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{"type":"standard","title":"Rouché's theorem","displaytitle":"Rouché's theorem","namespace":{"id":0,"text":""},"wikibase_item":"Q1191862","titles":{"canonical":"Rouché's_theorem","normalized":"Rouché's theorem","display":"Rouché's theorem"},"pageid":1141150,"thumbnail":{"source":"https://upload.wikimedia.org/wikipedia/commons/thumb/3/33/Gamma_abs_3D.png/330px-Gamma_abs_3D.png","width":320,"height":250},"originalimage":{"source":"https://upload.wikimedia.org/wikipedia/commons/3/33/Gamma_abs_3D.png","width":1280,"height":1000},"lang":"en","dir":"ltr","revision":"1289162085","tid":"4c418435-2abe-11f0-b8ed-8d720f38c9d5","timestamp":"2025-05-06T21:08:46Z","description":"Theorem about zeros of holomorphic functions","description_source":"local","content_urls":{"desktop":{"page":"https://en.wikipedia.org/wiki/Rouch%C3%A9's_theorem","revisions":"https://en.wikipedia.org/wiki/Rouch%C3%A9's_theorem?action=history","edit":"https://en.wikipedia.org/wiki/Rouch%C3%A9's_theorem?action=edit","talk":"https://en.wikipedia.org/wiki/Talk:Rouch%C3%A9's_theorem"},"mobile":{"page":"https://en.m.wikipedia.org/wiki/Rouch%C3%A9's_theorem","revisions":"https://en.m.wikipedia.org/wiki/Special:History/Rouch%C3%A9's_theorem","edit":"https://en.m.wikipedia.org/wiki/Rouch%C3%A9's_theorem?action=edit","talk":"https://en.m.wikipedia.org/wiki/Talk:Rouch%C3%A9's_theorem"}},"extract":"Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions f and g holomorphic inside some region with closed contour , if |g(z)| < |f(z)| on , then f and f + g have the same number of zeros inside , where each zero is counted as many times as its multiplicity. This theorem assumes that the contour is simple, that is, without self-intersections. Rouché's theorem is an easy consequence of a stronger symmetric Rouché's theorem described below.","extract_html":"

Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions $f$ and $g$ holomorphic inside some region $\text{[math]}$ with closed contour $\text{[math]}$ , if $| g (z)| < | f (z)|$ on $\text{[math]}$ , then $f$ and $f + g$ have the same number of zeros inside $\text{[math]}$ , where each zero is counted as many times as its multiplicity. This theorem assumes that the contour $\text{[math]}$ is simple, that is, without self-intersections. Rouché's theorem is an easy consequence of a stronger symmetric Rouché's theorem described below.

"}

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