Chapter 1

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

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

Let \(a \in \mathbb{C}^{*}\) be given. For which \(z_{0} \in \mathbb{C}\) converges the sequence \(\left(z_{n}\right)\), which is recursively defined by $$ z_{n+1}=\frac{1}{2}\left(z_{n}+\frac{a}{z_{n}}\right) \quad \text { for } n \geq 0 ? $$

Investigate the continuity and complex differentiability of the following functions \(f\). Find the derivatives at points where they exist. (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 \(\exp ^{\prime}=\exp\).

Let \(f: \mathbb{C} \rightarrow \mathbb{C}\) be defined by \(f(z)=x^{3} y^{2}+i x^{2} y^{3}\). Show: \(f\) is complex differentiable exactly on the coordinate axes, and there is no open subset \(D \subset \mathbb{C}\) such that \(f \mid D\) is analytic.

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" $$ || z|-| w|| \leq|z-w|, \quad z, w \in \mathbb{C} $$

If the function \(f: \mathbb{C} \rightarrow \mathrm{C}\) is complex differentiable at all points \(z \in \mathrm{C}\) and takes 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: $$ \begin{aligned} |z+w|^{2} &=|z|^{2}+2\langle z, w\rangle+|w|^{2} & & \text { (cosine law) } \\\ |z-w|^{2} &=|z|^{2}-2\langle z, w\rangle+|w|^{2}, & & \\ |z+w|^{2}+|z-w|^{2} &=2\left(|z|^{2}+|w|^{2}\right) & \text { (parallelogran } \end{aligned} $$ (parallelogram law). Further, show that for each pair \((z, w) \in \mathbb{C}^{*} \times \mathbb{C}^{*}\) there is a unique real number \(\omega:=\omega(z, w) \in]-\pi, \pi]\) with $$ \cos \omega=\cos \omega(z, w)=\frac{\langle z, w\rangle}{|z||w|} $$ I Differential Calculus in the Complex Plane \(\mathbb{C}\) $$ \sin \omega=\sin \omega(z, w)=\frac{(\mathrm{i} z, w\rangle}{|z||w|} $$ \(\omega=\omega(z, w)\) 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 \(M \subset \mathbb{R}^{p}\). A point \(a \in \mathbb{R}^{p}\) is called an accumulation point of \(M\) if for each \(\varepsilon\)-ball \(U_{\varepsilon}(a)\) there holds $$ U_{c}(a) \cap(M \backslash\\{a\\}) \neq \emptyset $$ In each \(\varepsilon\)-ball 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 \(A \subset \mathbb{R}^{p}\) 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 \(B \subset \mathbb{R}^{p}\) with \(B \supset A\) we have \(B \supset \bar{A}\). \(\bar{A}\) is called the closure (or closed hull) of \(A\).

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 \(\mathbb{R}^{P} . \mathrm{A}\) point \(a \in \mathbb{R}^{P}\) is called an accumulation value of the sequence \(\left(x_{n}\right)\) if for each \(e\)-ball \(U_{c}(a)\) there are infinitely many indices \(n\) such that \(x_{n} \in U_{c}(a)\) Show (BOLZANO-WEIERSTRASS Theorem): Any bounded sequence \(\overline{\left(x_{n}\right)}, x_{n} \in\) \(\mathbb{R}^{P}\) has an accumulation value. A subset \(K \subset \mathbb{R}^{p}\) is called sequence compact if each sequence \(\left(x_{n}\right)_{n} \geq 0\) with \(x_{n} \in K\) has (at least) one accumulation value in \(K\) Show: For a subset \(K \subset \mathbb{R}^{p}\) the following are equivalent: (a) \(K\) is compact, (b) \(K\) is sequence compact. Remark. This equivalence holds for any metric space.

Prove the following variant of the theorem of invertible functions: Let \(D\) and \(D^{\prime} \subset \mathbb{C}\) be open and \(f: D \rightarrow \mathbb{C}\) and \(g: D^{\prime} \rightarrow \mathbb{C}\) continuous functions with \(f(D) \subset D^{\prime}\) 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 $$ f^{\prime}(a)=\frac{1}{g^{\prime}(b)} $$

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

Sketch the following subsets of \(\mathrm{C}\) 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) Consider \(a, c \in \mathbb{R}\) 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) \(L:=\left\\{z \in \mathbb{C} ;\left|z-\frac{\sqrt{2}}{2}\right| \cdot\left|z+\frac{\sqrt{2}}{2}\right|=\frac{1}{2}\right\\}\)

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 (c) \(\cosh ^{2} z-\sinh ^{2} z=1\) for all \(z \in \mathbb{C}\) (d) sinh and cosh have the period \(2 \pi \mathrm{i}\), i.e. \(\sinh (z+2 \pi i)=\sinh z\) \(\cosh (z+2 \pi i)=\cosh z$$\quad\) 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, \(D \subset \mathbb{C}\) 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.

