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. This theorem assumes that the contour ∂ K {\displaystyle \partial K} is simple, that is, without self-intersections. Rouché's theorem is an easy consequence of a stronger symmetric Rouché's theorem described below.
Usage The theorem is usually used to simplify the problem of locating zeros, as follows. Given an analytic function, we write it as the sum of two parts, one of which is simpler and grows faster than (thus dominates) the other part. We can then locate the zeros by looking at only the dominating part. For example, the polynomial z 5 + 3 z 3 + 7 {\displaystyle z^{5}+3z^{3}+7} has exactly 5 zeros in the disk | z | < 2 {\displaystyle |z|<2} since | 3 z 3 + 7 | ≤ 31 < 32 = | z 5 | {\displaystyle |3z^{3}+7|\leq 31<32=|z^{5}|} for every | z | = 2 {\displaystyle |z|=2} , and z 5 {\displaystyle z^{5}} , the dominating part, has five zeros in the disk.
Geometric explanation
It is possible to provide an informal explanation of Rouché's theorem. One popular, informal way to summarize this argument is as follows: If a person were to walk a dog on a leash around and around a tree, such that the distance between the person and the tree is always greater than the length of the leash, then the person and the dog go around the tree the same number of times. Let C be a closed, simple curve (i.e., not self-intersecting). By Jordan curve theorem, it delimits a region called its interior that must not be confused with its topological interior, empty in this context. Let h(z) = f(z) + g(z). If f and g are both holomorphic on the interior of C, then h must also be holomorphic on the interior of C. Then, with the conditions imposed above, the Rouche's theorem in its original (and not symmetric) form says that
Notice that the condition |f(z)| > |h(z) − f(z)| means that for any z, the distance from f(z) to the origin is larger than the length of h(z) − f(z), which in the following picture means that for each point on the blue curve, the segment joining it to the origin is larger than the green segment associated with it. Informally we can say that the blue curve f(z) is always closer to the red curve h(z) than it is to the origin. The previous paragraph shows that h(z) must wind around the origin exactly as many times as f(z). The index of both curves around zero is therefore the same, so by the argument principle, f(z) and h(z) must have the same number of zeros inside C.
Applications
Bounding roots Consider the polynomial z 2 + 2 a z + b 2 {\displaystyle z^{2}+2az+b^{2}} with a > b > 0 {\displaystyle a>b>0} . By the quadratic formula it has two zeros at − a ± a 2 − b 2 {\displaystyle -a\pm {\sqrt {a^{2}-b^{2}}}} . Rouché's theorem can be used to obtain some hint about their positions. Since
| z 2 + b 2 | ≤ 2 b 2 < 2 a | z | for all | z | = b , {\displaystyle |z^{2}+b^{2}|\leq 2b^{2}<2a|z|{\text{ for all }}|z|=b,}
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