Complex Analysis/Differences from real differentiability

n-times Real Differentiability

The function

f:\mathbb {R} \rightarrow \mathbb {R} ,

with

x\mapsto f(x)=|x|\cdot x

,

can be differentiated once. However, its first derivative is no longer differentiable at 0.

Task

Sketch the graphs of the functions $f$ and $f'$ .

Can the function $f:\mathbb {R} \rightarrow \mathbb {R}$ be extended to a holomorphic function $F:\mathbb {C} \rightarrow \mathbb {C}$ , where $F_{|\mathbb {R} }=f$ (i.e., $f(x)=F(x)$ for all $x\in \mathbb {R}$ )? Justify your answer using the properties of holomorphic functions!

Show that the function

f_{n}:\mathbb {R} \rightarrow \mathbb {R} ,

with

x\mapsto f(x)=|x|\cdot x^{n}

,

can be differentiated

n

times. However, the

n+1

-th derivative is no longer differentiable at 0.

Remark

In complex analysis (Complex Analysis), one will see that a holomorphic function $f:U\to \mathbb {C}$ defined on $U\subseteq \mathbb {C}$ is automatically infinitely often complex differentiable if it is complex differentiable once (seeHolomorphy Criteria.

Page Information

You can display this page as real%20differentiability&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Differences%20from%20real%20differentiability&coursetitle=Complex%20Analysis Wiki2Reveal slides

Wiki2Reveal

The%20differentiability&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Differences%20from%20real%20differentiability&coursetitle=Complex%20Analysis Wiki2Reveal slides were created for the Complex Analysis' and the Link for the Wiki2Reveal Slides was created with the link generator.

from real differentiability This page is designed as a PanDocElectron-SLIDE document type.

Translation and Version Control

This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Unterschiede zur reellen Differenzierbarkeit Wikiversity source page] and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

Source: Kurs:Funktionentheorie/Unterschiede zur reellen Differenzierbarkeit - URL:

https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Unterschiede zur reellen Differenzierbarkeit

Date: 12/17/2024