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 $𝐂$ is a field. Prove the following arithmetic properties for complex numbers:

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

Task (Equations, 5 Points)

Find all $z\in 𝐂$ 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 $𝐂\to 𝐂$ are differentiable. If they are, provide their derivative.

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

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• Date: 01/14/2024

de:Kurs:Funktionentheorie/Übungen/1. Zettel