Square roots and the solvability of quadratic equations in \(\mathbb{C}\) Let \(c=a+\mathrm{i} b \neq 0\) be a given complex number. By splitting it into its real and imaginary part show that there are exactly two complex numbers \(z_{1}\) and \(z_{2}\) such that $$ z_{1}^{2}=z_{2}^{2}=c . \text { One has } z_{2}=-z_{1} $$ \(\left(z_{1}\right.\) and \(z_{2}\) are called the square roots of \(c\) ) For example, determine the square roots of $$ 5+7 \mathrm{i}, \quad \text { and } \quad \sqrt{2}+\mathrm{i} \sqrt{2} $$ Use also 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 \(z_{1}, z_{2} \in \mathrm{C}\).

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) \(\cos z=\cos (x+i y)=\cos x \cosh y-i \sin x \sinh y\) \(\sin z=\sin (x+i y)=\sin x \cosh y+i \cos x \sinh y\) 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 $$ |\sin (\mathrm{i} n)|>10000 $$

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 \(e>0\) 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) \(D=\mathbb{C}^{*}\) and \(u: D \rightarrow \mathbb{R}\) with \(u(x, y)=\frac{x}{x^{2}+y^{2}}\). (c) \(D=\mathbb{C}\) 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 \(u(x, y)=\sqrt{\frac{x+\sqrt{x^{2}+y^{2}}}{2}}\).

Existence of \(n^{\text {th }}\) roots Assume \(a \in \mathbb{C}\) and \(n \in \mathbb{N} .\) A complex number \(z\) is called (an) \(n^{\text {th }}\) 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) \(n^{\text {th }}\) 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 get Theorem I.1.7.

Definition of the tangent and cotangent 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: $$ \begin{gathered} \tan z=\frac{1}{\mathrm{i}} \frac{\exp (2 \mathrm{i} z)-1}{\exp (2 \mathrm{i} z)+1}, \quad \cot z=\mathrm{i} \frac{\exp (2 \mathrm{i} z)+1}{\exp (2 \mathrm{i} z)-1} \\ \tan (z+\pi / 2)=-\cot z, \quad \tan (-z)=-\tan z, \quad \tan z=\tan (z+\pi) \\\ \tan z=\cot z-2 \cot (2 z), \quad \cot (z+\pi)=\cot z \end{gathered} $$

For any subsets \(A, B \subset C\) $$ d(A, B):=\inf \\{|z-w| ; \quad z \in A, w \in B\\} $$ 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 \(A \subset \mathbb{C}\) 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 \(A \subset \mathbb{C}\) is a closed subset and \(B \subset \mathbb{C}\) is compact, then there are elements \(a \in A\) and \(b \in B\) such that $$ d(A, B)=|a-b| $$

The Laplace operator in polar coordinates 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 \(D \subset \mathbb{R}^{2} \backslash\\{(0,0)\\}\) 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 Maps \(\left(\mathbb{N}_{0}, \mathrm{C}\right)\) be the set of all maps of \(\mathbb{N}_{0}\) into \(\mathbb{C}(=\) the set of all complex sequences). Show: The map $$ \begin{aligned} \sum: \operatorname{Maps}\left(\mathbb{N}_{0}, \mathrm{C}\right) & \longrightarrow \operatorname{Maps}\left(\mathbb{N}_{0}, \mathbb{C}\right) \\ \left(a_{n}\right)_{n \geq 0} \longmapsto\left(S_{n}\right)_{n \geq 0} \text { with } S_{n}:=a_{0}+a_{1}+\cdots+a_{n} \end{aligned} $$ is bijective (telescope trick). The theories of sequences and of infinite series are therefore in principle the same.

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

Determine all harmonic functions $$ u: \mathbb{C}^{*}=\mathbb{R}^{2} \backslash\\{(0,0)\\} \longrightarrow \mathbb{R} $$ 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 complex conjugate numbers.

Show: (a) There is no continuous function \(f: \mathbb{C}^{*} \rightarrow \mathbb{C}^{*}\) such that $$ (f(z))^{2}=z \text { for all } z \in \mathbb{C}^{*} $$ (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 \(D \subset \mathbb{C}\) be open and \(D^{\prime} \subset \mathbb{C}\) 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: \(z \in \mathrm{H} \Longleftrightarrow-1 / z \in \mathbb{H} .\) (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 $$ |z|<1 \Longleftrightarrow\left|\frac{z-a}{\bar{a} z-1}\right|<1 \quad \text { and } \quad|z|=1 \Longleftrightarrow\left|\frac{z-a}{\bar{a} z-1}\right|=1 $$

