$\displaystyle f_{1}(z) = z^{1/2}$ to get the half disc map

$\displaystyle f_{2}(z) = \frac{1-z}{1+z}$ this maps the half disc onto the 4th quadrant.

$\displaystyle f_{3}(z) = z^2$ maps the 4th quadrant onto $\displaystyle \Im(z) < 0$

$\displaystyle f_{4}(z) = iz$ to rotate the region onto $\displaystyle \Re(z) > 0$

$\displaystyle f_{5}(z) = \frac{1-z}{1+z}$ maps the region onto the unit disc.

$\displaystyle f_5 \circ f_4 \circ f_3 \circ f_2 \circ f_1 = \frac{1 - i \left( \frac{1 - \sqrt{z}}{1+\sqrt{z}} \right)^2}{1 + i \left( \frac{1 - \sqrt{z}}{1+\sqrt{z}} \right)^2}$ I'm sure that can be simplified is it any good TPH?

Bobak