Editing Argument principle (section)

==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 <math>{1\over 2\pi i}\oint_{C} {f'(z) \over f(z)}\, dz</math> 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 Xi function|Riemann's <math>\xi(s)</math> function]] inside a rectangle intersecting the critical line. The argument principle can also be used to prove [[Rouché's theorem]], which can be used to bound the roots of polynomials.

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

:<math> \frac{1}{2\pi i} \oint_C g(z)\frac{f'(z)}{f(z)}\, dz = \sum_a n(C,a)g(a) - \sum_b n(C,b)g(b).</math>

For example, if ''f'' is a [[polynomial]] having zeros ''z''<sub>1</sub>, ..., ''z''<sub>p</sub> inside a simple contour ''C'', and ''g''(''z'') = ''z''<sup>k</sup>, then
:<math> \frac{1}{2\pi i} \oint_C z^k\frac{f'(z)}{f(z)}\, dz = z_1^k+z_2^k+\cdots+z_p^k,</math>
is [[power sum symmetric polynomial]] of the roots of ''f''.

Another consequence is if we compute the complex integral:

: <math>\oint_C f(z){g'(z) \over g(z)}\, dz</math>

for an appropriate choice of ''g'' and ''f'' we have the [[Abel–Plana formula]]:

: <math>  \sum_{n=0}^{\infty}f(n)-\int_{0}^{\infty}f(x)\,dx= f(0)/2+i\int_{0}^{\infty}\frac{f(it)-f(-it)}{e^{2\pi t}-1}\, dt\, </math>

which expresses the relationship between a discrete sum and its integral.

The argument principle is also applied in [[control theory]]. In modern books on feedback control theory, it is commonly used as the theoretical foundation for the [[Nyquist stability criterion]]. Moreover, a more generalized form of the argument principle can be employed to derive [[Bode's sensitivity integral]] and other related integral relationships.<ref>{{Cite journal |last1=Xu |first1=Yong |last2=Chen |first2=Gang |last3=Chen |first3=Jie |last4=Qiu |first4=Li |date=2023 |title=Argument Principle and Integral Relations: Hidden Links and Generalized Forms |url=https://ieeexplore.ieee.org/document/9736615 |journal=IEEE Transactions on Automatic Control |volume=68 |issue=3 |pages=1831–1838 |doi=10.1109/TAC.2022.3159565 |issn=0018-9286|url-access=subscription }}</ref>