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How can i find an example of a

  • Bijective conformal map between $$S= \{x+iy: 0 < x < 1, 0<y<1\}$$ onto $\mathfrak{H} = \{re^{i\theta}: 0<r<1, \ 0 <\theta<\pi\}$

I know conformal maps preserves angles. Any insight of how to deal with such question would be helpful.

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Unfortunately, such a map cannot be expressed by a simple formula. It will involve the inverse of an incomplete elliptic integral of the first kind. This integral is given explicitly in the Wikipedia article linked in the answer below. – user31373 Sep 26 '12 at 1:09

These might help: Conformal Map, Schwarz-Christoffel mapping.

It says, a holomorphic $f:U\to\mathbb C$ function ($U\subseteq\mathbb C$ open subset), is conformal iff $f'(z)\ne 0$ for $z\in U$. So, for example our favorit $z\mapsto e^z$ function is conformal, and so is $z\mapsto c\cdot z$ for any $c\ne 0$, and $z\mapsto 1/z$ if $0\notin U$.

Good, now consider $f_1:z\mapsto e^{\pi z}$, this maps $z=x+iy$ to the number with length $e^x$ and angle $\pi y$. It is getting closer what we want. (Where will $S$ go by $f_1$?)

So, before $f_1$ we should need a holomorphic function $f_0$ with nonvanishing differentiate that takes $S$ to a semi-infinite strip, preferaribly to $S':=\{x+iy \mid x<0,\ 0<y<1\}$. Then $f_1\circ f_0$ will be good.

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