Complex Analysis/Exercises/Paper 1

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  

  1. Show that   is a field, i.e.:

    1.   is commutative and associative and has a neutral element  . Additionally, every element of   is invertible with respect to  .

    2.   is commutative and associative and has a neutral element  . Additionally, every element of   is invertible with respect to  .

    3. 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.

  2. 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  .

  1. Show that for all  :  .
  2. Show that  .
  3. Compute:   for the follwing equatios:
    1.  .
    2.  .
    3.  .

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

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Prove the following:

  1. For all  ,   and  .
  2. For all  ,   and  .
  3. For all  ,   and (if  )  .
  4. For all  ,   and (if  )  .
  5. 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

  • Date: 01/14/2024