Elementary area

An area called elementary region (also stem area ) if and only if each of holomorphic function has a master function, that is, on applies the statement of the integral set of Cauchy . ${\displaystyle D\subseteq \mathbb {C} }$ ${\displaystyle D}$ ${\displaystyle D}$

characterization

The following characterizations apply to an elementary area : ${\displaystyle D}$

• ${\displaystyle D}$is simply connected , that is, every closed curve in is nullhomotop, that is, it is continuously contractible to the starting point. This clearly means that it has no holes. ${\displaystyle D}$ ${\displaystyle D}$

• ${\displaystyle D}$is homologous simply connected, that is, every cycle in is zero homologous, that is, the interior of the cycle is completely in . ${\displaystyle D}$ ${\displaystyle D}$

• ${\displaystyle D}$is conformally equivalent to whole or to the unit disk , that is, there is a biholomorphic mapping of to or to , compare: Riemann's mapping theorem . ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {E} }$ ${\displaystyle D}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {E} }$

properties

• If and are elementary areas whose intersection is connected and not empty, then is also an elementary area. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A\cup B}$
• If there is a sequence of elementary areas for which applies, then is also an elementary area. ${\displaystyle (A_{i})_{i\in \mathbb {N} }}$ ${\displaystyle A_{1}\subset A_{2}\subset A_{3}\subset \ldots }$ ${\displaystyle \cup _{i=1}^{\infty }A_{i}}$

With these two operations all elementary areas can be generated from circular disks.

example

The following areas are elementary areas:

• ${\displaystyle \mathbb {C} }$ and ${\displaystyle \mathbb {E} }$
• each star region
• the slotted plane ${\displaystyle \mathbb {C} \setminus \{z\in \mathbb {R} :z\leq 0\}}$

The following area is not an elementary area:

• ${\displaystyle \mathbb {C} \setminus \{0\}}$

literature

• Eberhard Freitag, Rolf Busam: Function theory 1, Springer-Verlag, Berlin, ISBN 3-540-67641-4