Applications and Consequences
The argument principle can be used to efficiently locate zeros or poles of meromorphic functions on a computer. Even with rounding errors, the expression will yield results close to an integer; by determining these integers for different contours C one can obtain information about the location of the zeros and poles. Numerical tests of the Riemann hypothesis use this technique to get an upper bound for the number of zeros of Riemann's function inside a rectangle intersecting the critical line.
The proof of Rouché's theorem uses the argument principle.
Modern books on feedback control theory quite frequently use the argument principle to serve as the theoretical basis of the Nyquist stability criterion.
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 function 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.
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