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Concept
Argument principle
Summary
In complex analysis, the argument principle (or Cauchy's argument principle) relates the difference between the number of zeros and poles of a meromorphic function to a contour integral of the function's logarithmic derivative.
Specifically, if f(z) is a meromorphic function inside and on some closed contour C, and f has no zeros or poles on C, then
where Z 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, indicate. This statement of the theorem assumes that the contour C is simple, that is, without self-intersections, and that it is oriented counter-clockwise.
More generally, suppose that f(z) is a meromorphic function on an open set Ω in the complex plane and that C is a closed curve in Ω which avoids all zeros and poles of f and is contractible to a point inside Ω. For each point z ∈ Ω, let n(C,z) be the winding number of C around z. Then
where the first summation is over all zeros a of f counted with their multiplicities, and the second summation is over the poles b of f counted with their orders.
The contour integral can be interpreted as 2πi times the winding number of the path f(C) around the origin, using the substitution w = f(z):
That is, it is i times the total change in the argument of f(z) as z travels around C, explaining the name of the theorem; this follows from
and the relation between arguments and logarithms.
Let zZ be a zero of f. We can write f(z) = (z − zZ)kg(z) where k is the multiplicity of the zero, and thus g(zZ) ≠ 0. We get
and
Since g(zZ) ≠ 0, it follows that g' (z)/g(z) has no singularities at zZ, and thus is analytic at zZ, which implies that the residue of f′(z)/f(z) at zZ is k.
Let zP be a pole of f. We can write f(z) = (z − zP)−mh(z) where m is the order of the pole, and
h(zP) ≠ 0. Then,
and
similarly as above. It follows that h′(z)/h(z) has no singularities at zP since h(zP) ≠ 0 and thus it is analytic at zP. We find that the residue of
f′(z)/f(z) at zP is −m.
Official source
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