Chapter 1

Find the real and imaginary parts of each of the following complex numbers: $$ \frac{\mathrm{i}-1}{\mathrm{i}+1} ; \quad \frac{3+4 \mathrm{i}}{1-2 \mathrm{i}} ; \quad \mathrm{i}^{n}, n \in \mathbb{Z} ; \quad\left(\frac{1+\mathrm{i}}{\sqrt{2}}\right)^{n}, n \in \mathbb{Z} $$ $$ \left(\frac{1+\mathrm{i} \sqrt{3}}{2}\right)^{n}, n \in \mathbb{Z} ; \quad \sum_{\nu=0}^{7}\left(\frac{1-\mathrm{i}}{\sqrt{2}}\right)^{\nu} ; \frac{(1+\mathrm{i})^{4}}{(1-\mathrm{i})^{3}}+\frac{(1-\mathrm{i})^{4}}{(1+\mathrm{i})^{3}} $$

Let \(z_{0}=x_{0}+\mathrm{i} y_{0} \neq 0\) be a given complex number. Define the sequence \(\left(z_{n}\right)_{n \geq 0}\) recursively by $$ z_{n+1}=\frac{1}{2}\left(z_{n}+\frac{1}{z_{n}}\right), \quad n \geq 0 $$ Show: If \(x_{0}>0\), then \(\lim _{n \rightarrow \infty} z_{n}=1\) If \(x_{0}<0\), then \(\lim _{n \rightarrow \infty} z_{n}=-1\) If \(x_{0}=0, y_{0} \neq 0\), then \(\left(z_{n}\right)_{n \geq 0}\) is undefined or divergent. Hint. Consider \(w_{n+1}=\frac{z_{n+1}-1}{z_{n+1}+1}\).

Calculate the absolute value (modulus) and an argument for each of the following complex numbers: $$ \begin{array}{r} -3+\mathrm{i} ; \quad-13 ; \quad(1+\mathrm{i})^{17}-(1-\mathrm{i})^{17} ; \quad \mathrm{i}^{4711} ; \quad \frac{3+4 \mathrm{i}}{1-2 \mathrm{i}} \\ \frac{1+\mathrm{i} a}{1-\mathrm{i} a}, a \in \mathbb{R} ; \quad \frac{1-\mathrm{i} \sqrt{3}}{1+\mathrm{i} \sqrt{3}} ; \quad(1-\mathrm{i})^{n}, n \in \mathbb{Z} . \end{array} $$

Let \(a \in \mathbb{C}^{\bullet}\) be given. For which \(z_{0} \in \mathbb{C}\) converges the series \(\left(z_{n}\right)\), which is recursively defined by $$ z_{n+1}=\frac{1}{2}\left(z_{n}+\frac{a}{z_{n}}\right) \quad \text { for } n \geq 0 \quad ? $$ Remark: Both exercises 1 and 2 are special complex instances of NEWTON's approximation method for zeros (of the polynomials \(\left.z^{2}-a\right) .\) See also Exercise 7 in I.4.

Investigate the continuity and complex differentiability, finding the derivative at points where it is differentiable, of the functions \(f\) : (a) $$ \begin{array}{ll} f(z)=z \operatorname{Re}(z), & f(z)=\bar{z} \\ f(z)=z \bar{z}, & f(z)=z /|z|, z \neq 0 \end{array} $$ (b) The exponential function exp is differentiable, and we have \(\exp ^{\prime}=\exp\).

Prove the "Triangle Inequality" $$ |z+w| \leq|z|+|w|, \quad z, w \in \mathbb{C} $$ and discuss when it becomes an equality; also prove the "Triangle Inequality" $$ || z|-| w|| \leq|z-w|, \quad z, w \in \mathbb{C}. $$

Set \(D \subseteq \mathbb{R}^{p}\) A point \(a \in D\) is called an interior point \((\) of \(D\) ) if together with \(a\) there exists a \(\varepsilon\)-disk \(U_{\varepsilon}(a):=\) \(\left\\{x \in \mathbb{R}^{p} ;|x-a|<\varepsilon\right\\}\) which is contained in \(D .\) Show: \(D\) is open \(\Longleftrightarrow\) each point of \(D\) is an interior point. A subset \(U \subseteq \mathbb{R}^{p}\) is called a neighborhood of \(a \in \mathbb{R}^{p}\) if \(U\) contains an \(\varepsilon\)-disk \(U_{\varepsilon}(a) .\) Show: \(D\) is open \(\Longleftrightarrow D\) is a neighborhood of each point \(a \in D\). Let \(\stackrel{\circ}{D}:=\\{x \in D ; \quad D\) neighborhood of \(x\\}\) Show: \(D\) is open \(\Longleftrightarrow D=\stackrel{\circ}{D}\). \(\stackrel{\circ}{D}\) is always open, and for each open subset \(U \subseteq \mathbb{R}^{p}\) with \(U \subseteq D\) we have \(U \subseteq \stackrel{\circ}{D}\).

