52.1.2 Rouche’s Theorem
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 ∑
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
Rouche’s theorem allows the comparison of Zh − Ph for h = f,g. It is a wonderful and
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
, no zeros or poles are on γ∗, and for all z ∈
either equals 0 or 1. Let Zf and Pf denote respectively the numbers of zeros and
poles of f, which have the property that the winding number equals 1, counted
according to order, with Zg and Pg being defined similarly. Also suppose that for
z ∈ γ∗
Proof: From the hypotheses,
which shows that for all z ∈ γ∗,
Letting l denote a branch of the logarithm defined on ℂ ∖ [0,∞), it follows that l
a primitive for the function,
Therefore, by the argument principle,
This proves the theorem.
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
for z ∈ γ∗. Then the conclusion is the same.
Proof: The new condition implies
0] and so you can do the same argument with a branch of the logarithm.