Fourier Analysis and Its Applications
This book presents the theory and applications of Fourier series and integrals, Laplace Transforms, eigenfunction expansions, and related topics. It deals almost exclusively with those aspects of Fourier analysis that are useful in physics and engineering. Using ideas from modern analysis, it discusses the concepts and reasoning behind the techniques without getting bogged down in the technicalities of rigorous proofs. A wide variety of applications are included, in addition to discussions of integral equations and signal analysis.
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Orthogonal Sets of Functions
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analytic function apply approximation Bessel functions Bessel's equation boundary conditions boundary value problems bounded calculation Chapter complex number consider constant continuous function contour convergence convolution coordinates defined delta function denote derivatives Dirichlet problem eigenfunctions eigenvalues endpoints example Exercise expansion exponential fact Figure finite Fourier coefficients Fourier inversion Fourier series Fourier sine series Fourier transform Green's function half-plane heat equation Hence Hint homogeneous infinite inhomogeneous initial conditions inner product integrable function interval inversion formula kuxx Laplace transform Lemma linear combination Moreover multiples norm obtain ordinary differential equations orthogonal set orthonormal basis orthonormal set partial sums periodic function piecewise continuous piecewise smooth pointwise positive zeros Proof region result satisfies separation of variables sequence Show singular solution solve string Sturm-Liouville problem Suppose tends to zero test function Theorem vanishes vectors vibrating wave equation