If the function \(f: \mathbb{C} \rightarrow \mathbb{C}\) is complex differentiable at all points \(z \in \mathbb{C}\) and takes on only real or pure imaginary values, then \(f\) is constant.

Write the following functions in the form \(f=u+\mathrm{i} v\) and give explicit formulas for \(u\) and \(v\). (a) \(f(z)=\sin z\), (b) \(f(z)=\cos z\) (c) \(f(z)=\sinh z\), (d) \(f(z)=\cosh z\), (e) \(f(z)=\exp \left(z^{2}\right)\), (f) \(f(z)=z^{3}+z\). Show that in every case the CAUCHY-RIEMANN equations are satisfied (for all \(z \in \mathbb{C})\), and conclude that these functions are analytic in \(\mathbb{C}\).

For \(z=x+\mathrm{i} y, w=u+\mathrm{i} v\), with \(x, y, u, v \in \mathbb{R}\), the standard scalar product in the \(\mathbb{R}\)-vector space \(\mathbb{C}=\mathbb{R} \times \mathbb{R}\) with respect to the basis \((1, \mathrm{i})\) is defined by $$ \langle z, w\rangle:=\operatorname{Re}(z \bar{w})=x u+y v $$ Verify by direct calculation that, for \(z, w \in \mathbb{C}\) $$ \langle z, w\rangle^{2}+\langle\mathrm{i} z, w\rangle^{2}=|z|^{2}|w|^{2} $$ and infer from this the CAUCHY-SCHWARZ Inequality in \(\mathbb{R}^{2}\) : $$ |\langle z, w\rangle|^{2}=|x u+y v|^{2} \leq|z|^{2}|w|^{2}=\left(x^{2}+y^{2}\right)\left(u^{2}+v^{2}\right) $$ In addition, show the following identities for \(z, w \in \mathbb{C}\) by direct calculation: In addition, show the following identities for \(z, w \in \mathbb{C}\) by direct calculation: \(\begin{aligned}|z+w|^{2} &=|z|^{2}+2\langle z, w\rangle+|w|^{2} & &(\text { cosine law }) \\\|z-w|^{2} &=|z|^{2}-2\langle z, w\rangle+|w|^{2}, & & \\\|z+w|^{2}+|z-w|^{2} &=2\left(|z|^{2}+|w|^{2}\right) & & \text { (parallelogram law) } \end{aligned}\) (parallelogram law). Further, show that for each pair \((z, w) \in \mathbb{C}^{\bullet} \times \mathbb{C}^{\bullet}\) there is a unique real number \(\omega:=\omega(z, w) \in]-\pi, \pi]\) with $$ \cos \omega=\cos \omega(z, w)=\frac{\langle z, w\rangle}{|z||w|} $$ and $$ \sin \omega=\sin \omega(z, w)=\frac{\langle\mathrm{i} z, w\rangle}{|z||w|} $$ \(\omega=\omega(z, w)\) is called the oriented angle between \(z\) and \(w\) and will often be denoted by \(\angle(z, w)\) Show: \(\quad \angle(1, \mathrm{i})=\pi / 2, \angle(\mathrm{i}, 1)=-\pi / 2=-\angle(1, \mathrm{i}) .\)

Prove the following inequalities. (a) For all \(z \in \mathbb{C}\) we have $$ |\exp (z)-1| \leq \exp (|z|)-1 \leq|z| \exp (|z|) $$ (b) For all \(z \in \mathbb{C}\) with \(|z| \leq 1\) we have $$ |\exp (z)-1| \leq 2|z|. $$

