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

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

    However I am having difficulties proving that if f is conformal at $\displaystyle z_0$ then f is analytic at $\displaystyle z_0$. 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 $\displaystyle z_0$ is preserved by the mapping $\displaystyle f(z)$ and so is the orientation, the f is analytic at $\displaystyle z_0$
    Last edited by Haven; Nov 10th 2010 at 01:07 PM.
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