Let $$f= z+ie^{iz^*}$$ where $z=x+iy$, and $*$ means complex conjugate. This function is the sum of one holomorphic function and one anti-holomorphic function. Geometrically, these curves, for fixed $y$, are trochoids.
For $y\le 0$ I have single valued curves for fixed $y$. I would like to extend those curves into the region $y>0$. Specifically, the function $$g=z+e^{iz}$$ has the geometry I am looking for. However, I would like to write $g$ in terms of one function of $z$ and one of $z^*$. Is there any way to do this? This seems vaguely related to the Schwarz function.
That is, can I extend this function $f$, to regions where $y>0$, in a way so that it remains the sum of a holomorphic and anti-holomorphic function?