Comformal mapping asymptotics at a cusp

We describe the asymptotic behavior of the mapping function at an analytic cusp compared with Kaiser’s results for cusps with small perturbation of angles and the known explicit formulae for cusps with circular boundary curves. We propose a boundary curve parametrization by generalized power series which allows us to give explicit representations for locally univalent mapping functions with given asymptotic properties and for cusp boundary curves having an arbitrary order of tangency.