91Ó°ÊÓ

Let \(f: \mathbb{C} \rightarrow \bar{C}\) be a meromorphic function, such that all its poles are simple with integer residues. Then there exists a meromorphic function \(h: \mathbb{C} \longrightarrow \mathbb{C}\) with \(f(z)=h^{\prime}(z) / h(z)\).

Prove the following refinement of the MITTAG-LEFFLER theorem: Theorem. (Mittag-Leffler Anschmiegungssatz) Let \(S \subset \mathbb{C}\) be a discrete subset. Then one can construct an analytic function \(f: \mathbb{C} \backslash S \rightarrow \mathbb{C}\) which has at any \(s \in S\) finitely many prescribed coefficients for the LAURENT power series representation in \(s\). Guide for the proof. Consider a suitable product of a partial fraction series with a WEIERSTRASS product.

Let \(D, D^{*} \subset \mathbb{C}\) be conformally equivalent domains. Show that the groups of (conformal) automorphism \(\operatorname{Aut}(D)\) and \(\operatorname{Aut}\left(D^{*}\right)\) are isomorphic.

The Bohr-Mollerup Theorem (H. BOHR, J. MOLLERUP, 1922\()\). Let \(f\) : \(\mathbb{R}_{+}^{*} \rightarrow \mathbb{R}_{+}^{*}\) be a function with the following properties: (a) \(f(x+1)=x f(x)\) for all \(x>0\) and (b) \(\log f\) is convex. Then \(f(x)=f(1) \Gamma(x)\) for all \(x>0\)