6220-2024-08

State and prove Riemann’s theorem on removable singularities.

Let \(f:B(0,1)\to \mathbb {C}\) be holomorphic. Describe all such \(f\) that satisfy \[ f\left (\frac {1}{n}\right )=\frac {n^2-2n-1}{n^2} \] for every \(n\in \{2,3,\ldots \}\).

Let \(f\) be an entire function with \(f(0)=0\). Define

Show that \(M_f(r)\) is non-decreasing. What can you say about \(f\) if it is not strictly increasing?

Let \(f:\mathbb {C}\setminus \{0,1,2\}\to \mathbb {C}\) be a holomorphic function. Show that if \(f\) omits at least 4 values then it is constant.

Show that if \(f:B(0,1)\to \mathbb {C}\) is holomorphic and \(\operatorname {Re}(f'(z))>0\) for all \(z\in B(0,1)\) then \(f\) is injective.

State and prove Rouché’s theorem.