6220-2023-08

What is the group of all biholomorphisms of \(\mathbb {D}\) that fix 0? Justify your answer.

Compute the following integral using methods of complex analysis.

Let \(P(z)\) be a nonzero polynomial. Show that the equation

has infinitely many solutions \(z\in \mathbb {C}\).

Recall that the hyperbolic metric on \(\mathbb {H}\) is given by

Compute the hyperbolic distance between \(i\) and \(1+i\).

Let \(f\) be an entire function and suppose that for every positive integer \(n>1\) and every \(z\in \mathbb {C}\) with \(|z|<n\) we have

Does this imply that \(f\) is a polynomial?

1. (3 points) State Runge’s theorem.
2. (7 points) Does there exist a sequence of entire functions \(f_n\) that converges to 0 pointwise, but not uniformly on compact sets?