Show that \(f(z) = z^2\) is uniformly continuous in any open disk \(\abs{z} < R\), where \(R > 0\) is fixed, but it is not uniformly continuous on \(\mathbb{C}\).
Under this number the source also prints, as parts (1) and (2), the Cauchy problem for the Laplace equation with \(u(x,y) = \frac{e^{ny} - e^{-ny}}{2n^2}\sin nx\) on the unit disk, which is question 19 of the Fall 2009 list. It is a separate problem, carried by its own card.