I am facing a problem where I want to find the complex roots of $f(z)=z-sin(z)=0$ numerically.
There are infinitely many roots of the function, but I am only interested in the $N$ closest to the origin, or to put it more general, within a certain domain in the complex plane.
The Newton-Raphson method also works on complex functions, so given some initial $z_0$, I end up in some root of $f(z)$. However, this is not giving me all roots. I know that the roots will have a basin of attraction, so I will basically need a $z_0$ in every basin of attraction, except that I don't have this basin to start with. Smartest I can come up with, is starting with a grid of $z_0$'s, and to select the uniques. This seems very inefficient.
Therefor my question: is there an efficient method to find all complex roots of $f(z)$ within a domain of the complex plane?