Binomial series For \(\alpha \in \mathbb{C}\) and \(\nu \in \mathbb{N}\) let $$ \left(\begin{array}{c} \alpha \\ 0 \end{array}\right):=1 \quad \text { and } \quad\left(\begin{array}{l} \alpha \\ \nu \end{array}\right):=\prod_{j=1}^{\nu} \frac{\alpha-j+1}{j} $$ 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 \(b_{\alpha}(z):=\sum_{\nu=0}^{\infty}\left(\begin{array}{c}\alpha \\\ \nu\end{array}\right) z^{\nu} .\) 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 $$ (1+z)^{n}=\sum_{v=0}^{n}\left(\begin{array}{l} n \\ \nu \end{array}\right) z^{\nu} $$

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

Characterization of the exponential function by a differential equation 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 $$ \frac{1}{(1-z)^{k+1}}=\sum_{n=0}^{\infty}\left(\begin{array}{c} n+k \\ k \end{array}\right) z^{n}=\sum_{n=0}^{\infty}\left(\begin{array}{c} n+k \\ n \end{array}\right) z^{n} $$

Let \(\widetilde{\mathbb{C}}\) be another field of complex numbers. Determine all mappings \(\varphi: \mathbb{C} \rightarrow \widetilde{\mathbb{C}}\) with the following properties: (a) \(\varphi(z+w)=\varphi(z)+\varphi(w) \quad\) for all \(z, w \in \mathbb{C}\) (b) \(\varphi(z w)=\varphi(z) \varphi(w) \quad\) for all \(z, w \in \mathbb{C}\) \(\begin{array}{cll}(c) & \varphi(x)=x & \text { for all } x \in \mathbb{R} \text {. }\end{array}\) 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 \(\mathbb{R}\) elementwise fixed. 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 onto itself) admits the field of real numbers \(\mathbb{R}\) ? 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 in the case \(m=0\) we set \(A_{-1}=0\) (empty sum).

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

Let \(n \geq 2\) be a natural number. There is no continuous function \(q_{n}: \mathbb{C} \rightarrow \mathbb{C}\) such that $$ \left(q_{n}(z)\right)^{n}=z \text { for all } z \in \mathbb{C} $$

Let \(D=\\{z \in \mathbb{C} ; \quad-\pi<\operatorname{Im} z<\pi, 0<\operatorname{Re} z

(a) Consider the map $$ f: \mathbb{C}^{*} \longrightarrow \mathbb{C} \text { with } f(z)=1 / \bar{z} $$ Give a geometrical construction (with ruler and compass) for the image \(f(z)\) and justify calling this map "reflection at the unit circle". Find the image under \(f\) of each of \((\alpha) \quad D_{1}:=\\{z \in \mathbb{C} ; \quad 0<|z|<1\\}\) \((\beta) \quad D_{2}:=\\{z \in \mathbb{C} ; \quad|z|>1\\}\) (\gamma) \(D_{3}:=\\{z \in \mathbb{C} ; \quad|z|=1\\}\). (b) Now consider the map $$ g: \mathbb{C}^{*} \longrightarrow C \text { with } g(z)=1 / z(=\overline{f(z)}) $$ and give a geometrical construction for the image \(g(z)\) of \(z .\) Why is this map called "inversion at the unit circle"? What are the fixed points of \(g\), i.e. for which \(z \in \mathbb{C}^{\bullet}\) is it true that \(g(z)=z ?\)

The Joukowski function -named after the Russian mathematician N.J. JOUKOWSKI \((1847-1921)-\) $$ f: C^{*} \longrightarrow \mathbb{C}, \quad z \mapsto \frac{1}{2}\left(z+\frac{1}{z}\right) $$ is analytic, is not injective since \(f(z)=f(1 / z)\), but it is (locally) conformal because of \(f^{\prime}(z)=\frac{1}{2}\left(1-1 / z^{2}\right)\) in \(C^{*} \backslash\\{1,-1\\}\) Show (by introducing polar coordinates): (a) The image of a circle is \(C_{r}:=\\{z \in \mathbb{C} ; \quad|z|=r\\}, r>0\), under \(f\) is (i) in the case \(r \neq 1\), an ellipse with the foci \(\pm 1\) and semi-axes \(\frac{1}{2}\left(r+\frac{1}{r}\right)\) resp. \(\frac{1}{2}\left|r-\frac{1}{r}\right|\) (ii) and else \(f\left(C_{1}\right)=[-1,1]\).

Assume \(n \in \mathbb{N}\) and let \(W(n)=\left\\{z \in \mathbb{C} ; z^{n}=1\right\\}\) be the set of \(n^{\text {th }}\) roots of unity. Show: (a) \(W(n)\) is a subgroup of \(\mathbb{C}^{*}\) (and so is a group itself). (b) \(W(n)\) is a cyclic group of order \(n\), i.e. there is a \(\zeta \in W(n)\) such that $$ W(n)=\left\\{\zeta^{\nu} ; \quad 0 \leq \nu