Show that if \(f_k\to f\) uniformly on \(\Omega\) with \(f_k\) holomorphic then \(f\) is holomorphic.

Alternatively,

Show that if \(f_k \to f\) uniformly with \(f\) holomorphic then every \(n\)th derivative satisfies \(f_k^{(n)}\to f^{(n)}\) uniformly.

Alt:

Show that if \(f\) is holomorphic on \({\mathbb{D}}\) then \(f\) has a power series expansion that converges uniformly on every compact \(K\subset {\mathbb{D}}\).

Show that any holomorphic function \(f\) can be uniformly approximated by polynomials.

Show that if \(f\) is holomorphic on a connected region \(\Omega\) and \(f'\equiv 0\) on \(\Omega\), then \(f\) is constant on \(\Omega\).

Show that if \({\left\lvert {f} \right\rvert} = 0\) on \({{\partial}}\Omega\) then either \(f\) is constant or \(f\) has a zero in \(\Omega\).

Show that if \(\left\{{f_n}\right\}\) is a sequence of holomorphic functions converging uniformly to a function \(f\) on every compact subset of \(\Omega\), then \(f\) is holomorphic on \(\Omega\) and \(\left\{{f_n'}\right\}\) converges uniformly to \(f'\) on every such compact subset.

Show that if each \(f_n\) is holomorphic on \(\Omega\) and \(F \coloneqq\sum f_n\) converges uniformly on every compact subset of \(\Omega\), then \(F\) is holomorphic.

Show that if \(f\) is once complex differentiable at each point of \(\Omega\), then \(f\) is holomorphic.

## Polynomials

Show that if \(f\) is entire and \(f(z) \overset{z\to\infty}\longrightarrow \infty\) then \(f\) is a polynomial.

- Set \(g(z) \coloneqq f(1/z)\), so \(g(z) \overset{z\to 0}\longrightarrow \infty\) making \(z=0\) a singularity.
- This is not an essential singularity by Casorati-Weierstrass.
- So this is a pole and \(g(z) = \sum_{-N\leq k \leq 0} c_k z^k\) for \(N\) the order of the pole
- Thus \(f(z) = \sum_{0<k<N}c_k z^k\) is a polynomial.