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Why is the image of $g(w)=\log\left({1+w\over 1-w}\right)$ for $0<\arg(w)<\pi$ all lie on a straight line?

Sorry about the confusing statement in the beginning.

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    I don't think your statement is true the way it is stated. If you take $\omega_\alpha=\alpha i$ (so $\arg(\omega)=\pi/2$), then $g(\omega)=i\arctan(2\alpha/(1-\alpha^2))$, which does not form a straight line.2012-02-19
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    Note that the question has been changed after an answer was given. See also my comments under bgins' answer on problems in the formulation of the question.2012-02-19
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    I think the image is only on a straight line when we also assume that $w$ has unit modulus. See my (mostly) corrected post for details.2012-02-19
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    Possibly related: http://math.stackexchange.com/questions/118868/for-complex-z-z-1-implies-textre-left-frac1-z1z-right-02012-03-22

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