Abstract
This chapter discusses Fourier and Laplace transforms. The inversion of the Laplace transform is accomplished for analytic functions f (p) of order O (p-k) with k > 1 by means of the inversion integral. The inversion of the Fourier transform is accomplished by means of the inversion integral. The Fourier sine and cosine transforms of the function f(x), denoted by Fs (ξ) and Fc (ξ), respectively, are defined by the integrals. The functions f(x) and Fs( ξ ) are called a Fourier sine transform pair, and the functions f(x) and Fc( ξ ) a Fourier cosine transform pair, and knowledge of either Fs ( ξ ) or Fc ( ξ ) enables f(x) to be recovered. The inversions of the Fourier sine and Fourier cosine transform are accomplished by means of the inversion integral.
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