A bijective map between a horizontal strip and the unit disc.

Consider the map $\phi(z):= \tanh(\frac{\pi}{4}z)$, which is a conformal bijection from the strip $\{z\in\mathbb{C} ; |\text{Im} (z)|<1\}$ to the unit disc $\{z\in\mathbb{C} ; |z|<1\}$. Prove that it also maps $\mathbb{R}+i$ bijectively to $\{e^{i\theta} ; \theta\in(0,\pi) \}$ and $\mathbb{R}-i$ bijectively to $\{e^{i\theta} ; \theta\in(-\pi,0) \}$. 

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Erdos Erdos
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  • Erdos Erdos
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    This was a challenging question, and it took me some time to figure out and write an answer. Please consider offering slightly higher bounties in the future.

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