Chapter 2

Which of the following subsets of \(\mathbb{C}\) are domains? (a) \(\left\\{z \in \mathbb{C} ; \quad\left|z^{2}-3\right|<1\right\\}\) (b) \(\left\\{z \in \mathbb{C} ; \quad\left|z^{2}-1\right|<3\right\\}\), (c) \(\left\\{z \in \mathbb{C} ;\left.\quad|| z\right|^{2}-2 \mid<1\right\\}\), (d) \(\left\\{z \in \mathbb{C} ; \quad\left|z^{2}-1\right|<1\right\\}\) (e) \(\\{z \in \mathbb{C} ; \quad z+|z| \neq 0\\}\) (f) \(\\{z \in \mathbb{C} ; 0

$$ \text { Let } \alpha:[0, \pi] \rightarrow \mathbb{C} \text { be defined by } $$ $$ \alpha(t):=\exp (\mathrm{i} t) $$ and \(\beta:[0,2] \rightarrow \mathbb{C}\) by $$ \beta(t)= \begin{cases}1+t(-\mathrm{i}-1) & \text { for } t \in[0,1] \\\ 1-t+\mathrm{i}(t-2) & \text { for } t \in[1,2]\end{cases} $$ Sketch \(\alpha\) and \(\beta\), and calculate $$ \int_{\alpha} \frac{1}{z} d z \quad \text { and } \quad \int_{\beta} \frac{1}{z} d z $$

Let \(z_{0}, \ldots, z_{N} \in \mathbb{C}(N \in \mathbb{N}) .\) Define the segments connecting \(z_{\nu}\) to \(z_{\nu+1},(\nu=\) \(0,1, \ldots, N-1)\), by $$ \alpha_{\nu}:[\nu, \nu+1] \longrightarrow \mathbb{C} \text { with } \alpha_{\nu}(t)=z_{\nu}+(t-\nu)\left(z_{\nu+1}-z_{\nu}\right) $$ Then \(\alpha:=\alpha_{1} \oplus \alpha_{2} \oplus \cdots \oplus \alpha_{N-1}\) defines a curve \(\alpha:[0, N] \rightarrow \mathbb{C} . \alpha\) is a polygonal path, which connects \(z_{0}\) to \(z_{N}\) (through \(\left.z_{1}, z_{2}, \ldots, z_{N-1}\right)\). Show: An open set \(D \subseteq \mathbb{C}\) is connected (and thus a domain) if and only if any two points of \(D\) can be connected by a polygonal path \(\alpha\) inside \(D\) (i.e. Image \(\alpha \subset D)\).

Sketch the following curve \(\alpha\) ("figure eight") \(\alpha(t):=\left\\{\begin{aligned} 1-\exp (\text { it }) & & \text { for } t \in[0,2 \pi] \\\\-1+\exp (-\mathrm{i} t) & & \text { for } t \in[2 \pi, 4 \pi] \end{aligned}\right.\)

Let \(\emptyset \neq D \subseteq \mathbb{C}\) be open. The continuous function $$ f: D \longrightarrow \mathbb{C}, \quad z \longmapsto \bar{z} $$ has no primitive in \(D\).

Compute \(\int_{\alpha} z \exp \left(z^{2}\right) d z\) where (a) \(\alpha\) is the line between the point 0 and the point \(1+\mathrm{i}\), (b) \(\alpha\) is the piece of the parabola with equation \(y=x^{2}\), which lies between the points 0 and \(1+\mathrm{i}\).

$$ \text { For } \alpha:[0,1] \rightarrow \mathbb{C} \text { with } \alpha(t)=\exp (2 \pi \mathrm{i} t) \text { compute } $$ $$ \int_{\alpha} 1 /|z| d z, \quad \int_{\alpha} 1 /\left(|z|^{2}\right) d z, \quad \text { and show } \quad\left|\int_{\alpha} 1 /(4+3 z) d z\right| \leq 2 \pi $$

Let $$ D:=\\{z \in \mathbb{C} ; \quad 1<|z|<3\\} $$ and \(\alpha:[0,1] \rightarrow D\) be defined by \(\alpha(t)=2 \exp (2 \pi \mathrm{i} t)\). Calculate $$ \int_{\alpha} \frac{1}{z} d z $$

Show: If \(f: \mathbb{C} \rightarrow \mathbb{C}\) is analytic and if there is a real number \(M\) such that for all \(z \in \mathbb{C}\) $$ \operatorname{Re} f(z) \leq M $$ then \(f\) is constant.

