1. (Scherk's minimal surface) $M: e^z \cos x = \cos y$. Let $R$ be the region in the $xy$-plane on which $\cos x \cos y > 0$. $R$ is a checkerboard pattern of open squares, with vertices $(\pi/2 + m\pi, \pi/2 + n\pi)$. Show that:
(a) $M$ is a surface.
(b) For each point $(u, v)$ in $R$ there is exactly one point $(u, v, w)$ in $M$. The only other points of $M$ are entire vertical lines over each of the vertices of $R$.