Fourier–Mellin Transforms for Circular Domains

Generalized Fourier–Mellin transforms for analytic functions defined in simply connected circular domains are derived. Circular domains are taken to be those with boundaries that are a finite union of circular arcs, including straight line edges. The results are an extension to circular domains of the generalized Fourier transforms for convex polygons (having only straight line edges) derived by Fokas and Kapaev (IMA J Appl Math 68:355–408, 2003 ). First, a new, elementary derivation of the latter result for polygons is given based on Cauchy’s integral formula and a spectral representation of the Cauchy kernel. This rederivation extends in a natural way to the case of circular domains once an adapted spectral representation of the Cauchy kernel is established. Domains with boundaries that are a combination of circular arc and straight line edges can be treated similarly. The newly derived transforms are generalizations of the classical Fourier and Mellin transforms to general circular domains. It is shown by example how they can be used to solve boundary value problems for Laplace’s equation in such domains. The notions of spectral matrix and fundamental contour , which arise naturally in the formulation, are also introduced.