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.

See also

• Complex Analysis

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