Chapter 3

Find the convergence radius for each of the series: (a) \(\sum_{n=0}^{\infty} n ! z^{n}\), (b) \(\sum_{n=0}^{\infty} \frac{z^{n}}{e^{n}}\), (c) \(\sum_{n=1}^{\infty} \frac{n !}{n^{n}} z^{n}\), (d) \(\sum_{n=1}^{\infty} a_{n} z^{n}, \quad a_{n}:=\left\\{\begin{array}{ll}a^{n}, & n \text { pair, } \\ b^{n}, & n \text { impair, }\end{array} \quad b>a>0\right.\)

Let \(\left(a_{n}\right)\) and \(\left(b_{n}\right)\) be two sequences of complex numbers. Two power series are defined by $$ P(z):=\sum a_{n} z^{n} \quad \text { and } \quad Q(z):=\sum b_{n} z^{n} $$ Prove or refute: If the equation \(P(z)=Q(z)\) has infinitely many solutions, then \(P=Q\) and thus \(a_{n}=b_{n}\) for all \(n \in \mathbb{N o}\)

Let \(D \subseteq \mathbb{C}\) be open and \(a \in \mathbb{C} \backslash D\) an isolated singularity of the analytic function \(f: D \rightarrow \mathbb{C}\) Show: (a) The point \(a\) is a removable singularity for \(f\), iff each one of the following conditions is satisfied \((\alpha) f\) is bounded in a punctured neighborhood of \(a\) (RIEMANNian Removability Condition). ( \(\beta\) ) The limit \(\lim _{x \rightarrow a} f(z)\) exists. \((\gamma) \lim _{z \rightarrow a}(z-a) \overrightarrow{f(z)}=0\) (b) The point \(a\) is a simple pole of \(f\), iff \(\lim _{x \rightarrow a}(z-a) f(z)\) exists, and is \(\neq 0\).

Represent the function given by the formula \(f(z)=z /\left(z^{2}+1\right)\) in $$ \mathcal{A}=\\{z \in \mathbb{C} ; \quad 0<|z-\mathrm{i}|<2\\} $$ as a LAURENT series. What kind of singularity has \(f\) in \(a=\mathrm{i}\) ?

For the functions defined by the following expressions compute all residues in all singular points. (a) \(\frac{1-\cos z}{z^{2}}\), (b) \(\frac{z^{3}}{(1+z)^{3}}\) (c) \(\frac{1}{\left(z^{2}+1\right)^{3}}\), (d) \(\frac{1}{\left(z^{2}+1\right)(z-1)^{2}}\), (e) \(\frac{\exp (z)}{(z-1)^{2}}\) (f) \(z \exp \left(\frac{1}{1-z}\right)\), (g) \(\frac{1}{\left(z^{2}+1\right)(z-i)^{3}}\), (h) \(\frac{1}{\exp (z)+1}\) (i) \(\frac{1}{\sin \pi z}\),

Find the number of solutions of each of the following equations in the given domains: $$ \begin{aligned} 2 z^{4}-5 z+2 &=0 & & \text { in }\\{z \in \mathbb{C} ;&|z|>1\\} \\ z^{7}-5 z^{4}+i z^{2}-2 &=0 & & \text { in }\\{z \in \mathbb{C} ; \quad|z|<1\\} \\ z^{5}+\mathrm{i} z^{3}-4 z+i=0 & & \text { in }\\{z \in \mathbb{C} ; \quad 1<|z|<2\\} \end{aligned} $$

Show directly (without using theorem III.1.3): The power series \(\sum_{n=0}^{\infty} c_{n} z^{n}\) and the formally derived power series \(\sum_{n=1}^{\infty} n c_{n} z^{n-1}\) have the same radius of convergence \(r .\) Moreover, for all \(z \in U_{r}(0)\) one has \(P^{\prime}(z)=Q(z)\) Hint: For \(z, b \in U_{r}(0)\) $$ \begin{aligned} &P(z)-P(b)=\sum_{n=0}^{\infty} c_{n}\left(z^{n}-b^{n}\right)=(z-b) \sum_{n=1}^{\infty} c_{n} \varphi_{n}(z) \\ &\text { with } \varphi_{n}(z)=z^{n-1}+z^{n-2} b+\cdots+z b^{n-2}+b^{n-1} \end{aligned} $$

