argument principle

If a function f is meromorphicPlanetmathPlanetmath on the interior of a rectifiable simple closed curve C, then

12πiCf(z)f(z)𝑑z (1)

equals the differencePlanetmathPlanetmath between the number of zeros and the number of poles of f counted with multiplicityMathworldPlanetmath. (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 f is meromorphic on the interior of a rectifiable simple closed curve C and has m poles and n zeros on the interior of C, then the argumentPlanetmathPlanetmath of f increases by 2π(n-m) upon traversing C. The relationMathworldPlanetmathPlanetmath of this statement to the previous statement is easy to see. Note that f/f=(logf) and that log(z)=log|z|+iargz. Substituting this into formulaMathworldPlanetmathPlanetmath (1), we find


The first integral on the rightmost side of this equation equals zero because log|f| is single-valued. The second integral on the rightmost side equals the change in the argument as one traverses C. Cancelling the i from both sides, we conclude that the change in the argument equals 2π(n-m).

Note also that the integral (1) is the winding number, about zero, of the image curve fC.

Title argument principle
Canonical name ArgumentPrinciple
Date of creation 2013-03-22 14:34:28
Last modified on 2013-03-22 14:34:28
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 10
Author rspuzio (6075)
Entry type Algorithm
Classification msc 30E20
Synonym Cauchy’s argument principle
Defines argument principle