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Some Conformal Mappings to Try in Complex Paint

With Complex Paint, you can draw in either the domain or the range. Also, the Equation menu lets you both change the mapping and adjust the domain and range. Try

a) $z \to z^2$ :
First quadrant to the half space {Im(z) > 0}. The rectangular grid tool might be especially useful here.
b) $z \to e^z$ :
The strip $\{0 < Im(z) < \pi\}$ to the half space {Im(z) > 0}.
c) $z \to e^z$ :
The half-strip $\{0 < Im(z) < \pi$, Re(z) < 0} to a half disk.
d) $z \to e^{i \theta} \frac{z-z_0}{1-\bar{z}_0 z}$ :
The unit disk to itself, z0 to 0. What is the geometric effect of the $e^{i \theta}$ term? What happens to a small circle around z0?
e) $z \to \frac{1+z}{1-z}$ :
Upper half of unit disk to first quadrant. Check where the unit circle goes. How about the real axis? Compare the images of the quadrant 2 portion of the unit disk and the quadrant 1 portion.
f) $z \to \frac{z-1}{z+1}$ :
Halfplane $\{ Re(z) > 0 \}$ to unit disk. Increasing the domain size will help here.

g) $z \to \frac{1}{z}$ :
The inside of the unit disk to its exterior.
h) $z \to \sin{z}$ :
The half strip $\{ 0 < Re(z) < \frac{\pi}{2}$, Im(z) >0} .

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