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) \(\sum_{n=1}^{\infty} \frac{n !}{n^{n}} z^{n},(\mathrm{~d}) \quad \sum_{n=1}^{\infty} a_{n} z^{n}, \quad a_{n}:=\left\\{\begin{array}{ll}a^{n}, & n \text { even, } \\ b^{n}, & n \text { odd },\end{array} \quad b>a>0 .\right.\)

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 \(n \in \mathbb{N}_{0}\).

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

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

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) \(\frac{1}{\left(z^{2}+1\right)(z-\mathrm{i})^{3}}\) (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: $$ \begin{aligned} 2 z^{4}-5 z+2 &=0 & & \text { in }\\{z \in \mathbb{C} ;&|z|>1\\}, \\ z^{7}-5 z^{4}+\mathrm{i} z^{2}-2 &=0 & & \text { in }\\{z \in \mathbb{C} ;&|z|<1\\}, \\ z^{5}+\mathrm{i} z^{3}-4 z+\mathrm{i} &=0 & & \text { in }\\{z \in \mathbb{C} ;&1<|z|<2\\}. \end{aligned} $$

The power series \(\sum_{n=0}^{\infty} c_{n} z^{n}\) and the termwise differentiated 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 \(z, b \in U_{r}(0)\) $$ \begin{gathered} P(z)-P(b)=\sum_{n=0}^{\infty} c_{n}\left(z^{n}-b^{n}\right)=(z-b) \sum_{n=1}^{\infty} c_{n} \varphi_{n}(z) \\ \text { with } \varphi_{n}(z)=z^{n-1}+z^{n-2} b+\cdots+z b^{n-2}+b^{n-1}. \end{gathered} $$

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 \(f: \dot{U}_{r}(a) \rightarrow \mathbb{C}\) be analytic \((a \in \mathbb{C}, r>0) .\) 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 \(U_{\rho}(a) \subset U_{r}(a)\) 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 \(U_{\rho}(a) \subset U_{r}(a)\) 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 \(a\) : $$ M_{1}|z-a|^{-k} \leq|f(z)| \leq M_{2}|z-a|^{-k}. $$

Expand the function given by the formula \(f(z)=\frac{1}{(z-1)(z-2)}\) into 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 $$ \chi(\alpha ; a)=\frac{1}{2 \pi \mathrm{i}} \int_{\alpha} \frac{1}{\zeta-a} d \zeta $$ is always an integer. 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 \(t \in[0,1]\)

The polynomial \(P(z)=z^{4}-5 z+1\) has (a) one root \(a\) with \(|a|<\frac{1}{4}\). (b) and the other three roots in the annulus \(\frac{3}{2}<|z|<\frac{15}{8}\).

Let \(D \subset \mathbb{C}\) be open and let \(\left(f_{n}\right)\) be a sequence of analytic functions \(f_{n}: D \rightarrow \mathbb{C}\) with the property: For every closed disk \(K \subset D\) there is a real number \(M(K)\) such that \(\left|f_{n}(z)\right| \leq M(K)\) for all \(z \in K\) and all \(n \in \mathbb{N}\). Show: The sequence \(\left(f_{n}^{\prime}\right)\) has the analogous property.

Give examples of 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 \(f\) with respect to the center \(c=0\) are real, and one has: \(\overline{f(z)}=f(\bar{z})\).

Expand the function given by the formula \(f(z)=\frac{1}{z(z-1)(z-2)}\) into a LAURENT series in each of the annuli \(\mathcal{A}(0 ; 0,1), \mathcal{A}(0 ; 1,2)\) and \(\mathcal{A}(0 ; 2, \infty)\)

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

Show that the series $$ \sum_{\nu=1}^{\infty} \frac{z^{2 \nu}}{1-z^{\nu}} $$ converges normally in the unit disk \(\mathbb{E}=\\{z \in \mathbb{C} ; \quad|z|<1\\}\).

