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Math Help - Fixed points of holomorphic systems

  1. #1
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    Fixed points of holomorphic systems

    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)?
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  2. #2
    Super Member Rebesques's Avatar
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    ...Something like \int_{\gamma(z_0,z)}f maybe?
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