Argument Principle

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 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, respectively order, indicates. 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.