A power series with positive radius of convergence \(r<\infty\) converges absolutely either for all points or for no point on 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 \(\varepsilon>0\), such that \(U_{\varepsilon}(p) \cap M=\\{p\\}\), and \(M\) is closed in \(D\) (i.e. there exists a closed set \(A \subset \mathbb{C}\) 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 \(U_{\varepsilon}(z) \subset D\), such that \(M \cap U_{\varepsilon}(z)\) 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 to the uniqueness of the LAURENT expansion $$ \begin{aligned} 0 &=\frac{1}{z-1}+\frac{1}{1-z}=\frac{1}{z} \cdot \frac{1}{1-1 / z}+\frac{1}{1-z} \\ &=\sum_{n=1}^{\infty} \frac{1}{z^{n}}+\sum_{n=0}^{\infty} z^{n}=\sum_{n=-\infty}^{\infty} z^{n} ? \end{aligned} $$

Assume that \(f\) has at \(\infty\) an isolated singularity. We define $$ \begin{aligned} \operatorname{Res}(f ; \infty) &:=-\operatorname{Res}(\widetilde{f} ; 0), \quad \text { where we set } \\ \widetilde{f}(z) &:=\frac{1}{z^{2}} \widehat{f}(z)=\frac{1}{z^{2}} f\left(\frac{1}{z}\right) \end{aligned} $$ 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 big enough to ensure that \(f\) is analytic in the complement of the closed disk centered at 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 at \(\infty\) a removable singularity, but \(\operatorname{Res}(f ; \infty)=-1\) (not zero!). It seems that \(\infty\) plays a special role. The special role disappears if one defines the notion of the "residue" for the differential \(f(z) d z\) instead of the function \(f\). The notion of residue of \(f\) is related to the differential \(f(z) d z\), and the notion of order of \(f\) to the function \(f\) itself.

For \(n \in \mathbb{N}_{0}\) define $$ e_{n}(z)=\sum_{\nu=0}^{n} \frac{z^{\nu}}{\nu !} $$ For a given \(R>0\) there exists an \(n_{0}\), such that for all \(n \geq n_{0}\) the function \(e_{n}\) has no zero in \(U_{R}(0)\).

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

For the following functions \(f\) defined in a neighborhood of the point \(a \in \mathbb{C}\) determine the TAYLOR series at \(a\) and the convergence radius: (a) \(f(z)=\exp (z), \quad a=1\), (b) \(f(z)=\frac{1}{z}\), (c) \(f(z)=\frac{1}{z^{2}-5 z+6}, \quad a=0\) (d) \(f(z)=\frac{1}{(z-1)(z-2)}, \quad a=0\)

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

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: $$ \sum_{p \in \overline{\mathbb{C}}} \operatorname{Res}(f ; p)=0. $$

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 \(|f(z)|<1\) 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 \(\mathbb{E}\).

Let \(\sum_{n=0}^{\infty} a_{n} z^{n}\) be a power series with radius of convergence \(r\) Show: (a) If \(R:=\lim _{n \rightarrow \infty}\left|a_{n}\right| /\left|a_{n+1}\right|\) exists, then \(r=R\). (b) If \(\tilde{\rho} \lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|} \in[0, \infty]\) exists, then \(r=1 / \tilde{\rho} .\) Here we formally use the conventions \(1 / 0=\infty\) and \(1 / \infty=0 .(r=0\) for \(\tilde{\rho}=\infty\), and \(r=\infty\) for \(\tilde{\rho}=0 .)\) (c) If we set $$ \rho:=\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}:=\lim _{n \rightarrow \infty}\left(\sup \left\\{\sqrt[n]{\left|a_{n}\right|}, \sqrt[n+1]{\left|a_{n+1}\right|}, \sqrt[n+2]{\left|a_{n+1}\right|}, \ldots\right\\}\right), $$ then - following the same conventions as in (b)- one hasa $$ r=1 / \rho $$

If the analytic function \(f: D \rightarrow \mathbb{C}\) on the domain \(D\) is not constant zero, then the zeros of \(f\) are at most countable.

