Rouche's Theorem

Rouche's theorem is a fundamental result for locating zeros of holomorphic functions. It states that a "small" perturbation of a holomorphic function does not change the number of zeros inside a contour.

Statement

Theorem6.8Rouche's theorem (restated)

Let $f$ and $g$ be holomorphic inside and on a simple closed contour $\gamma$ , with $|g(z)| < |f(z)|$ on $\gamma$ . Then $f$ and $f + g$ have the same number of zeros inside $\gamma$ , counted with multiplicity.

RemarkSymmetric form

An equivalent symmetric formulation: if $|f(z) - g(z)| < |f(z)| + |g(z)|$ on $\gamma$ , then $f$ and $g$ have the same number of zeros inside $\gamma$ . This form is sometimes easier to verify.

Proof via the Argument Principle

Proof

Consider the homotopy $h_t(z) = f(z) + tg(z)$ for $t \in [0,1]$ . On $\gamma$ , $|h_t(z)| \geq |f(z)| - t|g(z)| > |f(z)| - |f(z)| = 0$ , so $h_t$ has no zeros on $\gamma$ for any $t \in [0,1]$ .

By the argument principle, the number of zeros of $h_t$ inside $\gamma$ is

$N(t) = \frac{1}{2\pi i}\oint_\gamma \frac{h_t'(z)}{h_t(z)}\,dz.$

Since $h_t$ depends continuously on $t$ and is nonvanishing on $\gamma$ , the integrand depends continuously on $t$ . Therefore $N(t)$ is a continuous integer-valued function, hence constant.

In particular, $N(1) = N(0)$ , i.e., $f + g$ and $f$ have the same number of zeros. $\blacksquare$

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Applications

ExampleZeros of a polynomial

Determine the number of zeros of $p(z) = z^7 - 5z^3 + 12$ in the annulus $1 < |z| < 2$ .

On $|z| = 1$ : $|12| = 12 > |z^7 - 5z^3| \leq 1 + 5 = 6$ . By Rouche, $p$ and $12$ have the same number of zeros in $|z| < 1$ : zero zeros.

On $|z| = 2$ : $|z^7| = 128 > |-5z^3 + 12| \leq 40 + 12 = 52$ . By Rouche, $p$ and $z^7$ have the same number of zeros in $|z| < 2$ : seven zeros.

Therefore $p$ has $7 - 0 = 7$ zeros in $1 < |z| < 2$ .

ExamplePerturbation of zeros

For small $\varepsilon > 0$ , the polynomial $p_\varepsilon(z) = z^n + \varepsilon q(z)$ (where $\deg q < n$ ) has all $n$ zeros near the origin. On $|z| = r$ for any fixed $r > 0$ , $|z^n| = r^n$ while $|\varepsilon q(z)| \leq \varepsilon C r^{n-1}$ . For $\varepsilon < r/C$ , Rouche applies, showing $p_\varepsilon$ has $n$ zeros in $|z| < r$ . As $\varepsilon \to 0$ , all zeros converge to the origin.

Hurwitz's Theorem

Theorem6.9Hurwitz's theorem

Let $\{f_n\}$ be a sequence of holomorphic functions on a domain $D$ converging uniformly on compact subsets to $f$ , where $f$ is not identically zero. If $f(z_0) = 0$ with multiplicity $m$ , then for sufficiently large $n$ , $f_n$ has exactly $m$ zeros (counted with multiplicity) in any sufficiently small disk around $z_0$ .

In particular, if each $f_n$ is nonvanishing on $D$ and $f_n \to f$ uniformly on compacts, then either $f \equiv 0$ or $f$ is also nonvanishing on $D$ .

RemarkSignificance of Hurwitz's theorem

Hurwitz's theorem is crucial in the proof of the Riemann mapping theorem: it ensures that the limit of injective holomorphic functions is either injective or constant. Combined with normal families theory, it provides the key tool for extracting convergent subsequences with the desired properties.