I'm working on the conformal mappings of complex analysis. I can find conformal mappings from&onto simple domains but cannot find one from the abnormal domains.
Can any one give me a hint? Is there a rule that I can apply to find such mappings?
($D(z;r)$ means the open disk centered at $z$ with radius $r$.)
On
I think there is no conformal map between your domains, because conformal mapping must preserve angle between two curves. If $P$ is one of the corner points of the domain $D(0,1)\setminus D(1,1)$, then the angle formed at $P$ is less than $\pi$ but the image of $P$ must be on the circle $C(0,1)$ and so the angle formed there is $\pi$; hence the angle is not preserved.
Hint: Note that the boundary of your region is a union of two circular arcs, which share two common end points. At a first step, you may choose a fractional linear transformation, which maps one end point to $0$ and the other end point to $\infty$. Then both circular arcs are mapped to rays with the same initial point $0$. Secondly, you may choose an appropriate power function to map the region bounded by the two rays to the upper half plane. Finally, you may use another fractional linear transformation to map the upper half plane to the unit disk.