# Extra Questions

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.