The Symmetric Versions of Rouché’s Theorem via ∂--Calculus
Keyword(s):
Let (f,g) be a pair of holomorphic functions. In this expositional paper we apply the ∂--calculus to prove the symmetric version “|f+g|<|f|+|g| on ∂K” as well as the homotopic version of Rouché's theorem for arbitrary planar compacta K. Using Eilenberg's representation theorem we also give a converse to the homotopic version. Then we derive two analogs of Rouché's theorem for continuous-holomorphic pairs (a symmetric and a nonsymmetric one). One of the rarely presented properties of the non-symmetric version is that in the fundamental boundary hypothesis, |f+g|≤|g|, equality is allowed.
1976 ◽
Vol 83
(3)
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pp. 186-187
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2009 ◽
Vol 211
(2)
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pp. 329-335
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1969 ◽
Vol 16
(4)
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pp. 329-331
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Keyword(s):
2012 ◽
Vol 23
(4)
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pp. 816-847
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