In complex analysis, the argument principle (or Cauchy's argument principle) states that if f(z) is a meromorphic function inside and on some closed contour C, with f having no zeros or poles on C, then the following formula holds
where N and P denote respectively the number of zeros and poles of f(z) inside the contour C, with each zero and pole counted as many times as its multiplicity and order respectively. This theorem assumes that the contour C is simple, that is, without self- ...
Let zN be a zero of f. We can write f(z) = (z − zN)kg(z) where k is the multiplicity of the zero, and thus g(zN) ≠ 0. We get
and
Since g(zN) ≠ 0, it follows that g′(z)/g(z) has no singularities at zN, and thus is analytic at zN, which implies that the residue of f′(z)/f(See also:
Formally, the winding number is defined as follows:
If γ is a closed rectifiable curve in C, and z0 is a point in C not on γ, then the winding number of γ with respect to z0 (alternately called the index of γ with respect to z0) is defined by the formula:
This is verifiable from applying the Cauchy integral formula — the integral will be a multiple of 2πi, since each time γ goes about z0, we have effectively calculated the integral again.
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This has consequences in considering the winding number of f(z) about the origin, say, if C is a closed contour centered on the origin. We see that the integral of f′(z)/f(z) about C is the change in values of log f(z). Since C is closed we only need consider the change in i arg f(z) over C − which will be some multiple of 2πi since C is closed (but may wind more than once about the origin). But since by the argument principle
the factors of 2πi ...
