Problem Set 1 - Complex Analysis
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Problem 1 (Field Structure of the Complex Numbers)
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We define the following links and .
Definition:for :
und
Show that is a field, i.e.:
is commutative and associative and has a neutral element . Additionally, every element of is invertible with respect to .
is commutative and associative and has a neutral element . Additionally, every element of is invertible with respect to .
For and distributive law holds.
Your proof should identify, and are and what to Inverse with respect to or. (in case ) the Inverse with respect to is.
show that ,the figure is an injective field homomorphism. This means is injective and satisfies:
and
Problem 2 (Arithmetic in the Complex Field)
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We now use und for the operations und auf defined in Problem 1.Additionally, for we simply write instead of .In this notation, .)Let .
- Show that for all : .
- Show that .
- Compute: for the follwing equatios:
- .
- .
- .
Problem 3 (Real and Imaginary Parts, Complex Conjugates, and Moduli)
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Prove the following:
- For all , and .
- For all , and .
- For all , and (if ) .
- For all , and (if ) .
- For all , .
Translation and Version Control
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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:
https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Übungen/1._Blatt