For \(a, b \in \mathbb{R}_{+}^{\bullet}\), let \(\alpha, \beta:[0,1] \rightarrow \mathbb{C}\) be defined by $$ \begin{aligned} &\alpha(t):=a \cos 2 \pi t+\mathrm{i} a \sin 2 \pi t \\ &\beta(t):=a \cos 2 \pi t+\mathrm{i} b \sin 2 \pi t \end{aligned} $$ (a) Show: $$ \int_{\alpha} \frac{1}{z} d z=\int_{\beta} \frac{1}{z} d z $$ (b) Show using (a) $$ \int_{0}^{2 \pi} \frac{1}{a^{2} \cos ^{2} t+b^{2} \sin ^{2} t} d t=\frac{2 \pi}{a b} $$

Let \(\omega\) and \(\omega^{\prime}\) be complex numbers which are linearly independent over \(\mathbb{R}\). Show: If \(f: \mathbb{C} \rightarrow \mathbb{C}\) is analytic and $$ f(z+\omega)=f(z)=f\left(z+\omega^{\prime}\right) \text { for all } z \in \mathbb{C} $$ then \(f\) is constant (J. LIOUVILLE, 1847).

Let \(R>0\) be a positive number. We consider the curve $$ \beta(t)=R \exp (\mathrm{i} t), \quad 0 \leq t \leq \frac{\pi}{4} $$ Show: $$ \left|\int_{\beta} \exp \left(\mathrm{i} z^{2}\right) d z\right| \leq \frac{\pi\left(1-\exp \left(-R^{2}\right)\right)}{4 R}<\frac{\pi}{4 R} $$

Let \(D_{1}, D_{2} \subseteq \mathbb{C}\) be star-shaped domains with the common star center \(z_{*}\). Then \(D_{1} \cup D_{2}\) and \(D_{1} \cap D_{2}\) are also domains which are star-shaped with respect to \(z_{*}\)

Let \(\alpha:[a, b] \rightarrow \mathbb{C}\) be continuously differentiable and assume \(f:\) Image \(\alpha \rightarrow \mathbb{C}\) is continuous. Show: For any \(\varepsilon>0\), there exists a \(\delta>0\) with the following property: If \(\left\\{a_{0}, \ldots, a_{N}\right\\}\) and \(\left\\{c_{1}, \ldots, c_{N}\right\\}\) are finite subsets of \([a, b]\) with $$ a=a_{0} \leq c_{1} \leq a_{1} \leq c_{2} \leq a_{2} \leq \cdots \leq a_{N-1} \leq c_{N} \leq a_{N}=b $$ and $$ a_{\nu}-a_{\nu-1}<\delta \text { for } \nu=1, \ldots, N $$ then $$ \left|\int_{\alpha} f(z) d z-\sum_{\nu=1}^{N} f\left(\alpha\left(c_{\nu}\right)\right) \cdot\left(\alpha\left(a_{\nu}\right)-\alpha\left(a_{\nu-1}\right)\right)\right|<\varepsilon $$ (Approximation of the line integral by a RIEMANNian sum.)

Which of the following domains are star-shaped? (a) \(\\{z \in \mathbb{C} ; \quad|z|<1\) and \(|z+1|>\sqrt{2}\\}\), (b) \(\\{z \in \mathbb{C} ; \quad|z|<1\) and \(|z-2|>\sqrt{5}\\}\), (c) \(\\{z \in \mathbb{C} ; \quad|z|<2\) and \(|z+\mathrm{i}|>2\\}\). In each case determine the set of all star centers.

Show that every rational function \(R\) (i.e. \(R(z)=P(z) / Q(z), P, Q\) polynomials, \(Q \neq 0)\) can be written as the sum of a polynomial and a finite linear combination, with complex coefficients, of "simple functions" of the form $$ z \mapsto \frac{1}{(z-s)^{n}}, \quad n \in \mathbb{N}, s \in \mathbb{C} $$ the so-called "partial fractions" (Partial fraction decomposition theorem), see also Chapter III, Appendix A to sections III.4 and III.5, Proposition A.7). Deduce: If the coefficients of \(P\) and \(Q\) are real, then \(f\) has "a real partial fraction decomposition" (by putting together pairs of complex conjugate zeros, or rather by putting together the corresponding partial fractions (see also Exercise 10 in I.1).

By splitting \(f\) into its real and imaginary parts, represent the complex line integral \(\int_{\alpha} f(z) d z\) in terms of real integrals. Result: If \(f=u+\mathrm{i} v, \alpha(t)=x(t)+\mathrm{i} y(t), t \in[a, b]\), then $$ \begin{aligned} \int_{\alpha} f(z) d z=\int_{\alpha}(u d x-v d y)+\mathrm{i} \int_{\alpha}(v d x+u d y) \\ =& \int_{a}^{b}\left[u(x(t), y(t)) x^{\prime}(t)-v(x(t), y(t)) y^{\prime}(t)\right] d t \\ &+\mathrm{i} \int_{a}^{b}\left[v(x(t), y(t)) x^{\prime}(t)+u(x(t), y(t)) y^{\prime}(t)\right] d t \end{aligned} $$

