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) \(\left\\{z \in C ; \quad\left|z^{2}-1\right|<3\right\\}\) (c) \(\left\\{z \in \mathbb{C} ;\left.\quad|| z\right|^{2}-2 \mid<1\right\\}\) (d) \(\left\\{z \in C ; \quad\left|z^{2}-1\right|<1\right\\}\) (e) \(\\{z \in \mathbb{C} ; \quad z+|z| \neq 0\\}\) (f) \(\\{z \in \mathbb{C} ; 0

Let \(\alpha:[0, \pi] \rightarrow \mathbb{C}\) 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 line segments connecting \(z_{\nu}\) with \(z_{2+1}\) \((\nu=0,1, \ldots, N-1) \mathrm{by}\) $$ \alpha_{\nu}:[\nu, \nu+1] \longrightarrow \mathrm{C} \text { with } \alpha_{\nu}(t)=z_{\nu}+(t-\nu)\left(z_{\nu+1}-z_{\nu}\right) $$ Then \(\alpha:=\alpha_{1} \oplus \alpha_{2} \oplus \cdots \oplus \alpha_{N-1}\) defines a curve \(\alpha:[0, N] \rightarrow \mathbb{C} .\) One calls \(\alpha\) the polygonal path, which joins \(z_{0}\) with \(z_{N}\) (along \(\left.z_{1}, z_{2}, \ldots, z_{N-1}\right)\). Show: An open set \(D \subset \mathbb{C}\) 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 \(\emptyset \neq D \subset \mathbb{C}\) 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 \(1+i\), (b) \(\alpha\) is the piece of the parabola with equation \(y=x^{2}\), which lies between the points 0 and \(1+\mathrm{i}\)

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

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 \(z \in \mathbb{C}\) $$ \text { Re } f(z) \leq M $$ then \(f\) is constant. Hint. Consider \(g:=\exp \circ f\) and apply LIOUVILLE's theorem to \(g .\)

Let \([a, b]\) and \([c, d]\) ( \(a

For \(a, b \in \mathbb{R}_{+}^{*}\), 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 $$ \beta(t)=R \exp (i t), \quad 0 \leq t \leq \frac{\pi}{4} $$ Show: $$ \left|\int_{\beta} \exp \left(\mathrm{i} z^{2}\right) d z\right| \leq \frac{\pi\left(1-\exp \left(-R^{2}\right)\right)}{4 R}<\frac{\pi}{4 R} $$

Gauss-Lucas Theorem (C.F. GAuss, 1816; F. LUCAS, 1879) Let \(P\) be a complex polynomial of degree \(n \geq 1\), with \(n\) not necessarily different zeros \(\zeta_{1}, \ldots, \zeta_{n} \in \mathbb{C} .\) Show that for all \(z \in \mathbb{C} \backslash\left\\{\zeta_{1}, \ldots, \zeta_{n}\right\\}\) $$ \frac{P^{\prime}(z)}{P(z)}=\frac{1}{z-\zeta_{1}}+\frac{1}{z-\zeta_{2}}+\cdots+\frac{1}{z-\zeta_{n}}=\sum_{\nu=1}^{n} \frac{\overline{z-\zeta_{\nu}}}{\left|z-\zeta_{\nu}\right|^{2}} $$ Deduce from this the GAUSS-LUCAS theorem: For each zero \(\zeta\) of \(P^{\prime}\) there are \(n\) real numbers \(\lambda_{1}, \ldots, \lambda_{n}\) with $$ \lambda_{1} \geq 0, \ldots, \lambda_{n} \geq 0, \quad \sum_{j=1}^{n} \lambda_{j}=1 \text { and } \zeta=\sum_{\nu=1}^{n} \lambda_{\nu} \zeta_{\nu} $$ Thus one can say: The zeros of \(P^{\prime}\) lie in the "convex hull" of the zero set of \(P\).

Let \(\alpha:[a, b] \rightarrow \mathbb{C}\) be continuously differentiable and assume that the function \(f:\) 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 $$ \left|\int_{\alpha} f(z) d z-\sum_{\nu=1}^{N} f\left(\alpha\left(c_{\nu}\right)\right) \cdot\left(\alpha\left(a_{\nu}\right)-\alpha\left(a_{\nu-1}\right)\right)\right|<\epsilon $$ (Approximation of the line integral by a RIEMANN 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.

Let \(\alpha:[a, b] \rightarrow \mathbb{C}\) be continuously differentiable and assume that the function \(f:\) 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 $$ \left|\int_{\alpha} f(z) d z-\sum_{\nu=1}^{N} f\left(\alpha\left(c_{\nu}\right)\right) \cdot\left(\alpha\left(a_{\nu}\right)-\alpha\left(a_{\nu-1}\right)\right)\right|<\epsilon $$ (Approximation of the line integral by a RIEMANN 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.

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 the property $$ |z| \geq \varrho:=\max \left\\{1, \frac{2}{\left|a_{n}\right|} \sum_{\nu=0}^{n-1}\left|a_{\nu}\right|\right\\} $$ we have. $$ \frac{1}{2}\left|a_{n}\right||z|^{n} \leq|P(z)| \leq \frac{3}{2}\left|a_{n}\right||z|^{n} $$ Corollary. Any root of the polynomial \(P\) lies in the open disk with radius \(\rho\) centered at the origin.

A proof of the Fundamental Theorem of Algebra Let \(P\) be a nonconstant polynomial of degree \(n\), $$ P(z)=a_{n} z^{n}+\cdots+a_{0}, \quad a_{\nu} \in \mathbb{C}, 0 \leq \nu \leq n, n \geq 1, a_{n} \neq 0 $$ We have \(P(z)=z\left(a_{n} z^{n-1}+\cdots+a_{1}\right)+a_{0}=z Q(z)+a_{0}\) Assumption: \(P(z) \neq 0\) for all \(z \in \mathbb{C} .\) Then for \(z \neq 0\) we have $$ \frac{1}{z}=\frac{P(z)}{z P(z)}=\frac{z Q(z)+a_{0}}{z P(z)}=\frac{Q(z)}{P(z)}+\frac{a_{0}}{z P(z)} $$ By integration along \(\alpha(t)=R \exp (\mathrm{i} t), 0 \leq t \leq 2 \pi, R>0\), it follows that $$ 2 \pi \mathrm{i}=\int_{\alpha} \frac{a_{0}}{z P(z)} d z $$ By using the lemma on growth of polynomials, derive a contradiction (consider the limit \(R \rightarrow \infty\) ).

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 the whole \(\mathbb{C}\), and $$ F^{\prime}(z)=-\int_{a}^{b} \exp (-z t) t f(t) d t $$

Let \(D \subset \mathbb{C}\) be a domain with the property $$ z \in D \Rightarrow-z \in D $$ 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 $$ \int_{a_{r}} f=0 \text { for } \alpha_{r}(t):=r \exp (2 \pi \mathrm{i} t), 0 \leq t \leq 1 $$

Let \(D \subset \mathbb{C}\) be a domain with the property $$ z \in D \Rightarrow-z \in D $$ 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 $$ \int_{a_{r}} f=0 \text { for } \alpha_{r}(t):=r \exp (2 \pi \mathrm{i} t), 0 \leq t \leq 1 $$