Complex Analysis/Exercises/Sheet 2

Exercise on Complex Analysis

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

$f_{1} : 𝐂 \to 𝐂$ , $z \mapsto z^{2}$
$f_{2} : 𝐂 \to 𝐂$ , $z \mapsto z \bar{z}$
$f_{3} : 𝐂 \to 𝐂$ , $z \mapsto R e z$
$f_{4} : 𝐂 \to 𝐂$ , $z \mapsto {\begin{matrix} 0 & z = 0 \frac{| z |^{2}}{z^{2} + {\bar{z}}^{2}} & z \neq 0 \end{matrix}$

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

Solution to Exercise 3 We consider a polynomial $p : 𝐂 \to 𝐂$ , given by

p (z) = \sum_{\binom{0 \leq κ \leq k}{0 \leq λ \leq ℓ}} a_{κ λ} x^{κ} y^{λ}

with $x = R e z$ and $y = I m z$ . Show that $p$ can also be expressed as a polynomial in $z$ and $\bar{z}$ by specifying the coefficients in

p (z) = \sum_{\binom{0 \leq μ \leq m}{0 \leq ν \leq n}} b_{μ ν} z^{μ} {\bar{z}}^{ν}

.

Solution to Exercise 4 Let $f, g : 𝐂 \to 𝐂$ 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.

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