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Argument principle
In complex analysis, the argument principle (or Cauchy's argument principle) is a theorem relating the difference between the number of zeros and poles of a meromorphic function to a contour integral of the function's logarithmic derivative.
If f 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 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 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.
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Argument principle
In complex analysis, the argument principle (or Cauchy's argument principle) is a theorem relating the difference between the number of zeros and poles of a meromorphic function to a contour integral of the function's logarithmic derivative.
If f 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 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 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.