Let \(M \subseteq \mathbb{R}^{p}\). A point \(a \in \mathbb{R}^{p}\) is called an accumulation point of \(M\) if for each \(\varepsilon\)-disk \(U_{\varepsilon}(a)\) there holds $$ U_{\varepsilon}(a) \cap(M \backslash\\{a\\}) \neq \emptyset $$ In each \(\varepsilon\)-disk for \(a\) there is therefore a point of \(M\) different from \(a\). Notation. \(M^{\prime}:=\left\\{x \in \mathbb{R}^{p} ; x\right.\) is an accumulation point of \(\left.M\right\\}\) Show: For a subset \(A \subseteq \mathbb{R}^{p}\) the following are equivalent: (a) \(A\) is closed, i. e. \(\mathbb{R}^{p}-A\) is open. (b) For each convergent sequence \(\left(a_{n}\right), a_{n} \in A\) we have \(\lim _{n \rightarrow \infty} a_{n} \in A\). (c) \(A \supset A^{\prime}\) Show that in addition: $$ \bar{A}:=A \cup A^{\prime} $$ is always closed, and for each closed set \(B \subseteq \mathbb{R}^{p}\) with \(B \supset A\) we have \(B \supset \bar{A}\). \(\bar{A}\) is called the closure (or closed hull) of \(A\).

Let \(f: D \rightarrow \mathbb{C}\) be complex differentiable at \(a \in D\) and \(D^{*}:=\\{z ; \bar{z} \in D\\}\) Then the function \(g: D^{*} \rightarrow \mathbb{C}\) defined by $$ g(z)=\overline{f(\bar{z})} $$ is complex differentiable at \(\bar{a}\), and we have $$ g^{\prime}(\bar{a})=\overline{f^{\prime}(a)} $$

The function \(f: \mathbb{C} \rightarrow \mathbb{C}\) $$ f(z)= \begin{cases}\exp \left(-1 / z^{4}\right) & \text { for } z \neq 0 \\ 0 & \text { for } z=0\end{cases} $$ satisfies the CAUCHY-RIEMANN equations for all \(z \in \mathbb{C}\) and is complex differentiable for all \(z \in \mathbb{C}^{\bullet}\), but not at the origin.

Suppose \(n \in \mathbb{N}\) and \(z_{\nu}, w_{\nu} \in \mathbb{C}\) for \(1 \leq \nu \leq n .\) Prove $$ \left|\sum_{\nu=1}^{n} z_{\nu} w_{\nu}\right|^{2}=\sum_{\nu=1}^{n}\left|z_{\nu}\right|^{2} \cdot \sum_{\nu=1}^{n}\left|w_{\nu}\right|^{2}-\sum_{1 \leq \nu<\mu \leq n}\left|z_{\nu} \bar{w}_{\mu}-z_{\mu} \bar{w}_{\nu}\right|^{2} $$ (the LAGRANGE Identity) and conclude from this the CAUCHY-SCHWARZ inequality in \(\mathbb{C}^{n}:\) $$ \left|\sum_{\nu=1}^{n} z_{\nu} w_{\nu}\right|^{2} \leq \sum_{\nu=1}^{n}\left|z_{\nu}\right|^{2} \cdot \sum_{\nu=1}^{n}\left|w_{\nu}\right|^{2}. $$

Determine, in each case, all the \(z \in \mathbb{C}\) with $$ \begin{aligned} \exp (z) &=-2, & \exp (z) &=\mathrm{i}, & & \exp (z) &=-\mathrm{i} \\ \sin z &=100, & \sin z &=7 \mathrm{i}, & \sin z &=1-\mathrm{i} \\ \cos z &=3 \mathrm{i}, & \cos z &=3+4 \mathrm{i}, & \cos z &=13. \end{aligned} $$

Let \(\left(x_{n}\right)_{n \geq 0}\) be a sequence in \(\mathbb{R}^{p} . a \in \mathbb{R}^{p}\) is called an accumulation value of the sequence \(\left(x_{n}\right)\) if for each \(\varepsilon\)-disk \(U_{\varepsilon}(a)\) there are infinitely many indices \(n\) such that \(x_{n} \in U_{\varepsilon}(a)\). Show (BOLZANO-WEIERSTRASS Theorem): Any bounded sequence \(\left(x_{n}\right), x_{n} \in\) \(\mathbb{R}^{p}\) has an accumulation point. A subset \(K \subseteq \mathbb{R}^{p}\) is called sequentially compact if each sequence \(\left(x_{n}\right)_{n \geq 0}\) with \(x_{n} \in K\) has (at least) one accumulation point in \(K\) Show: For a subset \(K \subseteq \mathbb{R}^{p}\) the following are equivalent: (a) \(K\) is compact, (b) \(K\) is sequentially compact. Remark. These equivalences hold for any metric space.

