Vol. 15, No. 4, 1965

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Development of the mapping function at a corner

Neil Marchand Wigley

Vol. 15 (1965), No. 4, 1435–1461

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(z𝜖). If τ = p∕q, a reduced fraction, then F(z) has an asymptotic expansion in powers of z, zτ, and zp log z, with error term o(z𝜖). 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.

Mathematical Subject Classification
Primary: 30.40
Secondary: 35.00
Received: 9 August 1964
Published: 1 December 1965
Neil Marchand Wigley