergence strong defining function theorem Transforms of Hyperfunctions uniform convergence upper and lower upper half-plane write yields zero