With the argument principle, it is possible to prove Rouche’s theorem . In the argument principle, denote by Zf the quantity ∑ k=1mrk and by Pf the quantity ∑ k=1nlk. Thus Zf is the number of zeros of f counted according to the order of the zero with a similar definition holding for Pf. Thus the conclusion of the argument principle is.
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Rouche’s theorem allows the comparison of Zh − Ph for h = f,g. It is a wonderful and amazing result.
Theorem 52.1.4 (Rouche’s theorem)Let f,g be meromorphic in an open set Ω. Also suppose γ∗ is a closed bounded variation curve with the property that for all z
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Then
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Proof: From the hypotheses,
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which shows that for all z ∈ γ∗,
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Letting l denote a branch of the logarithm defined on ℂ ∖ [0,∞), it follows that l
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Therefore, by the argument principle,
Often another condition other than 52.1.4 is used.
Corollary 52.1.5 In the situation of Theorem 52.1.4 change 52.1.4 to the condition,
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for z ∈ γ∗. Then the conclusion is the same.
Proof: The new condition implies