Fixed points of holomorphic systems

shawsend · #1 $Old$ December 2nd, 2008, 08:23 AM

Can any general statements be made about the fixed points of coupled systems of holomorphic functions?

Consider the system:

$z'(t)=f(z);\quad z\in \mathbb{C}$

where $f(z)$ is holomorphic.

For example:

$z'(t)=z\Rightarrow \begin{cases} x'(t)=x \\ y'(t)=y\end{cases}$

or:

$z'(t)=\sin(z)\Rightarrow \begin{cases} x'(t)=\textbf{Re}(\sin z) \\ y'(t)=\textbf{Im}(\sin z)\end{cases}$

Can I predict which types of fixed points I would encounter with these systems strictly on the analytic properties of $f(z)$ ?

Rebesques · #2 $Old$ May 5th, 2009, 06:39 PM

...Something like $\int_{\gamma(z_0,z)}f$ maybe?