Let $C$ be the unit circle $|z| = 1$ traversed once counterclockwise and then once clockwise, starting from $z = 1$. Construct a function $z(s, t)$ which deforms $C$ to the single point $z = 1$ in any domain $D$ containing the unit circle.

I'm not sure how to parametrize the unit circle accounting for ccw and cw direction. Also, all the examples in the text seems to have simple deformation functions $z(s, t)$ where you go from a bigger circle to a smaller one so I'm not too sure how it would work for this question.


1 Answer 1


The parameterization for the unit circle is just: $$ f(z) \mapsto f(e^{i\theta})ie^{i\theta}d\theta $$ for counterclockwise direction. $$ f(z) \mapsto f(e^{-i\theta})-ie^{-i\theta}d\theta $$ for clockwise direction. A circle of non-unit radius is parameterized by multiplying by $R$: $$ f(z) \mapsto f(Re^{i\theta})Rie^{i\theta}d\theta $$ ($R$ is the radius of the circle)

  • $\begingroup$ Yeah but what about the s, and t parameters $\endgroup$
    – DJ_
    Mar 17, 2013 at 1:35
  • $\begingroup$ $t=\theta$, $s(t) = e^{i\theta}$ $\endgroup$
    – Rustyn
    Mar 17, 2013 at 1:36

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