Example 22. Map the unit disk with a cut along the real axis from the center to the upper half-plane.
Solution. 1) Using the function $w_{1}=\sqrt{z}$, map the unit circle to the upper half-circle. In this process, the upper bank of the cut O remains in place, while the lower bank $O A^{\prime}$ will be mapped to the segment $[-1,0]$ on the $W_{1}$ plane.

2) Using the function

$$
w_{2}=\frac{w_{1}+1}{w_{1}-1}
$$

map the half-circle obtained in 1) to the first quadrant of the $W_{2}$ plane. In this process, the point $w_{1}=1$ will be mapped to the point $w_{2}=\infty$, and the point $w_{1}=-1$ will be mapped to the point $w_{2}=0$.

3) Finally, using the function $w=w_{2}^{2}$, map the first quadrant to the upper half-plane.

Ultimately, we obtain

$$
w=w_{2}^{2}=\left(\frac{w_{1}+1}{w_{1}-1}\right)^{2}=\left(\frac{\sqrt{z}+1}{\sqrt{z}-1}\right)^{2}
$$