Let \(\Omega\subseteq\CC\) be open, let \(M\) and \(m\) be meromorphic on \(\Omega\), and let \(\gamma\) be a piecewise smooth toy contour in \(\Omega\) whose interior \(\Omega_\gamma\) satisfies \(\overline{\Omega_\gamma}\subseteq\Omega\). Assume that \(M\) and \(m\) have no poles on \(\gamma\) and that \[ \abs{m(z)}<\abs{M(z)}\quad\text{for all } z\in\gamma. \] For a meromorphic function \(h\), write \(Z_h\) and \(P_h\) for the numbers of zeros and poles of \(h\) in \(\Omega_\gamma\), counted with multiplicity. Then the winding numbers of \(M\circ\gamma\) and \((M+m)\circ\gamma\) about \(0\) agree, and \[ Z_M-P_M=Z_{M+m}-P_{M+m}. \] In particular, if \(M\) and \(m\) are holomorphic on \(\Omega\), then \(M\) and \(M+m\) have the same number of zeros in \(\Omega_\gamma\).
Rouché's theorem
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If $\abs{m}<\abs{M}$ on $\gamma$, then $M\circ\gamma$ and $(M+m)\circ\gamma$ have the same winding number about $0$, so $M$ and $M+m$ have the same number of zeros minus poles inside $\gamma$.Suggest a slogan