Argument principle, Rouché's theorem, open mapping

In one sentence

The argument principle counts zeros minus poles by watching how far the image winds, Rouché turns that count into a comparison with a simpler function, and the open mapping theorem is the structural fact underneath both.

Why the exam asks it

Because "how many roots lie in this region" is a question with no elementary method and a two-line complex-analytic one, and because Rouché's hypothesis — a strict inequality holding on the whole contour — is exactly the sort of condition an option can weaken without looking different.

The idea

The argument principle

Let $f$ be meromorphic inside and on a simple closed contour $\gamma$, with no zeros or poles on $\gamma$. Then

$$\frac{1}{2\pi i}\oint_\gamma \frac{f'(z)}{f(z)}\,dz = Z - P,$$

the number of zeros minus poles inside, each counted with multiplicity.

The mechanism is worth seeing once. Near a zero of order $m$, $f(z) = (z-a)^m g(z)$ with $g(a)\ne0$, so

$$\frac{f'}{f} = \frac{m}{z-a} + \frac{g'}{g},$$

and the second term is holomorphic. So $f'/f$ has a simple pole with residue exactly $m$ at every zero of order $m$, and residue $-m$ at every pole of order $m$. The residue theorem does the rest.

The name comes from the other reading: $\oint f'/f$ is the total change in $\log f$, whose imaginary part is the change in $\arg f$. Dividing by $2\pi$ gives the number of times $f\circ\gamma$ winds around the origin. Counting zeros and watching the image wind are the same computation.

Rouché

If $|g| < |f|$ everywhere on $\gamma$, then $f$ and $f+g$ have the same number of zeros inside.

The picture is the proof: walking a dog around a lamp post, on a lead shorter than the distance to the post, the dog circles the post exactly as often as you do. $f$ is the walker, $f+g$ the dog, $|g|$ the lead.

Two hypotheses are doing work and both get removed in options.

Strict inequality. $|g| \le |f|$ is not enough: take $f(z) = z$ and $g(z) = -z$ on $|z|=1$, where $|g| = |f|$ and $f+g \equiv 0$ has infinitely many zeros.

On the whole contour. The inequality must hold at every point of $\gamma$, not merely somewhere or on average.

The standard use is bracketing roots by annulus. For $p(z) = z^5 + 3z + 1$:

• On $|z| = 2$: $|z^5| = 32$ and $|3z+1| \le 7$. Take $f = z^5$; all $5$ roots lie in $|z|<2$.
• On $|z| = 1$: $|3z| = 3$ and $|z^5 + 1| \le 2$. Take $f = 3z$; exactly $1$ root lies in $|z|<1$.

So four roots sit in the annulus $1 < |z| < 2$. Notice the dominant term changes between the two circles — choosing which term plays $f$ is the whole skill.

Rouché also gives the fundamental theorem of algebra: on a large enough circle the leading term dominates the rest, so a degree-$n$ polynomial has exactly $n$ roots.

Open mapping and its consequences

A non-constant holomorphic function on a domain maps open sets to open sets.

This is why $|f|$ can have no interior maximum: the image of a neighbourhood of $a$ is a neighbourhood of $f(a)$, so it contains points of larger modulus. The maximum modulus principle is a corollary rather than a separate fact.

Two more consequences:

• A holomorphic function that is injective has $f' \ne 0$ everywhere. Where $f'(a) = 0$ the map is locally $m$-to-one for $m \ge 2$.
• A holomorphic bijection has a holomorphic inverse — it is a conformal equivalence, and the inverse comes free.

A real-variable contrast worth carrying: $x \mapsto x^2$ maps the open interval $(-1,1)$ onto $[0,1)$, which is not open. Openness of the image is a complex phenomenon.

Where intuition breaks

"$|g| \le |f|$ on $\gamma$, so Rouché applies." Strict. $f = z$, $g = -z$ gives $f + g \equiv 0$. The weakened inequality is the commonest way this option is made wrong.

"Rouché needs $|g| < |f|$ inside the contour." On the contour only. Requiring it inside would make the theorem useless, since that is where the zeros are.

"The argument principle counts zeros." Zeros minus poles. For a meromorphic function the two cancel, and an option will supply a function with poles and ask for the winding number.

"Multiplicity does not matter." Everything here is counted with multiplicity. $z^3$ has one zero as a set and three as a count, and the integral returns three.

"$f' \ne 0$ implies $f$ is injective." Only locally. $e^{z}$ has non-vanishing derivative everywhere and is not injective on $\mathbb{C}$ — the same local-versus-global gap as the inverse function theorem.

"$f$ injective implies $f' \ne 0$." This one is true for holomorphic $f$, and is worth recognising among false options. It fails for real functions: $x^3$ is injective on $\mathbb{R}$ with $f'(0) = 0$.

"A non-constant holomorphic map can have a closed image." Its image is open. A non-constant holomorphic function on a domain never maps onto a closed set like $[0,1]$, and never takes only real values.

The exam's angle

1. For "how many roots in this region", reach for Rouché. Split the polynomial into a dominant term and a remainder, and check the inequality on the boundary circle.
2. Try the obvious split first, then the other one. Which term dominates changes with the radius, and comparing two circles brackets the roots into an annulus.
3. Count with multiplicity, and subtract poles. Options exploit both.
4. When an option asserts something is impossible for a holomorphic map, check openness. A non-open image, a real-valued non-constant function, an interior maximum of $|f|$ — all impossible for the same reason.
5. Compute $f'/f$ if the integrand looks like a logarithmic derivative. That form is the argument principle asking to be recognised.

The night before

• Argument principle: $\frac{1}{2\pi i}\oint f'/f = Z - P$, with multiplicity. Equivalently the winding number of $f\circ\gamma$ about $0$.
• $f'/f$ has residue $m$ at a zero of order $m$, and $-m$ at a pole of order $m$.
• Rouché: $|g| < |f|$ strictly, on the whole contour $\Rightarrow$ $f$ and $f+g$ have equally many zeros inside. $f=z$, $g=-z$ breaks the non-strict version.
• $z^5+3z+1$: five roots in $|z|<2$, one in $|z|<1$, so four in the annulus.
• Fundamental theorem of algebra follows from Rouché on a large circle.
• Open mapping: non-constant holomorphic maps open sets to open sets. Maximum modulus is a corollary.
• Injective $\Rightarrow f' \ne 0$, for holomorphic $f$. The converse is false — $e^z$.
• A holomorphic bijection has a holomorphic inverse.