Rouché's theorem

For the theorem in linear algebra, see Rouché–Capelli theorem.

Rouché's theorem, named after Eugène Rouché, states that if the complex-valued functions f and g are holomorphic inside and on some closed contour K, with |g(z)| < |f(z)| on K, then f and f + g have the same number of zeros inside K, where each zero is counted as many times as its multiplicity. This theorem assumes that the contour K is simple, that is, without self-intersections.

Symmetric version

Theodor Estermann (1902–1991) proved in his book Complex Numbers and Functions the following relation: Let $K\subset G$ be a bounded region with continuous boundary $\partial K$. Two holomorphic functions $f,\,g\in\mathcal H(G)$ have the same number of roots in $K$, if the strict inequality

$|f(z)-g(z)|<|f(z)| + |g(z)| \qquad \left(z\in \partial K\right)$

holds on the boundary $\partial K$.

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 $z^5 + 3z^3 + 7$ has exactly 5 zeros in the disk $|z| < 2$ since $|3z^3 + 7| < 32 = |z^5|$ for every $|z| = 2$, and $z^5$, the dominating part, has five zeros in the disk.

Geometric explanation

Since the distance between the curves is small, h(z) does exactly one turn around just as f(z) does.

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

First we need to slightly rephrase the theorem. Let h(z) = f(z) + g(z). If f and g are both holomorphic, then h must also be holomorphic. Then, with the conditions imposed above, the Rouche's theorem in its original (and not symmetric) form says that

If |f(z)| > |h(z) − f(z)| then f(z) and h(z) have the same number of zeros in the interior of C.

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.

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, and if the length of the leash is less than the minimum radius of the walk, then the person and the dog go around the tree an equal number of times. (Indeed, one may see that the converse of Rouche's theorem is false, insofar as the leash need only be less than the minimum circumference of the walk.)

Applications

Rouché's theorem can be used to give a short proof of the Fundamental Theorem of Algebra. Let

$p(z) = a_0 + a_1z + a_2 z^2 + \cdots + a_n z^n, \,$

and choose $R>1$ and so large that:

$|a_0 + a_1z + \cdots + a_{n-1}z^{n-1}| \le \sum_{j=1}^n |a_j| R^{n-1} < |a_n|R^n = |a_n z^n|\text{ for }|z| = R.$

Since $a_n z^n$ has $n$ zeros inside the disk $|z| < R$ (because $R>1$), it follows from Rouché's theorem that $p$ 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. We refer to the article for the proof.

Proof of symmetric form of Rouché's theorem

The hypothesis ensures both that $f$ and $g$ do not have any roots on the boundary $\partial K$ and that $\frac {f(z)}{g(z)}$ is not a negative real number for $z \in \partial K$. Thus the homotopy

$I(t):=\frac 1 {2 \pi i} \oint_{\partial K} \frac{F'(z)}{F(z)+t}dz$

is well defined for $t\ge 0$, where $F(z):=\frac {f(z)}{g(z)}$.

Clearly, $I(t) \to 0$ as $t \to \infty$. As $I(t)$ is continuous and integer valued, it follows that $I(0)=0$. By the argument principle, this winding number is given by

$0 = {1\over 2\pi i}\oint_{\partial K} {F'(z) \over F(z)}\,dz=N_F(K)-P_F(K)$

where N_F(K) is the number of zeroes of F inside K, P_F(K) is the number of poles inside K. Hence N_F = P_F. But F is the ratio of two holomorphic functions f and g inside K, and so the zeroes are those of f and the poles are the zeros of g (at least if f and g are coprime polynomials, which we can assume since otherwise, we are free to divide f and g by any common divisor). That is,

$0=N_F(K) - P_F(K) = N_{f}(K) - N_g(K),\,$

as required.

See also

• Hurwitz's theorem (complex analysis)
• Sturm's theorem
• Rational root theorem
• Properties of polynomial roots

References

• Beardon, Alan (1979). Complex Analysis: the Winding Number principle in analysis and topology. John Wiley and Sons. p. 131. ISBN 0-471-99672-6. 
• Titchmarsh, E. C. (1939). The Theory of Functions (2nd ed.). Oxford University Press. pp. 117–119,198–203. ISBN 0-19-853349-7. 

External link

Module for Rouche’s Theorem by John H. Mathews