6220-2023-05

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

Let \(f:\mathbb {D}\to \mathbb {D}\) be holomorphic and \(f(0)=a\). Show that \(f\) has no zeros in \(\{z\in \mathbb {C}\mid |z|<|a|\}\).

Prove the following version of Morera’s theorem. If \(f:\mathbb {D}\to \mathbb {C}\) is a continuous function such that for every rectangle \(R\subset \mathbb {D}\)

then \(f\) is holomorphic. Note: It does not follow from the assumption in any obvious way that integrals over triangles are 0.

Find all singularities of the function \(\frac {1}{e^z+1}\) in \(\mathbb {C}\). If they are poles, compute the residue.

Let \(f\) be holomorphic on \(\mathbb {C}\setminus \{0,2\}\) with simple poles at 0 and 2. Suppose that \(\int _{|z|=1} f(z)\,dz=2\pi i\) and \(\int _{|z|=3} f(z)\,dz=0\). Also suppose that \(f\) is bounded on \(|z| > 3\). List all such functions (with a proof that the list is complete).

Suppose a sequence of holomorphic functions \(f_n:\mathbb {D}\to \mathbb {C}\) converges pointwise to a function \(f:\mathbb {D}\to \mathbb {C}\), and that all \(f_n\) are uniformly bounded. Show that convergence is uniform on every disk

for \(r<1\).