ABSTRACT
Complex variables enter at various levels in the study of Fourier transforms. The simplest applications involve merely the evaluation of the relevant integrals by contour integration techniques. The integral for f(t) is usually done as a contour integral. Linear differential equations with constant coefficients are readily solved using Laplace transforms. A further change of variable in the Laplace transform leads to the so-called Mellin transform. The Laplace transform method can be profitably applied to initial value problems in heat conduction. More sophisticated applications involve the analytic continuation of the Fourier transform into the complex plane of the transform variable. In such applications dispersion relation methods and the so-called Wiener-Hopf method are powerful techniques for the solution of certain types of problems.
