PlanetMath: argument principle

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argument principle

(Algorithm)

If a function is meromorphic on the interior of a rectifiable simple closed curve , then

$\displaystyle {1 \over 2 \pi i} \oint_C {f'(z) \over f(z)} dz$

(1)

equals the difference between the number of zeros and the number of poles of

counted with multiplicity. (For example, a zero of order two counts as two zeros; a pole of order three counts as three poles.) This fact is known as the argument principle.

The principle may be stated in another form which makes the origin of the name apparent: If a function is meromorphic on the interior of a rectifiable simple closed curve and has poles and zeros on the interior of , then the argument of increases by $2 \pi (n - m)$ upon traversing . The relation of this statement to the previous statement is easy to see. Note that $f'/f = (\log f)'$ and that $\log (z) = \log \vert z\vert + i \arg z$ . Substituting this into formula (1), we find

$\displaystyle 2 \pi i (n - m) = \oint_C {f'(z) \over f(z)} dz = \oint_C d \log \vert f(z)\vert + i \oint_C d \arg (f(z))\,.$

The first integral on the rightmost side of this equation equals zero because $\log\vert f\vert$ is single-valued. The second integral on the rightmost side equals the change in the argument as one traverses

. Cancelling the

from both sides, we conclude that the change in the argument equals $2 \pi (n - m)$ .

Note also that the integral (1) is the winding number, about zero, of the image curve $f \circ C$ .

"argument principle" is owned by rspuzio. [ full author list (3) ]

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Other names:

Cauchy's argument principle

Also defines:

argument principle

Keywords:

argument, complex anaysis, contour integration

Attachments:: proof of argument principle (Proof) by rspuzio

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Cross-references: curve, image, winding number, equation, integral, easy to see, argument, order, zero of order, multiplicity, poles, number, difference, simple closed curve, rectifiable, interior, meromorphic, function
There are 7 references to this entry.

This is version 7 of argument principle, born on 2004-09-04, modified 2007-04-07.
Object id is 6133, canonical name is ArgumentPrinciple.
Accessed 6443 times total.

Classification:

AMS MSC:

30E20 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Integration, integrals of Cauchy type, integral representations of analytic functions)

Pending Errata and Addenda

None.[ View all 3 ]

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