Example of bijective map between $S_1$ and $S_2$ [closed]

Question

Example of a bijective conformal map between $$S_{1} = \{re^{i\theta} \ : \ 0 < r < 2 \ , 0 < \theta <\pi/4\}$$ and $$S_{2} = \{ x +iy : 0 < y <1\}$$

Answer 1

I assume that the question is to find a bijective conformal map, or a biholomorphic map, between the "pizza-slice" $S_1$ and the infinite strip $S_2$. You can do that basically in three steps.

(1) The complex logarithm $$f(z)=\log(z)=\log|z|+{\rm i}\arg(z)$$ will map $S_1$ to some semi-infinite strip.

(2) The hyperbolic cosine $$g(z)=\cosh(z)={1\over 2}({\rm e}^z+{\rm e}^{-z})=\cosh(x)\cos(y)+{\rm i}\sinh(x)\sin(y)$$ will map the semi-infinite strip $\{(x,y):x>0, 0<y<\pi\}$ to the upper half plane.

(3) The complex logarithm (again!) will map the upper halfplane to some infinite strip.

Now, you need to identify the semi-infinite strip in (1) and the infinite strip in (3). You also need to add some linear maps to bridge the steps, since the semi-infinite strip you get in (1) is not the one you start with in (2), and the infinite strip you get in (3) is not the same as $S_2$. Details are left to the interested reader.