Complex Analysis/Exercises/Sheet 2

Examine the following functions on ${\displaystyle \mathbf {C} }$ for partial and complex differentiability! Specify the points where differentiability exists.

1. ${\displaystyle f_{1}\colon \mathbf {C} \to \mathbf {C} }$, ${\displaystyle z\mapsto z^{2}}$
2. ${\displaystyle f_{2}\colon \mathbf {C} \to \mathbf {C} }$, ${\displaystyle z\mapsto z{\bar {z}}}$
3. ${\displaystyle f_{3}\colon \mathbf {C} \to \mathbf {C} }$, ${\displaystyle z\mapsto \mathop {\mathrm {Re} } z}$
4. ${\displaystyle f_{4}\colon \mathbf {C} \to \mathbf {C} }$, ${\displaystyle z\mapsto {\begin{cases}0&z=0\ {\frac {|z|^{2}}{z^{2}+{\bar {z}}^{2}}}&z\neq 0\end{cases}}}$

Determine the partial derivatives with respect to ${\displaystyle z}$ and ${\displaystyle {\bar {z}}}$ for the functions from the first task at the points where they exist.

Solution to Exercise 3 We consider a polynomial ${\displaystyle p\colon \mathbf {C} \to \mathbf {C} }$, given by

${\displaystyle p(z)=\sum _{0\leq \kappa \leq k \atop 0\leq \lambda \leq \ell }a_{\kappa \lambda }x^{\kappa }y^{\lambda }}$

with ${\displaystyle x=\mathop {\rm {Re}} z}$ and ${\displaystyle y=\mathop {\rm {Im}} z}$. Show that ${\displaystyle p}$ can also be expressed as a polynomial in ${\displaystyle z}$ and ${\displaystyle {\bar {z}}}$ by specifying the coefficients in

${\displaystyle p(z)=\sum _{0\leq \mu \leq m \atop 0\leq \nu \leq n}b_{\mu \nu }z^{\mu }{\bar {z}}^{\nu }}$

.

Solution to Exercise 4 Let ${\displaystyle f,g\colon \mathbf {C} \to \mathbf {C} }$ be continuously differentiable. Prove that

${\displaystyle {\frac {\partial }{\partial z}}(f\circ g)={\frac {\partial f}{\partial z}}\circ g\cdot {\frac {\partial g}{\partial z}}+{\frac {\partial f}{\partial {\bar {z}}}}\circ g\cdot {\frac {\partial {\bar {g}}}{\partial z}}}$

and

${\displaystyle {\frac {\partial }{\partial {\bar {z}}}}(f\circ g)={\frac {\partial f}{\partial z}}\circ g\cdot {\frac {\partial g}{\partial {\bar {z}}}}+{\frac {\partial f}{\partial {\bar {z}}}}\circ g\cdot {\frac {\partial {\bar {g}}}{\partial {\bar {z}}}}}$

hold.

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:

• Source: Kurs:Funktionentheorie/Übungen/2. Zettel - URL:

https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Übungen/2._Zettel

• Date: 01/14/2024