Complex Analysis/Exercises/Sheet 1

Task (Calculations, 5 Points)

Calculate the following complex numbers:

1. ${\displaystyle (1+4i)\cdot (3+5i)}$
2. ${\displaystyle {\frac {1+4i}{3+5i}}}$
3. ${\displaystyle i^{2018}}$
4. ${\displaystyle (1+i)^{8}}$
5. ${\displaystyle (1+i)^{2018}}$

Task ("Writing", 5 Points)

In the lecture, it was discussed that ${\displaystyle \mathbf {C} }$ is a field. Prove the following arithmetic properties for complex numbers:

1. Associativity of multiplication: ${\displaystyle \forall z,v,w\in \mathbf {C}$ :(z\cdot v)\cdot w=z\cdot (v\cdot w)}
2. Distributivity of multiplication over addition: ${\displaystyle \forall z,v,w\in \mathbf {C}$ :(z+v)\cdot w=z\cdot w+v\cdot w}

Task (Equations, 5 Points)

Find all ${\displaystyle z\in \mathbf {C} }$ that satisfy the following equations:

1. ${\displaystyle (3+i)z+(3+4i)=4+5i}$
2. ${\displaystyle (2+i)z^{2}-5iz+i=3}$
3. ${\displaystyle z^{4}=-1}$

Task (Differentiation, 5 Points)

Check whether the following functions ${\displaystyle \mathbf {C} \to \mathbf {C} }$ are differentiable. If they are, provide their derivative.

1. ${\displaystyle f\colon \mathbf {C} \to \mathbf {C} ,z\mapsto z^{2}+2}$
2. ${\displaystyle g\colon \mathbf {C} \to \mathbf {C} ,z\mapsto \left|z\right|^{2}}$

Translation and Version Control

This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

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

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

• Date: 01/14/2024