Zeros, Singularities, Residues: Rouche's Theorem
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The following theorem is closely related to the argument principle.
Rouche's Theorem: Let $f(z)$ and $g(z)$ be analytic inside and on a piecewise smooth simple closed curve $C$. Suppose that

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Then, the functions $f$ and $g$ have the same number of zeros inside the curve $C$.

Note: In the Rouche's Theorem, the strict inequality in MATH for $z \in C$ is important. If equality holds for some point on $C$, then the conclusion of Rouche's theorem is need not be true.

The following weaker version of Rouches thoerem was discovered by Irving Clicksberg in 1976:
Rouche's Theorem (Weaker Version): Let $f(z)$ and $g(z)$ be meromorphic inside a circle MATH. Further $f(z)$ and $g(z)$ are analytic and nonzero on $C$. If $Z_{f}$, $Z_{g}$ and $P_{f}$, $P_{g}$ denote the zeros and the poles of $f$ and $g$ respectively inside the curve $C$ counted according to their multipliciites and if MATH then, MATH.

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