Prove the following variant of the chain rule: Let \(D\) and \(D^{\prime} \subseteq \mathbb{C}\) be open and \(f: D \rightarrow \mathbb{C}\) and \(g: D^{\prime} \rightarrow \mathbb{C}\) continuous functions with \(f(D) \subseteq D^{\prime}\) and \(g(f(z))=z\) for all \(z \in D\) Show: If \(g\) is complex differentiable at \(b=f(a)\) and \(g^{\prime}(b) \neq 0\), then \(f\) is complex differentiable at \(a\), and we have $$ f^{\prime}(a)=\frac{1}{g^{\prime}(b)}. $$

Which is the maximal open set \(D \subseteq \mathbb{C}\), such that \(f: D \rightarrow \mathbb{C}, f(z):=\log \left(z^{5}+\right.\) 1), is well defined and analytic.

Sketch the following subsets of \(\mathbb{C}\) in the complex plane: (a) Assume \(a, b \in \mathbb{C}, b \neq 0\); $$ \begin{aligned} G_{0} &:=\left\\{z \in \mathbb{C} ; \quad \operatorname{Im}\left(\frac{z-a}{b}\right)=0\right\\} \\ G_{+} &:=\left\\{z \in \mathbb{C} ; \quad \operatorname{Im}\left(\frac{z-a}{b}\right)>0\right\\} \quad \text { and } \\ G_{-} &:=\left\\{z \in \mathbb{C} ; \quad \operatorname{Im}\left(\frac{z-a}{b}\right)<0\right\\} \end{aligned} $$ (b) Assume \(a, c \in \mathbb{R}\) and \(b \in \mathbb{C}\) with \(b \bar{b}-a c>0\), $$ K:=\\{z \in \mathbb{C} ; \quad a z \bar{z}+\bar{b} z+b \bar{z}+c=0\\} $$ (c) \(L:=\left\\{z \in \mathbb{C} ; \quad\left|z-\frac{\sqrt{2}}{2}\right| \cdot\left|z+\frac{\sqrt{2}}{2}\right|=\frac{1}{2}\right\\}\)

The (complex) hyperbolic functions cosh and sinh are defined similarly to the real ones. For \(z \in \mathbb{C}\) let $$ \cosh z:=\frac{\exp (z)+\exp (-z)}{2} \quad \text { and } \quad \sinh z:=\frac{\exp (z)-\exp (-z)}{2} $$ Show: (a) \(\sinh z=-\mathrm{i} \sin (\mathrm{i} z), \cosh z=\cos (\mathrm{i} z)\) for all \(z \in \mathbb{C}\). (b) Addition theorems $$ \begin{aligned} &\sinh (z+w)=\sinh z \cosh w+\cosh z \sinh w \\ &\cosh (z+w)=\cosh z \cosh w+\sinh z \sinh w \end{aligned} $$ (c) \(\cosh ^{2} z-\sinh ^{2} z=1\) for all \(z \in \mathbb{C}\). (d) sinh and cosh have the period \(2 \pi\) i, i.e. $$ \begin{aligned} &\sinh (z+2 \pi \mathrm{i})=\sinh z \\ &\cosh (z+2 \pi \mathrm{i})=\cosh z \end{aligned} \quad \text { for all } z \in \mathbb{C} $$ (e) For all \(z \in \mathbb{C}\) the series \(\sum \frac{z^{2 n}}{(2 n) !}\) and \(\sum \frac{z^{2 n+1}}{(2 n+1) !}\) are absolutely convergent, and one has $$ \cosh z=\sum_{n=0}^{\infty} \frac{z^{2 n}}{(2 n) !} \quad \text { and } \quad \sinh z=\sum_{n=0}^{\infty} \frac{z^{2 n+1}}{(2 n+1) !} $$

For all \(z \in \mathbb{C}\) $$ \lim _{n \rightarrow \infty}(1+z / n)^{n}=\exp (z) $$ More generally: For each sequence \(\left(z_{n}\right), z_{n} \in \mathbb{C}\) with \(\lim _{n \rightarrow \infty} z_{n}=z\) we have $$ \lim _{n \rightarrow \infty}\left(1+z_{n} / n\right)^{n}=\exp (z) $$

If \(f: D \rightarrow \mathbb{C}\) is analytic, \(D \subseteq \mathbb{C}\) is open, and one of the following conditions holds: (a) Re \(f=\) constant, (b) \(\operatorname{Im} f=\) constant, (c) \(|f|=\) constant, then \(f\) is locally constant.

