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I heard of a method called "argument variation" or "variation of argument" for finding the number of zeroes of a complex polynomial within a simple contour.

On Google, the only source I find is the Mathworld article, but it isn't very accessible to me, a physics undergraduate.

Where can I find an introduction or examples for this online? Most other concepts of complex analysis, e.g. Möbius transforms, have lots of resources, so I'm sure they're there somewhere. Maybe the technique is better known under a different name?

Edit: I'll add that what I don't understand is how real and imaginary lines are handled, equations 9 and 10 in the Mathworld article.

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Try googling for "argument principle". – Hans Lundmark Aug 17 '11 at 10:13

Perhaps you are looking for the Argument Principle. This says that if f is a meromorphic function in some region, and there is a closed contour in this region s.t. f has no zeroes or poles on this contour, then we have that

$$\oint_C \frac{f'(z)}{f(z)} \text{d}z = 2 \pi i (N - P)$$

Where N and P are the number of zeroes and poles, respectively, inside the contour. This is related to Rouche's Theorem, which might also suit your needs.

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