Decide, whether there are analytic functions \(f_{j}: \mathbb{E} \rightarrow \mathbb{C}, 1 \leq j \leq 4\), with (a) \(f_{1}\left(\frac{1}{2 n}\right)=f_{1}\left(\frac{1}{2 n-1}\right)=\frac{1}{n}, \quad n \geq 1\) (b) \(f_{2}\left(\frac{1}{n}\right)=f_{2}\left(-\frac{1}{n}\right)=\frac{1}{n^{2}}, \quad n \geq 1\) (c) \(f_{3}^{(n)}(0)=(n !)^{2}, \quad n \geq 0\) (d) \(f_{4}^{(n)}(0)=\frac{n !}{n^{2}}, \quad n \geq 0\).

Let \(f: U_{r}(a) \rightarrow \mathbb{C}\) be analytic \((a \in C, r>0) .\) Show that the following properties are equivalent: (a) The point \(a\) is a pole of \(f\) of order \(k \in \mathbb{N}\). (b) There exist an open neighborhood \(U_{\rho}(a) \subseteq U_{r}(a)\) and an analytic function \(h: U_{\rho}(a) \rightarrow \mathbb{C}\), such that \(h(a) \neq 0\) and \(f(z)=\frac{h(z)}{(z-a)^{k}}\) for all \(z \in \dot{U}_{\rho}(a)\) (c) There exists an open neighborhood \(U_{\rho}(a) \subseteq U_{r}(a)\) of \(a\), and an analytic function \(g: U_{\rho}(a) \rightarrow \mathbb{C}\) not vanishing in \(\dot{U}_{\rho}(a)\), which has a zero of order \(k\) in \(a\), such that \(f=1 / g\) in \(\dot{U}_{\rho}(a)\) (d) There exist positive constants \(M_{1}\) and \(M_{2}\), such that we have for all \(z\) in a punctured neighborhood of \(a:\) $$ M_{1}|z-a|^{-k} \leq|f(z)| \leq M_{2}|z-a|^{-k} $$

Develop the function given by the formula \(f(z)=\frac{1}{(z-1)(z-2)}\) as a LAURENT series in the annuli $$ \mathcal{A}(a ; r, R):=\\{z \in \mathbb{C} ; \quad r<|z-a|

Let \(D \subset \mathbb{C}\) be a domain, \(\alpha:[0,1] \rightarrow D\) a smooth closed curve, and \(a \notin\) Image \(\alpha .\) Show: The winding number $$ \chi(\alpha ; a)=\frac{1}{2 \pi i} \int_{\alpha} \frac{1}{\zeta-a} d \zeta $$ is always an integer number. Hint: Define for \(t \in[0,1]\). $$ G(t):=\int_{0}^{t} \frac{\alpha^{\prime}(s)}{\alpha(s)-a} d s \quad \text { and } \quad F(t):=(\alpha(t)-a) \exp (-G(t)) $$ then compute \(F^{\prime}(t)\), and finally show \(\alpha(t)-a=(\alpha(0)-a) \exp G(t)\) for all \(t \in[0,1]\)

Give examples for power series with finite radius of convergence \(r \neq 0\), which have respectively one of the following properties: (a) the power series converges on the full boundary of the convergence disk, (b) the power series diverges on the full boundary of the convergence disk, (c) there are at least two convergence points and at least two divergence points on the boundary of the convergence disk.

