
conformal map, sector
Find a conformal map of the sector $\displaystyle \{ \text{arg} z < \frac{\pi}{3} \}$ onto the open unit disk mapping 0 to 1 and $\displaystyle \infty$ to +1. Sketch the images of radial lines and of arcs of circles centered at 0. Is the map unique?
The back of the book says that $\displaystyle w=\frac{z^{\frac{3}{2}}s}{z^{\frac{3}{2}}+s}$, for any $\displaystyle s>0$ works. Though the map is not unique, the sketch is. I do not see how they got this map. Also, why isn't this map unique? I do not see why. Thank you.