advanced29 min studyLesson 367 in Mathematics

The argument principle and Rouché's theorem

Count zeros with multiplicity using logarithmic derivatives and boundary inequalities rather than solving for every root.

Before you begin

Residue calculus and real integrals

On this page

The question for today

What you’ll understand

Derive zero-minus-pole counts from the logarithmic derivative.
Apply a strict boundary comparison in Rouché's theorem.
Distinguish a zero count from a list of root locations.

A polynomial's roots may be impossible to express conveniently, yet a contour can reveal how many lie in a region. The argument principle converts that count into the winding of the function's boundary values around zero. Rouché's theorem makes the count accessible through an inequality: a smaller boundary perturbation cannot change it. These results count multiplicities and depend on excluding zeros and poles from the boundary. A plot suggesting that a curve barely misses zero is not a substitute for that exclusion.

A logarithmic derivative records multiplicity

Near a zero a of order m, factor f as $(z-a)^m h(z)$ with h holomorphic and nonzero. Then f′/f equals m/(z−a) plus the holomorphic logarithmic derivative h′/h. At a pole of order k the same reasoning gives residue −k. Applying the residue theorem to f′/f therefore counts zeros positively and poles negatively. Assume f is meromorphic on a neighborhood of the closed region enclosed by a simple positively oriented piecewise smooth contour and has neither zeros nor poles on that contour.

Along the contour, f never vanishes, so its argument can be followed continuously on the parameter interval even if a single global argument function is unavailable on its image. Its total change divided by 2π is an integer winding number. That integer is the zero count minus the pole count. Multiplicity matters: one double zero contributes two, while a simple pole subtracts one. Knowing the difference alone does not separately identify both counts unless one category is already known.

\frac1{2\pi i}\oint_\gamma\frac{f'(z)}{f(z)}\,dz=N-P=\operatorname{wind}(f\circ\gamma,0)

The stated meromorphic neighborhood and zero/pole-free boundary hypotheses apply. N and P count interior zeros and poles with multiplicity; the image curve may self-intersect but must avoid zero.↔ Scroll wide equations horizontally.

A strict smaller term preserves the count

Rouché's theorem concerns holomorphic functions f and g on a neighborhood of the same enclosed closed region. If $|g|<|f|$ everywhere on its boundary, then f and f+g have the same number of interior zeros, counted with multiplicity. The strict comparison makes f nonzero on the boundary and ensures that f+tg stays nonzero there for every real t between zero and one. Their boundary windings deform continuously without crossing zero and therefore remain the same integer.

The comparison is a boundary condition, not an interior pointwise domination requirement. It often uses a simple upper bound for a sum and a constant modulus for one monomial on a circle. Equality in the bound is insufficient for this version of the theorem: a boundary zero can enter exactly at the limiting case. Before selecting a dominant term, calculate the radius-dependent magnitudes and check that the inequality is genuinely strict at every boundary point.

|g(z)|<|f(z)|\quad(z\in\gamma)\quad\Longrightarrow\quad N_\gamma(f+g)=N_\gamma(f)

Both functions are holomorphic on a neighborhood of the enclosed closed region, and γ is a simple positively oriented piecewise C¹ contour. The conclusion counts interior zeros with multiplicity.↔ Scroll wide equations horizontally.

Use several contours to locate groups of roots

Different radii can make different terms dominant. Counting on an inner circle and an outer circle, with no boundary zeros, locates the difference of the counts in the intervening annulus. This can be much easier than finding approximate roots and then trying to prove that none were missed. It also gives a check on numerical solvers: a computed root list should match the certified multiplicity count inside each tested contour.

A count does not provide individual root coordinates, separation bounds, or simplicity unless further arguments supply them. Repeated roots can occupy the same location while contributing several units. For meromorphic functions the argument principle subtracts poles, whereas the basic Rouché statement used here compares holomorphic functions with no poles in the region. Keeping those hypotheses separate prevents a cancellation in a logarithmic-derivative integral from being misread as the absence of zeros.

Work it through

Count a seventh-degree polynomial's roots

For $p(z)=z^7+4z+2$ , count roots in |z|<1, in |z|<2, and in the intervening annulus. Include multiplicities.

On |z|=1, take f(z)=4z and g(z)=z⁷+2. Then |f|=4 while $|g|\le3<4$ . Rouché gives the same zero count as 4z: exactly one root in the unit disk.
On |z|=2, take f(z)=z⁷ and g(z)=4z+2. Now |f|=128 while $|g|\le10<128$ . There are seven roots inside the radius-2 disk, matching z⁷'s multiplicity count.
Both strict comparisons also exclude boundary zeros. Subtracting the inner count from the outer count gives six roots in $1<|z|<2$ . The argument does not claim that those roots are simple or provide their coordinates.

The result

There is one root in the unit disk and six in the annulus between radii 1 and 2, counting multiplicities. Boundary inequalities certify the result without a root formula.

The living notebook

Explore the model

Interactive lab

Interactive lab · waves

Inspect cancellation at the edge of strict domination

Pause and Reset; set Phase difference=180 degrees and Amplitude ratio=0.5. Predict resultant sine amplitude 0.5 from the underlying phasor coefficient 1+0.5e^(iπ). For a fresh comparison Reset and set phase=180,ratio=1: the two waves cancel completely. The real sine trace itself has zeros even in the first case; this phasor example is not a zero-free complex contour or a winding-number calculator.

SumFirst componentSecond component

Sum at x = 0: 0.866
Resultant amplitude: 1.732
Phase difference: 60°

Phase difference60°Amplitude ratio1

t = 0 s

Two waves share the same frequency and wave number. Playback advances t in sin(2x − t) + a sin(2x − t + phase). Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forStrictly smaller perturbations prevent complete phasor cancellation, illustrating why equality needs separate treatment in a boundary comparison.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

There is a double zero at zero and one simple pole at 1/4, with neither on the unit circle. The argument principle gives N−P=2−1=1.

Your turn

Try it yourself

For $q(z)=z^4+2z+5$ , use Rouché to count roots inside circles of radii 1 and 2. What does the difference imply?

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

At radius 1 compare the constant with the remaining terms. At radius 2 compare the fourth power with the lower-degree terms.

Compare with the solution

On |z|=1, $|z^4+2z|\le3<5$ , so q has the same zero count as the nonzero constant 5: none. On |z|=2, $|2z+5|\le9<16=|z^4|$ , so q has four zeros inside. Strictness excludes boundary roots, and therefore all four root multiplicities lie in $1<|z|<2$ . The count alone does not prove that the four roots are distinct.

Keep the idea

Take it with you

The argument principle counts zeros minus poles with multiplicity.
Rouché uses a strict boundary inequality for holomorphic functions.
Comparing counts on nested contours locates roots in an annulus without identifying each coordinate.

Read further

Original sources and deeper explanations to follow at your own pace.

Jeremy Orloff, MIT 18.04: Argument Principle

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