Let \(r>0\), and \(f: U_{r}(0) \rightarrow \mathbb{C}\) be analytic. For all \(z \in U_{r}(0) \cap \mathbb{R}\) assume \(f(z) \in \mathbb{R}\) Show: The TAYLOR coefficients of \(f\) at the development point \(c=0\) are real, and it holds: \(\overline{f(z)}=f(\bar{z})\).

Let \(\lambda>1\). Show that the functional equation \(\exp (-z)+z=\lambda\) has in the right open half-plane \(\\{z \in \mathbb{C} ; \quad\) Re \(z>0\\}\) exactly one solution, which is real.

A power series with positive radius of convergence \(r<\infty\) converges absolutely either for all points or for no points of the boundary of the convergence domain. Give examples for these cases.

Let \(D \subset \mathbb{C}\) be an open set. Show: For a subset \(M \subset D\) the following properties are equivalent: (a) \(M\) is discrete in \(D\), i.e. no accumulation point of \(M\) lies in \(D\). (b) For each \(p \in M\) there exists an \(e>0\), such that \(U_{e}(p) \cap M=\\{p\\}\), and \(\left.M\right]\) is closed in \(D\) (i.e. there exists a closed set \(A \subseteq \mathbb{C}\) with \(M=A \cap D)\). (c) For each compact subset \(K \subset D\) the intersection \(M \cap K\) is finite. (d) \(M\) is locally finite in \(D\), i.e. each point \(z \in D\) has an \(\varepsilon\)-neighborhood \(U_{c}(z) \subseteq D\), such that \(M \cap U_{c}(z)\) is finite.

Which of the following functions have a removable singularity at \(a=0\) ? (a) \(\frac{\exp (z)}{z^{17}}\), (b) \(\frac{(\exp (z)-1)^{2}}{z^{2}}\) (c) \(\frac{z}{\exp (z)-1}\), (d) \(\frac{\cos (z)-1}{z^{2}}\)

Does the following "identity" contradict the uniqueness of the LAURENT representation $$ \begin{aligned} 0 &=\frac{1}{z-1}+\frac{1}{1-z}=\frac{1}{z} \frac{1}{1-1 / z}+\frac{1}{1-z} \\\ &=\sum_{n=1}^{\infty} \frac{1}{z^{n}}+\sum_{n=0}^{\infty} z^{n}=\sum_{n=-\infty}^{\infty} z^{n} \quad ? \end{aligned} $$

Assume that \(f\) has in \(\infty\) an isolated singularity. We define $$ \begin{aligned} \operatorname{Res}(f ; \infty) &:=-\operatorname{Res}(\tilde{f} ; 0), \quad \text { where we set } \\ \tilde{f}(z):=\frac{1}{z^{2}} \widehat{f}(z)=\frac{1}{z^{2}} f\left(\frac{1}{z}\right) \end{aligned} $$ The factor \(z^{-2}\) is natural, as it will become transparent from the following computation rules, especially exercise \(5 .\) (a) Show: $$ \operatorname{Res}(f ; \infty)=-\frac{1}{2 \pi i} \oint_{\alpha_{R}} f(\zeta) d \zeta $$ where \(\alpha_{R}(t)=R \exp (i t), t \in[0,2 \pi]\), and \(R\) is chosen large enough to insure that \(f\) is analytic in the complement of the closed disk centered in 0 with radius \(R\) (b) The function $$ f(z)= \begin{cases}1 / z, & \text { if } z \neq \infty \\ 0, & \text { if } z=\infty\end{cases} $$ has in \(\infty\) a removable singularity, but \(\operatorname{Res}(f ; \infty)=-1\) (not zero !!). It looks like \infty plays a special role. The sensation of discomfort immediately disappears, if we (re)define the notion of "residue" by using the differential \(f(z) d z\), this being the more structural definition. The notion of residue of \(f\) is related to the differential \(f(z) d z\), the notion of order of \(f\) to the function \(f\) itself.

