Find a conformal map $\displaystyle w(z)$ of the strip $\displaystyle \{ \text{Im}(z)<\text{Re}(z)<\text{Im}(z)+2 \}$ onto the upper half-plane such that $\displaystyle w(0)=0, w(z) \rightarrow +1$, as $\displaystyle \text{Re}(z) \rightarrow -\infty$, and $\displaystyle w(z) \rightarrow -1$ as $\displaystyle \text{Re}(z) \rightarrow +\infty$. Sketch the images of the straight lines $\displaystyle \{ \text{Re}(z)=\text{Im}(z)+c \}$ in the strip. What is the image of the median line $\displaystyle \{ \text{Re}(z)=\text{Im}(z)+1 \}$ of the strip?

The back of the book says that $\displaystyle w=\frac{1+\zeta}{1-\zeta}$, where $\displaystyle \zeta=-e^{\pi(1+i)z/2}$ works. Also, the median is mapped to top half of unit circle. I do not see how they get this map. This is what I need help with. Thank you.