Ngô Quốc Anh

August 4, 2009

Another point of view of the Argument Principle


With Laurent series and the classi cation of singularities in hand, it is easy to prove the Residue Theorem. In addition to being a handy tool for evaluating integrals, the Residue Theorem has many theoretical consequences, for example, the Argument Principle, Rouché’s Theorem, the Local Mapping Theorem, the Open Mapping Theorem, the Hurwitz Theorem, the general Casorati-Weierstrass Theorem, and Riemann’s Theorem. This writeup presents the Argument Principle with an application.

The Argument Principle

Theorem (Argument Principle). Let \gamma be a simple closed counterclockwise curve. Let f be analytic and nonzero on \gamma and meromorphic inside \gamma. Let Z(f) denote the number of zeros of f inside \gamma, each counted as many times as its multiplicity, and let P(f) denote the number of poles of f inside \gamma, each counted as many times as its multiplicity. Then

\displaystyle Z\left( f \right) - P\left( f \right) = \frac{1} {{2\pi i}}\int_\gamma {\frac{{df}} {f}} .

The Winding Number

De finition. Let \gamma : [0,1] \to \mathbb C be any closed rectifiable path. By the usual abuse of notation, let \gamma also denote the corresponding subset of \mathbb C. Consider a complex-valued function on the complement of the path,

\displaystyle Ind\left( {\gamma ,\cdot} \right):\mathbb{C} - \gamma \to \mathbb{C}

where

\displaystyle Ind\left( {\gamma ,z} \right) = \frac{1}{{2\pi i}}\int_\gamma {\frac{{d\zeta }}{{\zeta - z}}} .

For any z \in \mathbb C- \gamma, the function Ind(\gamma, z) is the winding number of \gamma about z. For instance, if \gamma is a circle traversed once counterclockwise about z then its winding number about z is 1.

The Argument Principle Again

With the winding number in hand, we can rephrase the Argument Principle in a way that explains its name.

Theorem (Argument Principle, second version).

Let \gamma be a simple closed counterclockwise curve. Let f be analytic and nonzero on \gamma and meromorphic inside \gamma. Let Z(f) denote the number of zeros of f inside \gamma, each counted as many times as its multiplicity, and let P(f) denote the number of poles of f inside \gamma, each counted as many times as its multiplicity. Then

\displaystyle Z(f)-P(f) = Ind(f \circ \gamma, 0).

That is, the theorem is called the Argument Principle because the number of zeros minus poles of f inside \gamma is the number of times that the argument of f\circ \gamma increases by 2\pi.

For example, consider the polynomial

\displaystyle f(z)=z^4-8z^3+3z^2+8z+3.

To count the roots of f in the right half plane, let D be a large disk centered at the orgin, large enough to contain all the roots of f. Let \gamma be the boundary of the right half of D. Thus \gamma is the union of a segment of the imaginary axis and a semicircle. The values of f on the imaginary axis are

\displaystyle f(iy) = (y^4+3y^2+3) + i(8y^3+8y).

Therefore, \Re f(iy) is always positive, and so f takes the imaginary axis into the right half plane. On the semicircle, f(z) behaves qualitatively as z^4. Therefore the path \Gamma = f \circ \gamma winds twice around the origin, showing that f has two roots in the right half plane.

Source: http://people.reed.edu/~jerry/311/residue.pdf

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