Compute the following integrals: (a) \(I:=\oint_{|\zeta|=2} \frac{1}{(\zeta-3)\left(\zeta^{13}-1\right)} d \zeta\) (b) \(I:=\oint_{|\zeta|=10} \frac{\zeta^{3}}{\zeta^{4}-1} d \zeta\)

In which domain \(D \subset \mathbb{C}\) defines the following series $$ \sum_{n=1}^{\infty} \frac{\sin (n z)}{2^{n}} $$ an analytic function? \((\) 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 \(f, g: \mathbb{C} \rightarrow \mathbb{C}\) be two analytic functions. We assume $$ f(g(z))=0 \text { for all } z \in \mathbb{C} $$ Show: If \(g\) is non-constant, then \(f \equiv 0\)

For \(\nu \in \mathbb{Z}\), and \(w \in \mathbb{C}\) let \(\mathcal{J}_{\nu}(w)\) be the coefficient of \(z^{\nu}\) in the LAURENT power series expansion of $$ \begin{aligned} &f: \mathbb{C}^{\bullet} \longrightarrow \mathbb{C}, \quad f(z):=\exp \left(\frac{1}{2}\left(z-\frac{1}{z}\right) w\right), \quad \text { i.e. } \\ &f(z)=\sum_{\nu=-\infty}^{\infty} \mathcal{J}_{\nu}(w) z^{\nu} \end{aligned} $$ Show: (a) \(\mathcal{J}_{\nu}(-w)=\mathcal{J}_{-\nu}(w)=(-1)^{\nu} \mathcal{J}_{\nu}(w)\) for all \(\nu \in \mathbb{Z}\) and all \(w \in \mathbb{C}\). (b) \(\mathcal{J}_{\nu}(w)=\frac{1}{2 \pi} \int_{0}^{2 \pi} \cos (\nu t-w \sin t) d t=\frac{1}{\pi} \int_{0}^{\pi} \cos (\nu t-w \sin t) d t\) (c) The functions \(\mathcal{J}_{\nu}(w)\) are analytic at \(\mathbb{C}\). Their TAYLOR expansions around the origin have for \(\nu \geq 0\) the form $$ \mathcal{J}_{\nu}(w)=\sum_{\mu=0}^{\infty} \frac{(-1)^{\mu}\left(\frac{1}{2} w\right)^{2 \mu+\nu}}{\mu !(\nu+\mu) !}. $$

If \(f\) has at \(a \in \mathbb{C}\) 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 \(a_{1}, \ldots, a_{l} \in \mathbb{C}\) be pairwise different non-integral 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 $$ g(z):=\pi \cot (\pi z) f(z) \quad \text { and } \quad h(z):=\frac{\pi}{\sin \pi z} f(z) $$ Show: $$ \begin{gathered} \lim _{N \rightarrow \infty} \sum_{n=-N}^{N} f(n)=-\sum_{j=1}^{l} \operatorname{Res}\left(g ; a_{j}\right) \\ \lim _{N \rightarrow \infty} \sum_{n=-N}^{N}(-1)^{n} f(n)=-\sum_{j=1}^{l} \operatorname{Res}\left(h ; a_{j}\right) . \end{gathered} $$

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 odd \(n\).

Fix \(R>0\), and consider the closed disk \(\bar{U}_{R}(0):=\\{z \in \mathbb{C} ;|z| \leq R\\}\) and continuous functions \(f, g: \bar{U}_{R}(0) \rightarrow \mathbb{C}\) which are analytic on the open disk \(U_{R}(0)\), and such their absolute values coincide 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 \(f=\lambda g\)

Let us consider the function $$ f(z):=\frac{(z-1)^{2}(z+3)}{1-\sin (\pi z / 2)} $$ Find all singularities of \(f\) and determine for each one its type.

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 often!

The residue of an analytic function \(f\) at a singularity \(a \in \mathbb{C}\) is the uniquely determined complex number \(c\), such that the function $$ f(z)-\frac{c}{z-a} $$ admits a primitive in a punctured neighborhood of the point \(a\).

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 that \(f^{\prime}\) 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 radius of convergence of the TAYLOR series of \(1 / \mathrm{cos}\) with center \(a=0\). The numbers \(E_{2 n}\), defined by the formula $$ \frac{1}{\cos z}=\sum_{n=0}^{\infty} \frac{E_{2 n}}{(2 n) !} z^{2 n} $$ are called EULER numbers. Show that all \(E_{2 n}\) are natural numbers, and compute \(E_{2 \nu}\) for \(0 \leq \nu \leq 5\). Result: \(E_{0}=1=E_{2}, E_{4}=5, E_{6}=61, E_{8}=1385, E_{10}=50521, E_{12}=\) \(2702765 .\)