Let \(c=a+\mathrm{i} b \neq 0\) be a given complex number. By splitting it into its real and imaginary parts show that there are exactly two complex numbers \(z_{1}\) and \(z_{2}\) such that $$ z_{1}^{2}=z_{2}^{2}=c . \text { We have } z_{2}=-z_{1} $$ \(\left(z_{1}\right.\) and \(z_{2}\) are called the square roots of \(\left.c .\right)\) For example, determine the square roots of $$ 5+7 \mathrm{i}, \quad \text { and } \quad \sqrt{2}+\mathrm{i} \sqrt{2} $$ Use polar coordinates for this exercise. Furthermore, show that a quadratic equation $$ z^{2}+\alpha z+\beta=0, \quad \alpha, \beta \in \mathbb{C} \text { arbitrary } $$ always has at most two solutions \(z_{1}, z_{2} \in \mathbb{C}\).

For all \(z=x+\mathrm{i} y \in \mathbb{C}\) one has: (a) $$ \overline{\exp (z)}=\exp (\bar{z}), \quad \overline{\sin (z)}=\sin (\bar{z}), \quad \overline{\cos (z)}=\cos (\bar{z}) $$ (b) $$ \begin{aligned} &\cos z=\cos (x+\mathrm{i} y)=\cos x \cosh y-\mathrm{i} \sin x \sinh y \\ &\sin z=\sin (x+\mathrm{i} y)=\sin x \cosh y+\mathrm{i} \cos x \sinh y \end{aligned} $$ In the special case \(x=0, y \in \mathbb{R}\) we have $$ \cos (\mathrm{i} y)=\frac{1}{2}\left(e^{y}+e^{-y}\right)=\cosh y \quad \text { and } \quad \sin (\mathrm{i} y)=\frac{\mathrm{i}}{2}\left(e^{y}-e^{-y}\right)=\mathrm{i} \sinh y $$ Determine all the \(z \in \mathbb{C}\) with \(|\sin z| \leq 1\), and find an \(n \in \mathbb{N}\) such that $$ |\sin (\mathrm{in})|>10000. $$

Prove HEINE's theorem (E. HEINE, 1872 ): If \(K \subset \mathbb{C}\) is compact and \(f: K \rightarrow \mathbb{C}\) is continuous then \(f\) is uniformly continuous on \(K\), i. e. for each \(\varepsilon>0\) there exists a \(\delta>0\) so that for all \(z, z^{\prime} \in K\) with \(\left|z-z^{\prime}\right|<\delta\) $$ \left|f(z)-f\left(z^{\prime}\right)\right|<\varepsilon $$

For each of the harmonic functions given below construct an analytic function \(f: D \rightarrow \mathbb{C}\) with the given real part \(u:\) (a) \(D=\mathbb{C}\) and \(u: D \rightarrow \mathbb{R}\) with \(u(x, y)=x^{3}-3 x y^{2}+1\) (b) \(D=\mathbb{C}^{\bullet}\) and \(u: D \rightarrow \mathbb{R}\) with \(u(x, y)=\frac{x}{x^{2}+y^{2}}\) (c) \(D=\mathbb{C} \quad\) and \(u: D \rightarrow \mathbb{R}\) with \(u(x, y)=e^{x}(x \cos y-y \sin y)\). (d) \(D=\mathbb{C}_{-}\)and \(u: D \rightarrow \mathbb{R}\) with \(u(x, y)=\sqrt{\frac{x+\sqrt{x^{2}+y^{2}}}{2}}\).

Assume \(a \in \mathbb{C}\) and \(n \in \mathbb{N}\). A complex number \(z\) is called (an) \(n\)-th root of \(a\) if \(z^{n}=a\). Show: If \(a=r(\cos \varphi+\mathrm{i} \sin \varphi) \neq 0\), then \(a\) has exactly \(n\) (different) \(n\)-th roots, namely the complex numbers $$ z_{\nu}=\sqrt[n]{r}\left(\cos \frac{\varphi+2 \pi \nu}{n}+\mathrm{i} \sin \frac{\varphi+2 \pi \nu}{n}\right), \quad 0 \leq \nu \leq n-1 $$ In the special case \(a=1\) (thus \(r=1, \varphi=0)\), we have Theorem I.1.7.

