Example of analytic function that maps circle to self intersecting curve Is there an explicit example that an analytic function maps the unit circle to a self intersecting curve? As unit circle is not homogenous to self intersecting curve, I am considering finding an analytic function that is not homeomorphism to satisfy the requirement. Thanks for any suggestion.

The function $f(z) = (z+1/2)^2$ is an example. It first takes $D(0,1)$ to $D(1/2,1)$ and then applies the map $w\to w^2$ to it. The last map we know very well, and from it you can verify that $f$ maps the upper half of the unit circle bijectively to a curve that starts in the first quadrant, moves left to the fourth quadrant, comes down and intersects the negative real axis, continues downward and rightward under $0$ and then comes back up to end at $1/4.$ The bottom half of the circle gets mapped to the conjugate of this. The total curve intersects itself exactly once, on the negative real axis, and the curve is regular.

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