Abstract

This chapter discusses the Fourier and Laplace transformations. The Poisson Integral Formula is intimately related to the notion of Fourier series and a similar connection exists between the Fourier series and the Laurent series of a function f(z) analytic in an annulus r1<|z|<r2. The problem of convergence is of fundamental importance in the study of Fourier series. The main problem is to discover under what circumstances the values Û(ø) and U(ø) coincide, as then Û provides an inversion formula for the Fourier transform u. This has the effect of doubling the size of a given table of integrals, for if a closed form solution is known for the Fourier transform u(t), one is also known for its inverse. The Fourier integral theorem provides useful conditions under which Û(ø) and U(ø) agree.

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