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.

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• Fourier–Mellin Transforms for Circular Domains