Show that the sequence $$ \sum_{\nu=1}^{\infty} \frac{(-1)^{\nu}}{z-\nu} $$ converges locally uniformly, but not uniformly, in \(D=\mathbb{C}-\mathbb{N}\).

The functions defined by the following expressions have poles in \(a=0 .\) Find the orders of these poles. $$ \frac{\cos z}{z^{2}}, \frac{z^{7}+1}{z^{7}}, \frac{\exp (z)-1}{z^{4}} $$

Let us consider the recursively defined FIBONACCI sequence \(\left(f_{n}\right)\) with \(f_{0}=\) \(f_{1}=1\) and \(f_{n}:=f_{n-1}+f_{n-2}\) for \(n \geq 2\) Show: (a) The power series \(f(z):=\sum_{n=0}^{\infty} f_{n} z^{n}\) coincides with the rational function $$ z \mapsto \frac{1}{1-z-z^{2}} $$ (b) For all \(n \in \mathbb{N}_{0}\) we have the following formula of BINET for the FIBONACCI numbers $$ f_{n}=\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\right)^{n+1}-\frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^{n+1} $$

Let \(f: \overline{\mathbb{C}} \rightarrow \overline{\mathbb{C}}\) be a rational function. Show: $$ \sum_{p \in \bar{C}} \operatorname{Res}(f ; p)=0 \quad \text { (The exactity relation) } $$

Let \(f\) be analytic in an open set \(D\) containing the closed unit disk \(\overline{\mathbb{E}}=\\{z \in\) \(\mathbb{C} ; \quad|z| \leq 1\\} .\) Assume \(|\hat{f}(z)|<1\) for \(|z|=1\). For any \(n \in \mathbb{N}\) the equation \(f(z)=z^{n}\) has exactly \(n\) solutions in \(\mathbb{E}\). Especially, \(f\) has exactly one fixed point in \(\mathbb{E}\).

Show that the series $$ \sum_{\nu=1}^{\infty} \frac{1}{z^{2}-(2 \nu+1) z+\nu(\nu+1)} $$ converges normally in \(\mathrm{C}-\mathbb{N}_{0}\), and determine its limit.

Let \(\sum_{n=0}^{\infty} a_{n} z^{n}\) be a power series with radius of convergence \(r\). Show: (a) If \(\lim _{n \rightarrow \infty}\left|a_{n}\right| /\left|a_{n+1}\right|\) exists, then \(r=R\). (b) If \(\bar{\rho} \lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|} \in[0, \infty]\) exists, then \(r=1 / \bar{\rho} .\) Here we formally use the conventions \(1 / 0=\infty\) and \(1 / \infty=0 .(r=0\) for \(\bar{\rho}=\infty\), and \(r=\infty\) for \(\bar{\rho}=0 .)\) (c) If we set $$ \rho:=\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}:=\lim _{n \rightarrow \infty}\left(\sup \left\\{\sqrt[n]{\left|a_{n}\right|}, \sqrt[n+1]{\left|a_{n+1}\right|}, \sqrt[n+1]{\left|a_{n+1}\right|}, \ldots\right\\}\right) $$ then following the same conventions as in (b) there holds: \(r=1 / \rho\) (A.-L. CAUCHY, 1821;J. HADAMARD, 1892 )

In which domain \(D \subseteq \mathbb{C}\) the function defined by the following series is (well defined, and) analytic $$ \sum_{n=1}^{\infty} \frac{\sin (n z)}{2^{n}} ? $$ \((\) Answer: \(D=\\{z \in \mathbb{C} ; \quad|\operatorname{Im} z|<\log 2\\} .)\) Is there any domain, in which the series $$ \sum_{n=1}^{\infty} \frac{\sin (n z)}{n^{2}} $$ defines an analytic function?

