K. Koca and A. O. Çelebi

A Representation of Solutions for a System of Complex Differential Equations in the Plane and Periodic Solutions

abstract:
In this article, first we will obtain a representation of the solutions for the system of complex differential equations
\begin{align*}
w_z &= A\zzb w\\
w_{\ciz z} &= B\zzb w,\quad A,B\in\cog,
\end{align*}
which are defined in a simply-connected domain $G\subset\C$ containing $z_0=0$ and satisfying the functional relations
\[
w\left(z_1+z_2\right) = w\left(z_1\right)+w\left(z_2\right),\;
w\left(0\right)=1;\quad z_1,z_2,z_1+z_2\in G.
\]
Then we will discuss the conditions under which the solutions of the system are periodic.

Mathematics Subject Classification: 30D05

Key words and phrases: Functional equation, ordinary differential equation, exponential representation