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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 :)

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    @galadog no, the contour must be a closed loop in the plane. That integral calculates the number of zeros inside the loop.2012-06-01

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See Computing the Zeros of Analytic Functions for theory and code.

Despite the title, it handles computing zeros and poles of meromorphic functions.

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    @OldJohn, right, I just assumed the function could be extended to the complex plane.2012-06-01