Let \(f: D \rightarrow \mathbb{C}\) be an analytic function, defined on a domain \(D \subseteq \mathbb{C}, a \in D\), and let \(U_{R}(a)\) be the largest open disk inside \(D .\) Show: (a) If \(f\) is not bounded on \(U_{R}(a)\), then \(R\) is the radius of convergence of the TAYLOR series of \(f\) in \(a .\) (b) Give an example with \(r>R\), even in the case when there is no analytic continuation of \(f\) in a strictly larger domain.

Let \(f, g: \mathbb{C} \rightarrow \mathbb{C}\) be two analytic functions. We assume $$ f(g(z))=0 \text { for all } z \in \mathbb{C} $$ Show: If \(g\) is non-constant, then \(f \equiv 0\).

If \(f\) has in \(a \in \mathbb{C}\) a pole of order 1 , and if \(g\) is analytic in an open neighborhood of \(a\), then $$ \operatorname{Res}(f g ; a)=g(a) \operatorname{Res}(f ; a) $$

Let \(a_{1}, \ldots, a_{l} \in \mathbb{C} \backslash \mathbb{Z} .\) be pairwise different (non-integer) numbers, Let \(f\) be an analytic function in \(\mathbb{C} \backslash\left\\{a_{1}, \ldots, a_{l}\right\\}\), such that \(\left|z^{2} f(z)\right|\) is bounded outside a suitable compact set. We set $$ g(z):=\pi \cot (\pi z) f(z) \quad \text { and } \quad h(z):=\frac{\pi}{\sin \pi z} f(z) $$ Showz $$ \begin{gathered} \lim _{N \rightarrow \infty} \sum_{n=-N}^{N} f(n)=-\sum_{j=1}^{l} \operatorname{Res}\left(g ; a_{j}\right) \\ \lim _{N \rightarrow \infty} \sum_{n=-N}^{N}(-1)^{n} f(n)=-\sum_{j=1}^{l} \operatorname{Res}\left(h ; a_{j}\right) \end{gathered} $$

Let \(f\) be a continuous function on the closed unit disk $$ \overline{\mathrm{E}}:=\\{z \in \mathbb{C} ; \quad|z| \leq 1\\} $$ such that \(f \mid \mathbb{E}\) is analytic. Then $$ \oint_{|\zeta|=1} f(\zeta) d \zeta=0 $$ Hint: Consider, for \(0

Assume, that the power series \(P(z)=\sum_{n=0}^{\infty} a_{n} z^{n}\) has a positive radius of convergence, and that in the convergence disk the equality \(P(z)=P(-z)\), holds. Then \(a_{n}=0\) for all impair \(n\).

Fix \(R>0\), consider the closed disk \(\bar{U}_{R}(0):=\\{z \in \mathbb{C} ;|z| \leq R\\}\) and the continuous functions \(f, g: \bar{U}_{R}(0) \rightarrow \mathbb{C}\) which are analytic on the open disk \(U_{R}(0)\), and have coinciding absolute values on its boundary: $$ |f(z)|=|g(z)| \quad \text { for all }|z|=R $$ Show: If \(f\) and \(g\) have no zeros in \(\bar{U}_{R}(0)\), then there exists a constant \(\lambda \in \mathbb{C}\) with \(|\lambda|=1\) and \(f=\lambda g .\)

Show directly (without using more general propositions), that the function $$ f(z):=\exp \frac{1}{z} $$ takes in any punctured neighborhood \(\dot{U}_{r}(0)\) any value \(w \in \mathbb{C}^{\bullet}\) infinitely many times !

Determine in each case an entire function \(f: \mathbb{C} \rightarrow \mathbb{C}\), which satisfies (a) \(f(0)=1, f^{\prime}(z)=z f(z)\) for all \(z \in \mathbb{C}\), (b) \(f(0)=1, f^{\prime}(z)=z+2 f(z)\) for all \(z \in \mathbb{C}\).

Let \(f, g: \mathbb{E} \rightarrow \mathbb{E}\) be bijective analytic functions, which satisfy \(f(0)=g(0)\) and \(f^{\prime}(0)=g^{\prime}(0)\). Moreover, assume \(f^{\prime}\) and \(g^{\prime}\) have no common zero. Show: \(f(z)=g(z)\) for all \(z \in \mathbb{E}\).

