I just need some help with the penultimate question of my coursework:
Let $w=f(z)=\coth(z/2)$. Show that $w=f(z)=h(C) = (C+1)/(C-1)$ where $C=g(z)=e^z$. Find the image of the given points, boundary and region under $C=g(z)=e^z$, in the $\mathbb{C}$-plane (complex plane).
Hence find the image of these points, boundary and region under $w=f(z)=\coth(z/2)$
Thanks guys, any help will be appreciated!