Complex Analysis/Exercises/Sheet 2

Exercise on Complex Analysis

Task (Differentiability, 5 Points)

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

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

Task (Wirtinger, 5 Points)

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

Task (Working with Polynomials, 5 Points)

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

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

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

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

.

Task (Chain Rule, 5 Points)

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

{\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

{\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.

Translation and Version Control

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