For \(z \in \mathbb{C} \backslash\\{(k+1 / 2) \pi ; k \in \mathbb{Z}\\}\) let $$ \tan z:=\frac{\sin z}{\cos z} $$ and for \(z \in \mathbb{C} \backslash\\{k \pi ; k \in \mathbb{Z}\\}\) let $$ \cot z:=\frac{\cos z}{\sin z} . $$ Show: $$ \begin{array}{ll} \tan z=\frac{1}{\mathrm{i}} \frac{\exp (2 \mathrm{i} z)-1}{\exp (2 \mathrm{i} z)+1}, & \tan (z+\pi / 2)=-\cot z \\ \tan (-z)=-\tan z \\ \cot z=\mathrm{i} \frac{\exp (2 \mathrm{i} z)+1}{\exp (2 \mathrm{i} z)-1}, & \tan z=\tan (z+\pi) \end{array} $$ $$ \begin{aligned} &\tan z=\cot z-2 \cot (2 z) \\ &\cot (z+\pi)=\cot z. \end{aligned} $$

For any subsets \(A, B \subseteq \mathbb{C}\) $$ d(A, B):=\inf \\{|z-w| ; \quad z \in A, w \in B\\} $$ is called the distance between \(A\) and \(B .\) If \(B=\\{w\\}\), then one simply writes \(d(A, w)\) instead of \(d(A,\\{w\\})\). Show: (a) If \(A \subseteq \mathbb{C}\) is a closed subset and \(b \in \mathbb{C}\) is arbitrary, then there is an \(a \in A\) with $$ d(A, b)=|a-b| $$ (b) If \(A \subseteq \mathbb{C}\) is a closed subset and \(B \subset \mathbb{C}\) is compact, then there are elements \(a \in A\) and \(b \in B\) such that $$ d(A, B)=|a-b| $$

Let \(\mathbb{R}_{+}^{\bullet} \times \mathbb{R} \rightarrow \mathbb{R}^{2} \backslash\\{(0,0)\\}\) be the map defined by \((x, y)=(r \cos \varphi, r \sin \varphi)\). In addition, let \(D \subseteq \mathbb{R}^{2} \backslash\\{(0,0)\\}\) be an open subset and \(u: D \rightarrow \mathbb{R}\) a function which is twice continuously differentiable. Let \(\Omega:=\\{(r, \varphi) ;(x, y) \in D\\}\) and $$ U: \Omega \longrightarrow \mathbb{R}, \quad U(r, \varphi)=u(x, y) $$ Show: $$ (\Delta u)(x, y)=\left(U_{r r}+\frac{1}{r} U_{r}+\frac{1}{r^{2}} U_{\varphi \varphi}\right)(r, \varphi). $$

Determine all \(z \in \mathbb{C}\) such that \(z^{3}-\mathrm{i}=0\).

Let \(\operatorname{Maps}\left(\mathbb{N}_{0}, \mathbb{C}\right)\) be the set of all maps of \(\mathbb{N}_{0}\) into \(\mathbb{C}(=\) the set of all complex number sequences). Show: The map $$ \begin{aligned} \sum: \operatorname{Maps}\left(\mathbb{N}_{0}, \mathbb{C}\right) & \longrightarrow \operatorname{Maps}\left(\mathbb{N}_{0}, \mathbb{C}\right) \\ \left(a_{n}\right)_{n} \geq 0 & \longmapsto\left(S_{n}\right)_{n \geq 0} \text { with } S_{n}:=a_{0}+a_{1}+\cdots+a_{n} \end{aligned} $$ is bijective the (telescope trick). The theories of sequences and of infinite series are therefore in principle the same.

There does not exist a function \(f: \mathbb{C}^{\bullet} \rightarrow \mathbb{C}^{\bullet}\) with both following properties (a) \(\quad f(z w)=f(z) f(w)\) for all \(z, w \in \mathbb{C}^{\bullet}\), and (b) \(\quad(f(z))^{2}=z\) for all \(z \in \mathbb{C}^{\bullet}\).

Determine all the harmonic functions $$ u: \mathbb{C}^{\bullet}=\mathbb{R}^{2} \backslash\\{(0,0)\\} \longrightarrow \mathbb{R} $$ that depend only on \(r:=\sqrt{x^{2}+y^{2}} .\)

Let \(P\) be a polynomial with complex coefficients: $$ P(z):=a_{n} z^{n}+a_{n-1} z^{n-1}+\cdots+a_{0} \text { with } n \in \mathbb{N}_{0}, a_{\nu} \in \mathbb{C}, \text { for } 0 \leq \nu \leq n $$ A real or complex number \(\zeta\) is called a root or a zero of \(P\), if \(P(\zeta)=0\). Show: If all the coefficients \(a_{\nu}\) are real, then we have $$ P(\zeta)=0 \Longrightarrow P(\bar{\zeta})=0 $$ In other words, if the polynomial \(P\) has only real coefficients then the roots of \(P\) which are not real occur as pairs of conjugate complex numbers.