A somewhat more direct proof of the generalized CAUCHY integral formula (Theorem II.3.4) is obtained with the following Lemma: Let \(\alpha:[a, b] \rightarrow \mathbb{C}\) be a piecewise smooth curve and let \(\varphi\) : Image \(\alpha \rightarrow \mathbb{C}\) be continuous. For \(z \in D:=\mathbb{C} \backslash\) Image \(\alpha\) and \(m \in \mathbb{N}\) let $$ F_{m}(z):=\frac{1}{2 \pi \mathrm{i}} \int_{\alpha} \frac{\varphi(\zeta)}{(\zeta-z)^{m}} d \zeta $$ Then \(F_{m}\) is analytic in \(D\) and for all \(z \in D\) $$ F_{m}^{\prime}(z)=m F_{m+1}(z) $$ Prove this by direct estimate (without using the LEIBNIZ rule).

Let \(0

Let \(D \subseteq \mathbb{C}\) be open, and \(L \subset \mathbb{C}\) a line. If \(f: D \rightarrow \mathbb{C}\) is a continuous function, which is analytic at all points \(z \in D, z \notin L\), then \(f\) is analytic on all \(D\).

Lemma on polynomial growth Let \(P\) be a nonconstant polynomial of degree \(n\) : $$ P(z)=a_{n} z^{n}+\cdots+a_{0}, \quad a_{\nu} \in \mathbb{C}, 0 \leq \nu \leq n, n \geq 1, a_{n} \neq 0 $$ Then, for all \(z \in \mathbb{C}\) with $$ \begin{array}{|c} |z| \geq \varrho:=\max \left\\{1, \frac{2}{\left|a_{n}\right|} \sum_{\nu=0}^{n-1}\left|a_{\nu}\right|\right\\} \text { holds: } \\ \qquad \frac{1}{2}\left|a_{n}\right||z|^{n} \leq|P(z)| \leq \frac{3}{2}\left|a_{n}\right||z|^{n} . \end{array} $$ Corollary: Any root of the polynomial \(P\) lies in the open ball with radius \(\rho\) centered in origin.

Let \(f\) be a continuous function on the compact interval \([a, b]\). Show: The function defined by $$ F(z)=\int_{a}^{b} \exp (-z t) f(t) d t $$ is analytic on all \(\mathbb{C}\), and $$ F^{\prime}(z)=-\int_{a}^{b} \exp (-z t) t f(t) d t $$

Let \(D \subseteq \mathbb{C}\) be a domain and $$ f: D \longrightarrow \mathbb{C} $$ be an analytic function. Show: The function $$ \varphi: D \times D \longrightarrow \mathbb{C} $$ with $$ \varphi(\zeta, z):= \begin{cases}\frac{f(\zeta)-f(z)}{\zeta-z} & \text { if } \zeta \neq z \\ f^{\prime}(\zeta) & \text { if } \zeta=z\end{cases} $$ is a continuous function of two variables. For each given \(z \in D\) the function $$ \zeta \longmapsto \varphi(\zeta, z) $$ is analytic in \(D\).

Let \(D \subseteq \mathbb{C}\) be a domain with the property $$ z \in D \Longrightarrow-z \in D $$ and \(f: D \rightarrow \mathbb{C}\) a continuous and even function \((f(z)=f(-z))\). Moreover, for some \(r>0\) let the closed disk \(\bar{U}_{r}(0)\) be contained in \(D\). Then $$ \int_{\alpha_{r}} f=0 \text { for } \alpha_{r}(t):=r \exp (2 \pi \mathrm{i} t), 0 \leq t \leq 1 $$

Determine all pairs \((f, g)\) of entire functions with the property $$ f^{2}+g^{2}=1 $$ Result: \(f=\cos \circ h\) and \(g=\sin \circ h\), where \(h\) is an arbitrary entire function.

Continuous branches of the logarithm Let \(D \subset \mathbb{C}^{\bullet}\) be a domain which does not contain the origin. A continuous function \(l: D \rightarrow \mathbb{C}\) with \(\exp l(z)=z\) for all \(z \in D\) is called a continuous branch of the logarithm. Show: (a) Any other continuous branch of the logarithm \(\tilde{l}\) has the form \(\tilde{l}=\) \(l+2 \pi \mathrm{i} k, k \in \mathbb{Z}\) (b) Any continuous branch of the logarithm \(l\) is in fact analytic, and \(l^{\prime}(z)=1 / z\) (c) On \(D\) there exists a unique continuous branch of the logarithm only if the function \(1 / z\) has a primitive on \(D\). (d) Construct two domains \(D_{1}\) and \(D_{2}\) and continuous branches \(l_{1}: D_{1} \rightarrow\) \(\mathbb{C}, l_{2}: D_{2} \rightarrow \mathbb{C}\) of the logarithm, such that their difference is not constant on \(D_{1} \cap D_{2}\)