If two holomorphic functions on and inside a closed contour satisfy the inequality that one is everywhere strictly smaller in absolute value than the difference between the sum and that function, they have the same number of zeros inside the contour, counted with multiplicity. Named for Eugene Rouche, it is a practical tool for locating zeros of complex functions.
Facts
StatementFor two functions f and g holomorphic inside a region K with closed contour boundary, if the modulus of g is less than the modulus of f everywhere on the boundary, then f and f+g have the same number of zeros, counted with multiplicity, inside K. 1 Classification
Statement Form Connections
Has Statement Form
In Branch
Principal Result Of
Source Eugène Rouché (Wikipedia)
Proved By
Source Rouche's Theorem (Wikipedia)
Sources
1. Rouche's Theorem (Wikipedia)
Wikimedia FoundationLead section, statement sentenceQuote, Lead section, statement sentence
Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions f and g holomorphic inside some region K with closed contour ∂K, if |g(z)| < |f(z)| on ∂K, then f and f + g have the same number of zeros inside K, where each zero is counted as many times as its multiplicity.
Eugène Rouché (Wikipedia)
Principal Result Of: Eugène Rouché, Career, Rouche theoremQuote, Principal Result Of: Eugène Rouché, Career, Rouche theorem
He is best known for Rouché's theorem in complex analysis
Rouche's Theorem (Wikipedia)
Proved By: Eugène Rouché, Lead sentenceQuote, Proved By: Eugène Rouché, Lead sentence
Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions f and g holomorphic inside some region K {\displaystyle K} with closed contour ∂ K {\displaystyle \partial K} , if |g(z)| < |f(z)| on ∂ K {\displaystyle \partial K} , then f and f + g have the same number of zeros inside K {\displaystyle K} , where each zero is counted as many times as its multiplicity.
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