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On the Differentiability of Conformal Maps at the Boundary1)

Published online by Cambridge University Press:  22 January 2016

B.G. Eke
B.G. Eke*
Affiliation:
Queen’s University, Belfast, Northern Ireland
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Let S be a simply connected domain in the w + u + iv plane and let ∂S denote its boundary which we assume passes through w= ∞. Suppose that the segment L= {u ≧ u0; v = 0} of the real axis lies in S and that w∞ is the point of ∂ S accessible along L. Let z = z(w) = x(w) + iy(w) map S in a (1 — 1) conformal way onto ∑ = {z = x + iy: — ∞ < x < + ∞ } so that . The inverse map is w = w(z) = u(z) + iv(z). S is said to possess a finite angular derivative at w∞ if z(w) — w approaches a finite limit (called the angular derivative) as w→w∞ in certain substrips of S.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 41 , February 1971 , pp. 43 - 53
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1971

Footnotes

1)

Work supported partially by U.S.A.F. Contract AFOSR-68-1514, with the University of California, San Diego.

References

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