Let \(\left(a_{n}\right)_{n \geq 0}\) and \(\left(b_{n}\right)_{n \geq 0}\) be two sequences of complex numbers such that \(a_{n}=\) \(b_{n}-b_{n+1}, n \geq 0\) Show: The series \(\sum_{n=0}^{\infty} a_{n}\) is convergent if and only if the sequence \(\left(b_{n}\right)\) is convergent, and then $$ \sum_{n=0}^{\infty} a_{n}=b_{0}-\lim _{n \rightarrow \infty} b_{n+1} $$ Example: \(\sum_{n=0}^{\infty} \frac{1}{(n+1)(n+2)}=1 .\)

Show: (a) There is no continuous function \(f: \mathbb{C}^{\bullet} \rightarrow \mathbb{C}^{\bullet}\) such that $$ (f(z))^{2}=z \text { for all } z \in \mathbb{C}^{\bullet} $$ (b) There is no continuous function \(q: \mathbb{C} \rightarrow \mathbb{C}\) such that $$ (q(z))^{2}=z \text { for all } z \in \mathbb{C} $$

Let \(D \subseteq \mathbb{C}\) be open, and \(D^{\prime} \subseteq \mathbb{C}\) another open subset. Let \(\varphi: D \rightarrow D^{\prime}\) be analytic and even twice continuously differentiable, and \(\eta: D^{\prime} \rightarrow \mathbb{R}\) twice continuously partial differentiable. Show: $$ \Delta(\eta \circ \varphi)=((\Delta \eta) \circ \varphi)\left|\varphi^{\prime}\right|^{2} $$ Deduce: If \(\varphi\) is conformal then \(\eta\) is harmonic if and only if \(\eta \circ \varphi\) is harmonic.

(a) Let \(\mathbb{H}:=\\{z \in \mathbb{C} ; \quad \operatorname{Im} z>0\\}\) be the upper half-plane. Show: \(z \in \mathbb{H} \Longleftrightarrow-1 / z \in \mathbb{H}\). (b) Assume \(z, a \in \mathbb{C}\). Show: \(\quad|1-z \bar{a}|^{2}-|z-a|^{2}=\left(1-|z|^{2}\right)\left(1-|a|^{2}\right)\) Deduce: If \(|a|<1\), then $$ |z|<1 \Longleftrightarrow\left|\frac{z-a}{\bar{a} z-1}\right|<1 \quad \text { and } \quad|z|=1 \Longleftrightarrow\left|\frac{z-a}{\bar{a} z-1}\right|=1 $$

For \(\alpha \in \mathbb{C}\) and \(\nu \in \mathbb{N}\) let $$ \left(\begin{array}{l} \alpha \\ 0 \end{array}\right):=1 \quad \text { and } \quad\left(\begin{array}{l} \alpha \\ \nu \end{array}\right):=\prod_{j=1}^{\nu} \frac{\alpha-j+1}{j} $$ Show: \(\sum_{\nu=0}^{\infty}\left(\begin{array}{c}\alpha \\\ \nu\end{array}\right) z^{\nu}\) is absolutely convergent for all \(z \in \mathbb{C}\) with \(|z|<1\) Let \(b_{\alpha}(z):=\sum_{\nu=0}^{\infty}\left(\begin{array}{l}\alpha \\\ \nu\end{array}\right) z^{\nu}\) Show: For all \(z \in \mathbb{C}\) with \(|z|<1\) and arbitrary \(\alpha, \beta \in \mathbb{C}\) we have $$ b_{\alpha+\beta}(z)=b_{\alpha}(z) b_{\beta}(z) $$ Remark. We shall see later that for \(z \in \mathbb{C}\) with \(|z|<1\) there holds $$ b_{\alpha}(z)=(1+z)^{\alpha}:=\exp (\alpha \log (1+z)) $$ For \(\alpha=n \in \mathbb{N}_{0}\), one obtains the binomial formula $$ (1+z)^{n}=\sum_{\nu=0}^{n}\left(\begin{array}{l} n \\ \nu \end{array}\right) z^{\nu} $$.

