Let D be a domain in the plane
which is partially bounded by two curves Γ1 and Γ2 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,Γi admits a parametrization
x = xi(t), y = yi(t),0 ≦ t ≦ 1, where xi and yi have N-th derivatives which are
uniformly α-Hölder continuous, and |xi′(t)| + |yi′(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 zτ, with error
term o(zNτ−𝜖). If τ = p∕q, a reduced fraction, then F(z) has an asymptotic
expansion in powers of z, zτ, and zplogz, with error term o(zNτ−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.
