Results & Lemmas (3)
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Theorem 1.
Theorem 1. With the above notations, ˆF = ˜F ◦ˆf + C for some constant C.
Theorem 1. With the above notations, ˆF = ˜F ◦ˆf + C for some constant C.
Theorem 2.
Theorem 2. Let P be a polygon having vertices c1,..., cn given cyclically, and ordered by a positive orientation on ∂P. Let f be a…
Theorem 2. Let P be a polygon having vertices c1, . . . , cn given cyclically, and ordered by a positive orientation on ∂P. Let f be a univalent harmonic mapping of U such that f is the Poisson integral of a step function having the ordered sequence c1, . . . , cn as its values. Then the analytic dilatation a(ζ) of f is a finite Blaschke product of order at most n−2, f(U) = P, and if ϕ is a height function for f, then ϕ tends to +∞or −∞at points over the open segments making up the sides of P. If
Theorem 3.
Theorem 3. The solution f(ζ) to the Riemann mapping theorem above with D convex and (4.1) a(ζ) = ζn−2
Theorem 3. The solution f(ζ) to the Riemann mapping theorem above with D convex and (4.1) a(ζ) = ζn−2
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