Complex Analysis/Exercises/Paper 1

Problem Set 1 - Complex Analysis

Problem 1 (Field Structure of the Complex Numbers)

We define $ℂ:={ℝ}^2=\{\left(\begin{array}{c}a\\ b\end{array}\right);a,b\in ℝ\}$ the following links $\oplus$ and $\odot$. Definition:for $\left(\begin{array}{c}a\\ b\end{array}\right),\left(\begin{array}{c}c\\ d\end{array}\right)\in ℂ$:
$\left(\begin{array}{c}a\\ b\end{array}\right)\oplus \left(\begin{array}{c}c\\ d\end{array}\right):=\left(\begin{array}{c}a+c\\ b+d\end{array}\right)\in ℂ$ und $\left(\begin{array}{c}a\\ b\end{array}\right)\odot \left(\begin{array}{c}c\\ d\end{array}\right):=\left(\begin{array}{c}ac-bd\\ ad+bc\end{array}\right)\in ℂ$

1. Show that $(ℂ,\oplus ,\odot )$ is a field, i.e.:

   1. $\oplus$ is commutative and associative and has a neutral element $0\in ℂ$. Additionally, every element of $ℂ$ is invertible with respect to $\oplus$.

   2. $\odot$ is commutative and associative and has a neutral element $1\in ℂ$. Additionally, every element of $ℂ\setminus \{0\}$ is invertible with respect to $\odot$.

   3. For $\oplus$ and $\odot$ distributive law holds.

   Your proof should identify,$0$ and $1$ are and what to $\left(\begin{array}{c}a\\ b\end{array}\right)\in ℂ$ Inverse with respect to $\oplus$ or. (in case $\left(\begin{array}{c}a\\ b\end{array}\right)\ne 0$) the Inverse with respect to $\odot$ is.

2. show that ,the figure $\varphi :ℝ\to ℂ,\varphi (a)=\left(\begin{array}{c}a\\ 0\end{array}\right)$ is an injective field homomorphism. This means $\varphi$is injective and satisfies:
   $\varphi (a+b)=\varphi (a)+\varphi (b)$ and $\varphi (a\cdot b)=\varphi (a)\cdot \varphi (b)\text{for all }a,b\in ℝ$
   

Problem 2 (Arithmetic in the Complex Field)

We now use $+$ und $\cdot$ for the operations $\oplus$ und $\odot$ auf $ℂ$ defined in Problem 1.Additionally, for$a\in ℝ$ we simply write $a$ instead of $\varphi (a)=\left(\begin{array}{c}a\\ 0\end{array}\right)$.In this notation, $ℝ\subset ℂ$.)Let $i:=\left(\begin{array}{c}0\\ 1\end{array}\right)\in ℂ$.

1. Show that for all $a,b\in ℝ$: $a+ib=\left(\begin{array}{c}a\\ b\end{array}\right)$.
2. Show that $i^2=-1$.
3. Compute: $x\in ℂ$ for the follwing equatios:
   1. $x^2=1$.
   2. $x^2=-1$.
   3. $x^4=1$.

Problem 3 (Real and Imaginary Parts, Complex Conjugates, and Moduli)

Prove the following:

1. For all $x,y\in ℂ$, $\text{Re}(x+y)=\text{Re}(x)+\text{Re}(y)$ and $\text{Im}(x+y)=\text{Im}(x)+\text{Im}(y)$.
2. For all $x\in ℂ$, $\text{Re}(x)=\frac{x+\overline{x}}{2}$ and $\text{Im}(x)=\frac{x-\overline{x}}{2i}$.
3. For all $x\in ℂ$, $x\cdot \overline{x}=|x{|}^2$ and (if $x\ne 0$) $\frac{1}{x}=\frac{\overline{x}}{|x{|}^2}$.
4. For all $x,y\in ℂ$, $|x\cdot y|=|x|\cdot |y|$ and (if $y\ne 0$) $|\frac{x}{y}|=\frac{|x|}{|y|}$.
5. For all $x,y\in ℂ$, $|x+y|\ge |x|+|y|$.

Translation and Version Control

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• Source: Kurs:Funktionentheorie/Übungen/1._Blatt - URL:

https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Übungen/1._Blatt

• Date: 01/14/2024

de:Kurs:Funktionentheorie/Übungen/1._Blatt