Complex Analysis/Differences from real differentiability
n-times Real Differentiability
editThe function
- with ,
can be differentiated once. However, its first derivative is no longer differentiable at 0.
Task
edit- Sketch the graphs of the functions and .
- Can the function be extended to a holomorphic function , where (i.e., for all )? Justify your answer using the properties of holomorphic functions!
- Show that the function
- with ,
- can be differentiated times. However, the -th derivative is no longer differentiable at 0.
Remark
editIn complex analysis (Complex Analysis), one will see that a holomorphic function defined on is automatically infinitely often complex differentiable if it is complex differentiable once (seeHolomorphy Criteria.
See also
editTranslation and Version Control
editThis 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:
https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Unterschiede zur reellen Differenzierbarkeit
- Date: 12/17/2024