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Math Help - Conformal Proof

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    Member Haven's Avatar
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    Conformal Proof

    Here's an interesting question I can't seem to get my head around.

    A function f is called a diffeomorphism if J_f \neq 0, where J_f  = u_xv_y - u_yv_x.
    Let \alpha_1(t) and \alpha_2(t) denote contours  C_1 and  C_2 in a domain D, where  \alpha_1(a)= \alpha_2(b)=z_0.
    Let \Theta(C_1,C_2) = \arg{\alpha_1'(a)}-\arg{\alpha_2'(b)}.
    A function f is called conformal at a point if \Theta(C_1,C_2) = \Theta(f(C_1),f(C_2)), and f is called angle-preserving at a point |\Theta(C_1,C_2)| = |\Theta(f(C_1),f(C_2))|.

    I have shown a function f is conformal at z_0 if and only if J_f(z_0)>0 and is angle-preserving at z_0.

    However I am having difficulties proving that if f is conformal at z_0 then f is analytic at z_0. I would greatly appreciate any hints, but please don't solve the problem for me.

    EDIT: Without using all the theta terms, we wish to show that if the angle between two curves on the z-plane, at z_0 is preserved by the mapping f(z) and so is the orientation, the f is analytic at z_0
    Last edited by Haven; November 10th 2010 at 01:07 PM.
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