Complex Analysis/Exercises/Sheet 2/Exercise 4

Problem (Chain Rule, 5 Points)

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Let be continuously differentiable functions. Prove that

and

apply.

Solution

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We recall (Fischer/Lieb, Page 21 at the bottom): For a differentiable function , the partial derivatives with respect to and are characterized as follows: Let be continuous functions such that

so is and .

We will use this description of the Wirtinger derivatives. Let . There in is differentiable , we have continuous functions so that

applies.This means

Now set

Da is differentiable ,there exist continuous functions , so,that

if we insert, this results in

Da and of cotinous functions are continuous,is partially differentiable and

Continuing above,this lead to

and claimed. Analogously follows

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/Aufgabe_4

  • Date: 01/14/2024