Nusret Balcı https://balcinet.com/notes/ Mathematics and mathematics education. quarto-1.10.18 Wed, 09 Sep 2026 21:00:00 GMT The Argument Principle Nusret Balcı https://balcinet.com/notes/argument-principle.html

Setting and notation

Throughout, is a positively oriented simple closed contour in , and is an open set containing together with its interior. We write for that interior.

Definition 1 (Order of a zero or pole) Let be meromorphic near and not identically zero. There is a unique and a function , holomorphic and non-vanishing near , with We call the order of at . If then is a zero of order ; if then is a pole of order ; if then .

The whole proof rests on one computation, which is worth isolating.

Lemma 1 (Logarithmic derivative) With and as in Definition 1 and , the logarithmic derivative is meromorphic near with at worst a simple pole there, and

Proof. Differentiating Equation 1, and dividing by — legitimate on a punctured neighbourhood of , where does not vanish — gives Since is holomorphic and non-vanishing near , the term is holomorphic there and contributes nothing to the residue. The residue is therefore the coefficient of .

Notice how little Lemma 1 asks for: no estimate, no growth condition. Passing from to turns multiplicative structure — the factor — into additive structure, a simple pole of residue . Everything else follows from the residue theorem.

The principle

Theorem 1 (Argument principle) Let be meromorphic on with no zeros and no poles on . Then has finitely many zeros and poles in , and where and count the zeros and the poles of inside , each with multiplicity.

Proof. First, finiteness. The set is compact. If had infinitely many zeros in they would accumulate at some ; since , the identity theorem rules out an accumulation of zeros at a point where is holomorphic, and an accumulation at a pole is impossible because poles are isolated. The same argument applies to the poles, which are isolated by definition.

So let be the zeros and poles of inside , with orders . By Lemma 1, is holomorphic on a neighbourhood of except at the , where it has simple poles with residues . As has no zeros or poles on , the integrand is continuous there and the residue theorem applies: Splitting the sum by sign — positive orders are zeros, negative orders are poles — gives exactly .

Why “argument”?

The name comes from reading Equation 4 backwards. Substituting turns the integral into one over the image curve : the winding number of about the origin. And since a local branch of has returning to its starting value around a closed curve, only the argument can have changed: where is the net change in as traverses once. Counting zeros and poles is the same as counting turns.

A worked example

Take Inside the numerator vanishes at the three cube roots of unity, each simply, so ; the only pole is , of order , so . The argument principle predicts so the image curve should wind exactly once about the origin.

Left: the circle of radius 2 in the complex plane, containing three zeros marked as open circles at the cube roots of unity and one double pole marked as a cross at one half. Right: the image of that circle under f, a single closed loop encircling the origin exactly once.

Left: the circle of radius 2 in the complex plane, containing three zeros marked as open circles at the cube roots of unity and one double pole marked as a cross at one half. Right: the image of that circle under f, a single closed loop encircling the origin exactly once.

Figure 1: The contour and its image. Three zeros () and a double pole () lie inside , so — and duly encircles the origin exactly once.

We can check Equation 4 numerically. Parametrising by and integrating by the trapezoid rule should return :

import numpy as np

A = 0.5

def f(z):
    return (z**3 - 1) / (z - A) ** 2

def f_prime(z):
    return 3 * z**2 / (z - A) ** 2 - 2 * (z**3 - 1) / (z - A) ** 3

t  = np.linspace(0, 2 * np.pi, 200_001)
z  = 2 * np.exp(1j * t)
dz = 2j * np.exp(1j * t)          # dz/dt

I = np.trapezoid(f_prime(z) / f(z) * dz, t) / (2j * np.pi)

print(f"contour integral / 2*pi*i = {I.real:+.10f}{I.imag:+.10f}i")
print(f"Z - P                     = {3 - 2}")
contour integral / 2*pi*i = +1.0000000000+0.0000000000i
Z - P                     = 1

Agreement to ten decimal places, as it should be: the left-hand side is an integer in disguise.

Rouché’s theorem as a corollary

The argument principle is most often used through its best-known consequence.

Corollary 1 (Rouché’s theorem) Let be holomorphic on and suppose Then and have the same number of zeros inside , counted with multiplicity.

Proof. The hypothesis forces on , and also there, since . So the argument principle applies to both. Write . On we have , so lies in the open disc of radius about — a disc that misses the origin entirely. A curve confined to a disc not containing cannot wind around , so . Since both functions are zero-free on and is well defined there, winding numbers add:

TipWhere this gets used

Rouché is the standard tool for locating roots: to see that has exactly one root in , take and , and check on .

Exercises

  1. Let . Show that has exactly one zero in and exactly three in .
  2. Prove the fundamental theorem of algebra from Theorem 1 by applying it to a monic polynomial of degree on a sufficiently large circle.
  3. Suppose is holomorphic and injective on a neighbourhood of , the closed unit disc. Show that for , the number is if and otherwise — the argument-principle proof of the open mapping theorem.
  4. Where exactly does the proof of Theorem 1 use that has no zeros on ? Give an example showing the conclusion can fail without it.

Further reading

Beardon (1979) takes the argument principle as the organising idea of an entire book and is the natural next stop. The standard treatments are in Ahlfors (1979) (Ch. 4), Conway (1978) (Ch. V) and Stein and Shakarchi (2003) (Ch. 3).

References

Ahlfors, Lars V. 1979. Complex Analysis. 3rd ed. McGraw-Hill.
Beardon, Alan F. 1979. Complex Analysis: The Argument Principle in Analysis and Topology. John Wiley & Sons.
Conway, John B. 1978. Functions of One Complex Variable i. 2nd ed. Vol. 11. Graduate Texts in Mathematics. Springer.
Stein, Elias M., and Rami Shakarchi. 2003. Complex Analysis. II. Princeton Lectures in Analysis. Princeton University Press.
]]> complex analysis lecture notes https://balcinet.com/notes/argument-principle.html Wed, 09 Sep 2026 21:00:00 GMT