Complex Analysis/Exercises/Sheet 2

Exercise on Complex Analysis

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Task (Differentiability, 5 Points)

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Examine the following functions on   for partial and complex differentiability! Specify the points where differentiability exists.

  1.  ,  
  2.  ,  
  3.  ,  
  4.  ,  

Task (Wirtinger, 5 Points)

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Determine the partial derivatives with respect to   and   for the functions from the first task at the points where they exist.

Task (Working with Polynomials, 5 Points)

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Solution to Exercise 3 We consider a polynomial  , given by

 

with   and  . Show that   can also be expressed as a polynomial in   and   by specifying the coefficients in

 

.

Task (Chain Rule, 5 Points)

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Solution to Exercise 4 Let   be continuously differentiable. Prove that

 

and

 

hold.

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/2._Zettel

  • Date: 01/14/2024