There is no continuous function \(\varphi: \mathbb{C}^{\bullet} \rightarrow \mathbb{R}\) such that $$ z=|z| \exp (\mathrm{i} \varphi(z)) \text { for all } z \in \mathbb{C}^{\bullet}. $$

Let \(D=\mathbb{R}\) or \(D=\mathbb{C} .\) Let \(C \in \mathbb{C}\) be a constant and \(f: D \rightarrow \mathbb{C}\) differentiable with $$ f^{\prime}(z)=C f(z) \quad \text { for all } \quad z \in D $$ If \(A=f(0)\), then $$ f(z)=A \exp (C z) \quad \text { for all } \quad z \in D $$

Verify for \(z=x+\mathrm{i} y \in \mathbb{C}\) the inequalities $$ \frac{|x|+|y|}{\sqrt{2}} \leq|z|=\sqrt{x^{2}+y^{2}} \leq|x|+|y| $$ and $$ \max \\{|x|,|y|\\} \leq|z| \leq \sqrt{2} \max \\{|x|,|y|\\}. $$

For \(k \in \mathbb{N}_{0}\), and \(z \in \mathbb{C}\) with \(|z|<1\), show $$ \frac{1}{(1-z)^{k+1}}=\sum_{n=0}^{\infty}\left(\begin{array}{c} n+k \\ k \end{array}\right) z^{n}=\sum_{n=0}^{\infty}\left(\begin{array}{c} n+k \\ n \end{array}\right) z^{n} $$

There is no continuous function \(l: \mathbb{C}^{\bullet} \rightarrow \mathbb{C}\) such that $$ \exp (l(z))=z \text { for all } z \in \mathbb{C}^{\bullet} $$

Let \(\widetilde{\mathbb{C}}\) be another field of complex numbers. Determine all mappings \(\varphi: \mathbb{C} \rightarrow \widetilde{\mathbb{C}}\) with the following properties; \(\begin{aligned}(a) & \varphi(z+w) &=\varphi(z)+\varphi(w) & & \text { for all } z, w \in \mathbb{C} \\\\(b) & \varphi(z w) &=\varphi(z) \varphi(w) & & \text { for all } z, w \in \mathbb{C} \\\\(c) & \varphi(x) &=x & & \text { for all } x \in \mathbb{R} \end{aligned}\) Remark. It turns out that such mappings exist, and they are automatically bijective; thus they give isomorphisms \(\mathbb{C} \rightarrow \widetilde{\mathbb{C}}\) that leave \(\mathbb{R}\) fixed element by element. The field of complex numbers is therefore essentially uniquely determined. In the special case \(\mathbb{C}=\widetilde{\mathbb{C}}\) we get automorphisms of \(\mathbb{C}\) with the fixed field \(\mathbb{R}\). Remark: What automorphisms (i.e. isomorphisms with itself) does the real field \(\mathbb{R}\) have ? Hint: Such an automorphism of \(\mathbb{R}\) must preserve the ordering of \(\mathbb{R} !\)

Let \(\left(a_{n}\right)_{n \geq 0}\) and \(\left(b_{n}\right)_{n \geq 0}\) be two sequences of complex numbers and $$ A_{n}:=a_{0}+a_{1}+\cdots+a_{n}, \quad n \in \mathbb{N}_{0}. $$ Show: For each \(m \geq 0\) and each \(n \geq m\) we have $$ \sum_{\nu=m}^{n} a_{\nu} b_{\nu}=\sum_{\nu=m}^{n} A_{\nu}\left(b_{\nu}-b_{\nu+1}\right)-A_{m-1} b_{m}+A_{n} b_{n+1} $$ (ABEL's partial summation, N. H. ABEL, 1826) where if \(m=0\) we set by definition (convention) the coefficient \(a_{-1}=0\) (corresponding to an empty sum).

Let \(n \geq 2\) be a natural number. There is no continuous function \(f: \mathbb{C}^{\bullet} \rightarrow \mathbb{C}^{\bullet}\) with the two properties (a) \(\quad f(z w)=f(z) f(w)\) for all \(z, w \in \mathbb{C}^{\bullet}, \quad\) and (b) \(\quad(f(z))^{n}=z\) for all \(z \in \mathbb{C}^{\bullet} \quad(n \in \mathbb{N}, n \geq 2)\).

Sketch the following level lines for the map \(f: \mathbb{C} \rightarrow \mathbb{C}, z \mapsto z^{3}\) $$ \\{z \in \mathbb{C} ; \operatorname{Re} f(z)=c\\}, \quad\\{z \in \mathbb{C} ; \operatorname{Im} f(z)=c\\}, \quad\\{z \in \mathbb{C} ;|f(z)|=c\\} $$ for \(c \in \mathbb{Z}\) with \(|c| \leq 5\). Go on to find the images under \(f\) of these level lines and the images of the lines parallel to the real axis (resp. the imaginary axis).