= Tangent-circle crescent conformal map
{title2=$z\mapsto e^{4\pi/z}+i\mapsto (e^{4\pi/z}+i-i)/(e^{4\pi/z}+i+i)$}
For the crescent inside $|z-2i|<2$ and outside $|z-i|\leq1$, inversion $w=1/z$ gives the horizontal strip $-1/2<\operatorname{Im}w<-1/4$. The affine map $\zeta=4w+2i$ sends it to $0<\operatorname{Im}\zeta<1$, and $e^{\pi\zeta}$ maps it bijectively onto the upper half-plane. A Cayley transform then maps it onto the unit disc. Tangency, rather than a separate enclosed hole, is why this domain is simply connected.
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