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Question about Christoffel-Schwarz mappings from Ahlfors

  1. Apr 21, 2010 #1
    Suppose [tex]\zeta[/tex] lies in the upper half [tex]S_k[/tex] of an origin-centered disc in the complex plane and consider the function [tex]g(\zeta) = f(\zeta^{\alpha_k} + z_k)[/tex] where [tex]0 < \alpha_k < 2[/tex] is a real constant, [tex]z_k[/tex] is a complex constant, and [tex]f[/tex] is a bijective conformal map from its domain (which includes the points [tex]\zeta^{\alpha_k} + z_k[/tex]) to the open unit disc. (For the meanings of these constants, see Pg. 235 of Ahlfors if you have access to it.)

    The reflection principle of Schwarz guarantees that [tex]g[/tex] has an analytic continuation to the full disc, and hence it is analytic at the origin and has the Taylor expansion

    f(z_k + \zeta^{\alpha_k}) = f(z_k) + \sum_{m = 1}^{\infty} a_m \zeta^m.

    This is all fine, but now Ahlfors asserts that [tex]a_1[/tex] is nonzero because if it were zero then the half-disc [tex]S_k[/tex] would not be mapped into the unit disc by [tex]g[/tex].

    I'm not seeing why. But this is a crucial step. Any ideas?
  2. jcsd
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