Chapter 4

Which of the following products are absolutely convergent? Find the corresponding values, when they exist. (a) \(\prod_{\nu=2}^{\infty}\left(1-\frac{1}{\nu}\right)\), (b) \(\prod_{\nu=2}^{\infty}\left(1-\frac{1}{\nu^{2}}\right)\), (c) \(\prod_{i=2}^{\infty}\left(1-\frac{2}{\nu(\nu+1)}\right)\), (d) \(\prod_{\nu=2}^{\infty}\left(1-\frac{2}{\nu^{3}+1}\right)\).

Let \(D=\\{z \in \mathbb{C} ; \quad|z|>1\\}\). Can there exist any conformal map from \(D\) onto the punctured plane \(\mathbb{C}^{\bullet} ?\)

The product \(\prod_{\nu=0}^{\infty}\left(1+z^{2^{2}}\right)\) is absolutely convergent, iff \(|z|<1 .\) If this is the case, then $$ \prod_{\nu=0}^{\infty}\left(1+z^{2^{v}}\right)=\frac{1}{1-z} $$

Show that the sequence \(\left(\gamma_{n}\right)\) defined by the expression $$ \gamma_{n}:=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\log n $$ is (strictly) decreasing, and bounded from below by 0 . Hence the following limit exists: $$ \gamma:=\lim _{n \rightarrow \infty} \gamma_{n} \approx 0,577215664901532860606512090082402431042159 \ldots $$

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 \(f(z)=h^{\prime}(z) / h(z)\).

Find a meromorphic function in \(\mathbb{C}\) meromorphe function \(f\), which has simple poles in $$ S=\\{\sqrt{n} ; \quad n \in \mathbb{N}\\} $$ with corresponding residues \(\operatorname{Res}(f ; \sqrt{n})=\sqrt{n}\), and is analytic in \(\mathbb{C} \backslash S\).

Show for \(z \in \mathbb{C} \backslash S, S:=\\{0,-1,-2,-3, \ldots\\}\) $$ \lim _{n \rightarrow \infty} \frac{\Gamma(z+n)}{n^{x} \Gamma(n)}=1 $$

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.

Determine the image of $$ D=\\{z \in \mathbb{C} ; \quad|\operatorname{Re} z||\operatorname{Im} z|>1,0<\operatorname{Re} z, \operatorname{Im} z\\} $$ by the map \(\varphi(z)=z^{2}\).

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.

Let \(D \subset \mathrm{C}\) be an elementary domain, and let \(f: D \rightarrow \mathbb{E}\) be a conformal map. If \(\left(z_{n}\right)\) is a sequence in \(D\) with \(\lim _{n \rightarrow \infty} z_{n}=r \in \partial D\), then the sequence \(\left(\left|f\left(z_{n}\right)\right|\right)\) converges to 1. Give an example of a sequence \(\left(z_{n}\right)\) converging to a boundary point of \(D\), such that the image sequence \(\left(f\left(z_{n}\right)\right)\) by a conformal map \(f: D \rightarrow \mathbb{E}\) does not converge to a boundary point of \(\mathbb{E}\).

A characterization of \(\Gamma\) using the Doubling Formula. Let \(f: \mathbb{C} \rightarrow \mathbb{C}\) be a meromorphic function, and assume \(f(x)>0\) for all \(x>0\). Also assume $$ f(z+1)=z f(z) \quad \text { and } \quad \sqrt{\pi} f(2 z)=2^{2 z-1} f(z) f\left(z+\frac{1}{2}\right) $$ Then \(f(z)=\Gamma(z)\) for all \(z \in \mathbb{C}\). For the proof, use the following auxiliary result If \(g: \mathbb{C} \rightarrow \mathbb{C}\) is an analytic function which satisfies \(g(z+1)=g(z), g(2 z)=\) \(g(z) g\left(z+\frac{1}{2}\right)\) for all \(z \in \mathbb{C}\), and \(g(x)>0\) for all \(x>0\), then \(g(z)=a e^{b z}\) with suitable constants \(a\) and \(b .\)

The (most) general shape of a conformal map \(f: \mathbb{H} \rightarrow \mathbb{E}\) is $$ z \longmapsto e^{5 \varphi} \frac{z-\lambda}{z-\bar{\lambda}} \quad \text { with } \lambda \in \mathbb{H}, \varphi \in \mathbb{R} $$ In the special case \(\varphi=0, \lambda=\mathrm{i}\), we obtain the so-called CAYLEY map.

The EuLER beta function. Let \(D \subset \mathbb{C}\) be the half-plane \(\operatorname{Re} z>0\). For \(z, w \in D\) let $$ B(z, w):=\int_{0}^{1} t^{z-1}(1-t)^{w-1} d t $$ \(B\) is called EuLER's beta function. (Following A.M. LEGENDRE, 1811 , it is EuLER's integral of the first kind.) Showz (a) \(B\) is continuous (as a function of the total argument \((z, w) \ni D \times D\) to \(\mathbb{C}\) !). (b) For any fixed \(w \in D\) the map \(D \rightarrow \mathbb{C}, z \mapsto B(z, w)\), is analytic. For any fixed \(z \in D\) the map \(D \rightarrow \mathbb{C}, w \mapsto B(z, w)\), is analytic. (c) For all \(z, w \in D\) $$ B(z+1, w)=\frac{z}{z+w} \cdot B(z, w), \quad B(1, w)=\frac{1}{w} $$

The Gauss \(\psi\)-function is defined by \(\psi(z):=\Gamma^{\prime}(z) / \Gamma(z)\). Show: (a) \(\psi\) is meromorphic in \(\mathbb{C}\) with simple poles in \(S:=\left\\{-n ; \quad n \in \mathbb{N}_{0}\right\\}\) and \(\operatorname{Res}(\psi ;-n)=-1\) (b) \(\psi(1)=-\gamma .(\gamma\) is the EuLER-MASCHERONI constant). (c) \(\psi(z+1)-\psi(z)=\frac{1}{z}\) (d) \(\psi(1-z)-\psi(z)=\pi \cot \pi z\) (e) \(\psi(z)=-\gamma-\frac{1}{z}-\sum_{\nu=1}^{\infty}\left(\frac{1}{z+\nu}-\frac{1}{\nu}\right)\) (f) \(\psi^{\prime}(z)=\sum_{\nu=0}^{\infty} \frac{1}{(z+\nu)^{2}}\), where the series in the right member normally converges in \(\mathbb{C}\). (g) For any positive \(x\) $$ (\log \Gamma)^{\prime \prime}(x)=\sum_{\nu=0}^{\infty} \frac{1}{(x+\nu)^{2}}>0 $$ the real \(\Gamma\)-function is thus logarithmically convex.

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 \(x>0\)

For \(\alpha \in \mathbb{C}\), and \(n \in \mathbb{N}\) let $$ \left(\begin{array}{c} \alpha \\ n \end{array}\right):=\frac{\alpha(\alpha-1) \cdots(\alpha-n-1)}{n !}, \quad\left(\begin{array}{l} \alpha \\ 0 \end{array}\right):=1 $$ Show that for all \(\alpha \in \mathbb{C} \backslash \mathbb{N}_{0}\) $$ \left(\begin{array}{l} \alpha \\ n \end{array}\right)=\frac{(-1)^{n} \Gamma(n-\alpha)}{\Gamma(-\alpha) \Gamma(n+1)} \sim \frac{(-1)^{n}}{\Gamma(-\alpha)} n^{-\alpha-1} \quad \text { for } n \rightarrow \infty $$ i.e. the quotient of the expressions separated by \(\sim\) converges to 1 for \(n \rightarrow \infty\).