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Let $P(z)={\prod }_{\ell }(z-r_{\ell })$ have distinct roots. Choose a disc around $r_j$ with radius $\delta$ smaller than every separation from $r_j$. On its boundary, $|P(z)|\ge \delta {\prod }_{\ell \ne j}(|r_j-r_{\ell }|-\delta )$. If a holomorphic perturbation $H$ satisfies the displayed inequality, Rouché's theorem shows that $P+H$ has exactly one root in that disc, counted with multiplicity. Discs confined to specified half-planes or quadrants give direct geometric root-location proofs without needing a global root-continuation argument.

By Jeremy Orloff, named on the first page of the notes for MIT 18.04, Complex Variables with Applications, Spring 2018. The credit here refers to this authored set of notes.

Official course resource: ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/resources/mit18_04s18_topic11/

Online text: ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/44f1db513a6a17d655abe0b6ff7748fc_MIT18_04S18_topic11.pdf

Cited passages: §11.2, Theorems 11.1 and 11.4, pp. 1–3, gives the argument principle and its residue theorem explanation; §11.2.3 covers Rouché's theorem; §11.3.6 explains the Nyquist stability criterion, including the sign from the clockwise contour used there. Our rational-function and sparse-sampling examples and images are independently calculated.