How $Not$ to Compute a Fourier Transform

Abstract

We revisit the Fourier transform of a Hankel function, of considerable importance in the theory of knife edge diffraction. Our approach is based directly upon the underlying Bessel equation, which admits manipulation into an alternate second order differential equation, one of whose solutions is precisely the desired transform, apart from an $\textit{a priori}$ unknown constant, and a second, undesired solution of logarithmic type. A modest amount of analysis is then required to exhibit that constant as having its proper value, and to purge the logarithmic accompaniment. The intervention of this analysis, which relies upon an interplay of asymptotic and close-in functional behaviors, prompts our somewhat ironic, mildly puckish caveat, our negation $\textit{not}$ in the title. In a concluding section we show that this same transform is still more readily exhibited as an easy byproduct of the inhomogeneous wave equation in two dimensions satisfied by the Green's function $\textit{G}$, itself proportional to a Hankel function. This latter discussion lapses of course into the argot of physicists and, in its r$\hat{{\rm{o}}}$le of a mere afterthought, makes no claim whatsoever to any originality.