6220-2022-08

Let \(\Omega \) be a simply connected domain in \(\mathbb {C}\). Suppose that \(f:\Omega \to \mathbb {C}\) is holomorphic and has finitely many zeros in \(\Omega \). What is the supremum of \(m\in \mathbb {N}\) so that \(f(z)=g(z)^m\) for some \(g\) holomorphic in \(\Omega \)? Identify this value explicitly in terms of the orders of the zeros of \(f\) and prove that it is correct.

Show that if \(f\) is holomorphic on an open neighborhood of \(\overline {\mathbb {D}}\) and \(|f(z)|=1\) on \(|z|=1\) then \(f\) is a rational function.

Let \(\Omega \) be a bounded domain in \(\mathbb {C}\) and \(f_n:\Omega \to \mathbb {C}\) a sequence of holomorphic functions such that \[ \int _\Omega |f_n-f_m|^2\,dA(z)\to 0\text { as }n,m\to \infty . \] Here \(dA(z)\) is the Lebesgue area integral. Show that there is \(f\) holomorphic on \(\Omega \) such that \(f_n\to f\) locally uniformly on \(\Omega \).

Let \(a, b > 0\). Using the residue calculus evaluate the improper integral

Prove the following special case of Runge’s theorem: Suppose that \(f:D(0,4)\setminus (\overline {D(1,.1)}\cup \overline {D(-1,.1)})\to \mathbb {C}\) is holomorphic and \(\epsilon >0\) is given. Then there is a rational function \(q\) such that \(|f(z)-q(z)|\le \epsilon \) for \(z\in K=\) \[ \overline {D(0,2)}\setminus (\overline {D(1,.2)}\cup \overline {D(-1,.2)}). \]

Let \(f\) be an entire function with the property that for all \(z\in \mathbb {C}\) there is \(n\) such that the \(n\)-th derivative \(f^{(n)}(z)=0\). Show that \(f\) is a polynomial.