On the Differentiability of Conformal Maps at the Boundary

Source abstract

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 ≧ u 0 ; 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.