Let D be a domain in the plane
which is partially bounded by two curves Γ_{1} and Γ_{2} which meet at the origin and
form there an interior angle πτ > 0. Let N be an integer ≧ 2 and let α be a real
number such that 0 < α < 1. Suppose that for i = 1,2,Γ_{i} admits a parametrization
x = x_{i}(t), y = y_{i}(t), 0 ≦ t ≦ 1, where x_{i} and y_{i} have Nth derivatives which are
uniformly αHölder continuous, and x_{i}′(t) + y_{i}′(t) > 0. Let F(z) map the upper
half plane conformally onto D in such a way that F(0) = 0. Then if τ is
irrational F(z) has an asymptotic expansion in powers of z and z^{τ}, with error
term o(z^{Nτ−𝜖}). If τ = p∕q, a reduced fraction, then F(z) has an asymptotic
expansion in powers of z, z^{τ}, and z^{p} log z, with error term o(z^{Nτ−𝜖}). In
both cases 𝜖 is an arbitrarily small positive number. Furthermore expansions
for derivatives of F(z) of order ≦ N may be obtained by differentiating
formally.
