Find the image of certain holomorphic function on unit circle The problem asks to show that
$$w=\log\left(\frac{1+z}{1-z}\right)+\frac{2z}{1+z^2}$$
maps the unit circle $\mathbb{D}\subset \mathbb{C}_z$ one to one and onto the $w$-space $\mathbb{C}_w$ with four half-line deleted by some properly chosen branch of logarithm, and asks me to find the half-lines.
It seems unrealistic to discuss by letting $z=x+\mathrm{i}y$ since the curve will be too complicated, and I tried to find the boundary information by letting $z=e^{\mathrm{i}\theta}$, and observed $\mathrm{i},-\mathrm{i},-1$ may be three of the endpoints (while the image of $z=1$ cannot even be approximated) but I do not know how to grasp the graph of $z\mapsto w$, is there any better approach?
 A: If we write $z = e^{it}$, we get
$$\begin{align}
w &= \log \left(\frac{1+e^{it}}{1-e^{it}}\right) + \frac{2e^{it}}{1+e^{2it}}\\
&= \log \left( \frac{e^{-it/2}+e^{it/2}}{e^{-it/2}-e^{it/2}}\right) + \frac{2}{e^{-it}+e^{it}}\\
&= \log \left(i\cot (t/2)\right) + \frac{1}{\cos t}\\
&= \log (i\cot(t/2)) + \frac{\cos^2(t/2) + \sin^2(t/2)}{\cos^2(t/2) - \sin^2(t/2)}\\
&= \log (iu) + \frac{u^2+1}{u^2-1}\\
&= i\arg (iu) + \log \lvert u\rvert + \frac{u^2+1}{u^2-1}
\end{align}$$
with $u = \cot(t/2)$.
For $-\pi < t < -\pi/2$, $u$ decreases from $0$ to $-1$. Choosing the principal branch of the logarithm, $\arg (iu) = -\pi/2$ then, and the real part $\log \lvert u\rvert + \frac{u^2+1}{u^2-1}$ of $w$ increases from $-\infty$ until it reaches a maximum for $u = 1 - \sqrt{2}$, and then decreases to $-\infty$.
For $-\pi/2 < t < 0$, $u$ decreases from $-1$ to $-\infty$, the real part of $w$ decreases from $+\infty$ until it reaches a minimum for $u = -1-\sqrt{2}$, and then increases again towards $+\infty$. The argument is still $-\pi/2$.
For $0 < t < \pi/2$, $u = \cot (t/2)$ decreases from $+\infty$ to $1$, and the real part of $w$ decreases from $+\infty$ until it reaches a minimum for $u = 1+\sqrt{2}$ and then increases again towards $+\infty$. The argument is $\pi/2$.
For $\pi/2 < t < \pi$, $u$ decreases form $1$ to $0$, the real part of $w$ increases from $-\infty$ until it reaches a maximum for $u = \sqrt{2}-1$, and then decreases towards $-\infty$ again. The argument is also $\pi/2$ here.
Choice of a different branch of the logarithm changes the imaginary part of $w$ only (by a constant multiple of $2\pi$).
