Argument Principle and Rouche Theorem

Let/be the meromorphic (i.e., it is analytic throughout the domain except possibly for poles) in the domain interior to a positively oriented simple closed contour C with N number of zeros and P number of poles in the interior of C. Then the argument principle states that 

Rouche theorem. For two analytic functions/(z) and g z), inside and on a simple closed contour C, if y(z) g(z) at each point on C, then the number of zeros of fiz) is the same as the number of zeros of f z) + g(z), counting multiplicities inside the contour C. 

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