Find a conformal map from $U = \left\{z \,\Big|\, Im(z)\geq \frac{\sqrt{3}}{2}\right\} \cap \left\{z \,\Big|\, |z| < 1\right\}$ onto the upper half plane.
I want to transform U to the upper half unit semicircle and then use the conformal mapping $\frac{-1}{2} (z + 1/z)$. I am not sure how to do the first step.
I also want to know whether there is canonical method to find conformal mappings. I have observed that a few well-known mappings are known and the rest are obtained as their composition. What are the must-know conformal mappings?
Best Answer
For domains which are delimited by two generalized circles (i.e. circles or lines) it is always a good start to map the intersection points of the circles to $0$ and $\infty$ with a Möbius transformation. The domain is then mapped to a sector of the form $$ S_{\alpha, \beta} = \{ w = r e^{i\phi}\mid r > 0, \alpha < \phi< \beta \} $$ and that can be mapped to the upper half-plane with a rotation and $w \mapsto w^\lambda$ with a suitable real positive $\lambda$.
In your case the circle $|z|=1$ and the line $\operatorname{Im} z = \sqrt 3/2$ intersect at the points $$ a = -\frac 12 + \frac{\sqrt 3}{2}i \, , \, b= \frac 12 + \frac{\sqrt 3}{2}i $$ so that $T(z) = (z-a)/(z-b)$ maps the domain to a sector. The angles $\alpha$, $\beta$ can (for example) be determined by computing $T(i)$ and $T(i\sqrt 3/2)$, which lie on the boundary lines of the sector.