Rouché's theorem

Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions f and g holomorphic inside some region

, where each zero is counted as many times as its multiplicity.

Rouché's theorem is an easy consequence of a stronger symmetric Rouché's theorem described below.

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.

, the dominating part, has five zeros in the disk.

It is possible to provide an informal explanation of Rouché's theorem.

Let C be a closed, simple curve (i.e., not self-intersecting).

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.

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. 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.

Rouché's theorem can be used to obtain some hint about their positions.

Rouché's theorem says that the polynomial has exactly one zero inside the disk

This sort of argument can be useful in locating residues when one applies Cauchy's residue theorem.

also has the same number of zeros inside the disk.

One advantage of this proof over the others is that it shows not only that a polynomial must have a zero but the number of its zeros is equal to its degree (counting, as usual, multiplicity).

Another use of Rouché's theorem is to prove the open mapping theorem for analytic functions.

A stronger version of Rouché's theorem was published by Theodor Estermann in 1962.

be a bounded region with continuous boundary

have the same number of roots (counting multiplicity) in

The original version of Rouché's theorem then follows from this symmetric version applied to the functions

always holds by the triangle inequality, this is equivalent to saying that

never pass through the origin and never point in the same direction as

must wind around the origin the same number of times.

be a simple closed curve whose image is the boundary

, hence by the argument principle, the number Nf(K) of zeros of f in K is

i.e., the winding number of the closed curve

around the origin; similarly for g. The hypothesis ensures that g(z) is not a negative real multiple of f(z) for any z = C(x), thus 0 does not lie on the line segment joining f(C(x)) to g(C(x)), and

The winding number is homotopy-invariant: the function