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Math Help - Conformal Mapping

  1. #1
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    Conformal Mapping

    (1) Let f be a one-to-one analytic function on a domain D and suppose that there is an analytic function h on D with h^2 = f. Show that h is also one-to-one.

    Not really important but if anyone could help me out, I would appreciate it, thanks!
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  2. #2
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    Quote Originally Posted by shadow_2145 View Post
    (1) Let f be a one-to-one analytic function on a domain D and suppose that there is an analytic function h on D with h^2 = f. Show that h is also one-to-one.

    Not really important but if anyone could help me out, I would appreciate it, thanks!
    I may be wrong, but it doesn't depend on analysticity:
    If x,y\in D are such that h(x)=h(y) then, squaring both sides, h(x)^2=h(y)^2, i.e. f(x)=f(y), and hence x=y because f is one-to-one. This shows that h is one-to-one as well.

    A related question would be to show that h is locally one-to-one if f is locally one-to-one. This is easy if you know that f is locally one-to-one if and only if, for x\in D, f'(x)\neq 0. Derivating h(x)^2=f(x), we get that 2h(x)h'(x)\neq 0 hence h'(x)\neq 0 and h is thus locally one-to-one as well.
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