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.

This statement of the theorem assumes that the contour C is simple, that is, without self-intersections, and that it is oriented counter-clockwise.

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.

The argument principle can be used to efficiently locate zeros or poles of meromorphic functions on a computer.

function inside a rectangle intersecting the critical line.

A consequence of the more general formulation of the argument principle is that, under the same hypothesis, if g is an analytic function in Ω, then For example, if f is a polynomial having zeros z1, ..., zp inside a simple contour C, and g(z) = zk, then is power sum symmetric polynomial of the roots of f. Another consequence is if we compute the complex integral: for an appropriate choice of g and f we have the Abel–Plana formula: which expresses the relationship between a discrete sum and its integral.

In modern books on feedback control theory, it is commonly used as the theoretical foundation for the Nyquist stability criterion.

According to the book by Frank Smithies (Cauchy and the Creation of Complex Function Theory, Cambridge University Press, 1997, p. 177), Augustin-Louis Cauchy presented a theorem similar to the above on 27 November 1831, during his self-imposed exile in Turin (then capital of the Kingdom of Piedmont-Sardinia) away from France.

This theorem by Cauchy was only published many years later in 1874 in a hand-written form and so is quite difficult to read.

Cauchy published a paper with a discussion on both zeroes and poles in 1855, two years before his death.