6220-2025-08

What is the group of all biholomorphisms of \(\mathbb {D}\setminus \{0\}\)?

Show that there is no proper holomorphic map \(f:\mathbb {D}\to \mathbb {C}\). Recall that “proper” means that \(|z_n|\to 1\) implies \(|f(z_n)|\to \infty \) for any sequence \(z_n\in \mathbb {D}\).

Using residue calculus evaluate

Consider the upper half-plane

equipped with the hyperbolic metric. Show that the hyperbolic distance from \(n+in\) to the positive imaginary axis does not depend on \(n\in \{1,2,3,\ldots \}\).

Let \(f_1,f_2,\ldots :\mathbb {C}\to \mathbb {C}\) be a sequence of non-vanishing entire functions that converge uniformly on compact sets to an entire function \(f_\infty \). Show that \(f_\infty \) is either identically 0 or non-vanishing.

Let \(A_R=\{z\in \mathbb {C}\mid 1<|z|<R\}\) for \(R>1\). If \(A_R\) and \(A_{R'}\) are biholomorphic, show that \(R=R'\).