Funktionentheorie/Gemischte Definitionsabfrage/9/Aufgabe/Lösung

${\displaystyle {}U(a,r,R)=U{\left(a,R\right)}\setminus B\left(a,r\right)\,,}$

mit ${\displaystyle {}a\in {\mathbb {C} }}$ und ${\displaystyle {}0\leq r<R}$ reelle Zahlen.

${\displaystyle {}f}$

${\displaystyle {}{\frac {\partial f}{\partial {\overline {z}}}}:={\frac {1}{2}}\cdot {\frac {\partial f}{\partial x}}+{\frac {\mathrm {i} }{2}}\cdot {\frac {\partial f}{\partial y}}\,.}$

${\displaystyle {}\{0,1\}}$

${\displaystyle {}\gamma _{1}}$

${\displaystyle {}\gamma _{2}}$

${\displaystyle H\colon I\times I\longrightarrow X,}$

die die folgenden Eigenschaften erfüllt.

1. ${\displaystyle {}H(s,0)=\gamma _{1}(s)}$ für alle ${\displaystyle {}s\in I}$.
2. ${\displaystyle {}H(s,1)=\gamma _{2}(s)}$ für alle ${\displaystyle {}s\in I}$.
3. ${\displaystyle {}H(0,t)=\gamma _{1}(0)}$ für alle ${\displaystyle {}t\in I}$.
4. ${\displaystyle {}H(1,t)=\gamma _{1}(1)}$ für alle ${\displaystyle {}t\in I}$.

${\displaystyle {}\Gamma }$

${\displaystyle {}k}$

${\displaystyle {}G_{k}(\Gamma ):=\sum _{z\in \Gamma '}{\frac {1}{z^{k}}}\,}$