I have some function $F(\omega): \mathbb R\to\mathbb C$. The function $F(\omega)$ has both roots and singularities. Fortunately, I can calculate positions of singularities analytically.
So my problem is to find roots inside domain between singularities. Here I have difficulty, because I cannot estimate if there is one, many or none roots inside the domain. Now I scan $\omega$ through the domain for change of simultaneous change of real and imaginary part of $F(\omega)$, what is both inaccurate and ineffective.
If there some method to estimate number of roots inside a domain?
I don't know if this is a right place to post questions about computational math, so don't hesitate to show me the right place to post :)