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

The back of the book says that w=\frac{1+\zeta}{1-\zeta}, where \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.