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I'm reading Conway's complex analysis book and on page 53 the author found the analytic function $f:G\to \mathbb C$, where $G=\{z:\text{Re}(z)>0\}$ and $f(G)=D=\{z:|z|<1\}$ defined by $$f(z)=\frac{z-1}{z+1}$$

Afterwards he states the following:

Could someone help me to find where are in the book the results the author is talking about?

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It is not as specific as it stated. But it is on page 46, when the author is talking about the function $e^z$.

Notice that this function $e^z$ maps the infinite strip $\{z: |\Im z|<\frac{\pi }{2}\}$ onto the right half plane.

Consider $e^{x+yi}$. In the strip $|y|<\frac{\pi}{2}$. Then $e^{x+yi}=e^x \cos y+i e^x \sin y$ where the real part is $>0$, whereas the imaginary part can assume any value, hence the right half plane.

Now the map $g(z)=\frac{e^z-1}{e^z+1}$ is the composite function of $e^z$ and the $f(z)$. So the maps are also composited.

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I think he is referring to point 2.16 at the bottom of page 38, starting with the remarks that $|\exp(z)|=\exp(\hbox{Re } z)$ and $\arg\exp(z)=\hbox{Im } z.$

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