Show: $$ \int_{0}^{\infty} \frac{\sin ^{2} x}{x^{2}} d x=\frac{\pi}{2} $$

Let \(\mathcal{A}\) be the annulus $$ \mathcal{A}=\\{z \in \mathbb{C} ; \quad r<|z|

Let \(f\) be analytic in \(\dot{U}_{r}(0):=U_{r}(0) \backslash\\{0\\}, r>0\) Show: \(\operatorname{Res}\left(f^{\prime} ; 0\right)=0 .\)

Compute the integrals: $$ \int_{0}^{2 \pi} \frac{\cos 3 t}{5-4 \cos t} d t, \quad \int_{0}^{\pi} \frac{1}{(a+\cos t)^{2}} d t, \quad a \in \mathbb{R}, a>1 $$

Determine the maximum of \(|f|\) on \(\bar{E}:=\\{z \in \mathbb{C} ; \quad|z| \leq 1\\}\) for (a) \(f(z)=\exp \left(z^{2}\right)\), (b) \(f(z)=\frac{z+3}{z-3}\), (c) \(f(z)=z^{2}+z-1\), (d) \(f(z)=3-|z|^{2}\). In (d) the maximal modulus is attained in the (interior) point \(a=0\). Is there any contradiction to the maximum principle?

Show: $$ \int_{0}^{\infty} \frac{\sin ^{4} x}{x^{2}} d x=\frac{\pi}{4} $$

Show $$ \int_{0}^{2 \pi} \frac{\sin 3 t}{5-3 \cos t} d t=0, \quad \int_{0}^{2 \pi} \frac{1}{(5-3 \sin t)^{2}} d t=\frac{5 \pi}{32} $$

Determine for the TAYLOR series of tan : \(=\sin / \cos\) at the development point \(a=0\), the radius of convergence, and the first four coefficients.

Let \(u\) be a non-constant harmonic function on a domain \(D \subset \mathbb{R}^{2} .\) Show that \(u(D)\) is an open interval.

Variant of the maximum principle for bounded domains If \(D \subset \mathbb{C}\) is a bounded domain, and \(f: \bar{D} \rightarrow \mathbb{C}\) is a continuous function on the closure of \(D\) which is analytic on \(D\), then \(|f|\) takes its maximum on the boundary of \(D\). Using the example of the strip $$ S=\left\\{z \in \mathbb{C} ; \quad|\operatorname{Im} z|<\frac{\pi}{2}\right\\} $$ and the function \(f(z)=\exp (\exp (z))\) show the necessity of the boundedness of \(D .\)

The zero set of meromorphic function, defined in a domain \(D\), is discrete in \(D .\)

Show (a) $$ \int_{-\infty}^{\infty} \frac{x^{2}}{\left(x^{2}+a^{2}\right)^{2}} d x=\frac{\pi}{2 a}, \quad(a>0) $$ (b) $$ \int_{-\infty}^{\infty} \frac{d x}{\left(x^{2}+4 x+5\right)^{2}}=\frac{\pi}{2} $$ (c) \(\int_{0}^{\infty} \frac{d x}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)}=\frac{\pi}{2 a b(a+b)}, \quad(a, b>0)\).

Determine an entire function \(f: \mathbb{C} \rightarrow \mathbb{C}\) with $$ z^{2} f^{\prime \prime}(z)+z f^{\prime}(z)+z^{2} f(z)=0 \quad \text { for all } z \in \mathbb{C} $$ Result: One solution is the BESSEL function of order 0 , $$ f(z):=\mathcal{J}_{0}(z):=1+\sum_{n=1}^{\infty} \frac{(-1)^{n}}{(2 \cdot 4 \cdot 6 \cdots 2 n)^{2}} z^{2 n} $$