Complex Analysis/Exercises/Sheet 1

Exercise in Complex Analysis

Task (Calculations, 5 Points)

Calculate the following complex numbers:

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

Task ("Writing", 5 Points)

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

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

Task (Equations, 5 Points)

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

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

Task (Differentiation, 5 Points)

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

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

Translation and Version Control

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

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

• Date: 01/14/2024