Editing Argument principle

{{Short description|Theorem in complex analysis}}
[[Image:argument principle1.svg|frame|right|The simple contour ''C'' (black), the zeros of ''f'' (blue) and the poles of ''f'' (red). Here we have <math>\frac{1}{2\pi i}\oint_{C} {f'(z) \over f(z)}\, dz=4-5.\,</math>]]

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]].

==Formulation==
{{Complex analysis sidebar}}
If ''f'' is a meromorphic function inside and on some closed contour ''C'', and ''f'' has no zeros or poles on ''C'', then 
: <math>\frac{1}{2\pi i}\oint_{C} {f'(z) \over f(z)}\, dz=Z-P</math>
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 (mathematics)|multiplicity]] and [[Pole (complex analysis)|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 space|contractible]] to a point inside Ω.  For each point ''z'' ∈ Ω, let ''n''(''C'',''z'') be the [[winding number]] of ''C'' around ''z''.  Then
:<math>\frac{1}{2\pi i}\oint_{C} \frac{f'(z)}{f(z)}\, dz = \sum_a n(C,a) - \sum_b n(C,b)\,</math>
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.

==Interpretation of the contour integral==
The [[contour integral]] <math>\oint_{C} \frac{f'(z)}{f(z)}\, dz</math> can be interpreted as 2&pi;''i'' times the winding number of the path ''f''(''C'') around the origin, using the substitution ''w'' = ''f''(''z''): 
:<math>\oint_{C} \frac{f'(z)}{f(z)}\, dz = \oint_{f(C)} \frac{1}{w}\, dw</math>
That is, it is ''i'' times the total change in the [[argument (complex analysis)|argument]] of ''f''(''z'') as ''z'' travels around ''C'', explaining the name of the theorem; this follows from 
:<math>\frac{d}{dz}\log(f(z))=\frac{f'(z)}{f(z)}</math> 
and the relation between arguments and logarithms.

==Proof of the argument principle==
Let ''z''<sub>''Z''</sub> be a zero of ''f''. We can write ''f''(''z'') = (''z''&nbsp;&minus;&nbsp;''z''<sub>''Z''</sub>)<sup>''k''</sup>''g''(''z'') where ''k'' is the multiplicity of the zero, and thus ''g''(''z''<sub>''Z''</sub>) ≠&nbsp;0. We get

: <math>f'(z)=k(z-z_Z)^{k-1}g(z)+(z-z_Z)^kg'(z)\,\!</math>

and

: <math>{f'(z)\over f(z)}={k \over z-z_Z}+{g'(z)\over g(z)}.</math>

Since  ''g''(''z''<sub>''Z''</sub>) ≠ 0, it follows that ''g' ''(''z'')/''g''(''z'') has no singularities at ''z''<sub>''Z''</sub>, and thus is analytic at ''z''<sub>Z</sub>, which implies that the [[Residue (complex analysis)|residue]] of ''f''&prime;(''z'')/''f''(''z'') at ''z''<sub>''Z''</sub> is&nbsp;''k''.

Let ''z''<sub>P</sub> be a pole of ''f''. We can write ''f''(''z'') = (''z''&nbsp;&minus;&nbsp;''z''<sub>P</sub>)<sup>&minus;''m''</sup>''h''(''z'') where ''m'' is the order of the pole, and 
''h''(''z''<sub>P</sub>) ≠ 0. Then,

: <math>f'(z)=-m(z-z_P)^{-m-1}h(z)+(z-z_P)^{-m}h'(z)\,\!.</math>

and

: <math>{f'(z)\over f(z)}={-m \over z-z_P}+{h'(z)\over h(z)}</math>

similarly as above. It follows that ''h''&prime;(''z'')/''h''(''z'') has no singularities at ''z''<sub>P</sub> since ''h''(''z''<sub>P</sub>) ≠ 0 and thus it is analytic at ''z''<sub>P</sub>. We find that the residue of
''f''&prime;(''z'')/''f''(''z'') at ''z''<sub>P</sub> is &minus;''m''.

Putting these together, each zero ''z''<sub>''Z''</sub> of multiplicity ''k'' of ''f'' creates a simple pole for
''f''&prime;(''z'')/''f''(''z'') with the residue being ''k'', and each  pole ''z''<sub>P</sub> of order ''m'' of
''f'' creates a simple pole for ''f''&prime;(''z'')/''f''(''z'') with the residue being &minus;''m''. (Here, by a simple pole we
mean a pole of order one.) In addition, it can be shown that ''f''&prime;(''z'')/''f''(''z'') has no other poles,
and so no other residues.

By the [[residue theorem]] we have that the integral about ''C'' is the product of 2''πi'' and the sum of the residues.  Together, the sum of the ''k''{{'}}s for each zero ''z''<sub>''Z''</sub> is the number of zeros counting multiplicities of the zeros, and likewise for the poles, and so we have our result.

==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>

==Generalized argument principle==
There is an immediate generalization of the argument principle. Suppose that g is analytic in the region <math>\Omega</math>. Then
: <math>\frac{1}{2\pi i}\oint_{C} {f'(z) \over f(z)} g(z) \, dz = \sum_a g(a) n(C,a) - \sum_b g(b) n(C,b)\,</math>
where the first summation is again over all zeros ''a'' of ''f'' counted with their multiplicities, and the second summation is again over the poles ''b'' of ''f'' counted with their orders.

==History==
According to the book by [[Frank Smithies]] (''Cauchy and the Creation of Complex Function Theory'', Cambridge University Press, 1997, p.&nbsp;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. However, according to this book, only zeroes were mentioned, not poles. 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.

== See also ==

* [[Logarithmic derivative]]
* [[Nyquist stability criterion]]

==References==
{{Reflist}}
* {{cite book | last=Rudin | first=Walter | title = Real and Complex Analysis (International Series in Pure and Applied Mathematics) | publisher=McGraw-Hill | year=1986 |isbn=978-0-07-054234-1}}
* {{cite book | last=Ahlfors | first=Lars | title = Complex analysis: an introduction to the theory of analytic functions of one complex variable | publisher=McGraw-Hill | year=1979 |isbn=978-0-07-000657-7}}
* {{cite book | last1=Churchill | first1=Ruel Vance | last2=Brown | first2=James Ward | title = Complex Variables and Applications | publisher=McGraw-Hill | year=1989 |isbn=978-0-07-010905-6}}
* Backlund, R.-J. (1914) Sur les zéros de la fonction zeta(s) de Riemann, C. R. Acad. Sci. Paris 158, 1979–1982.

==External links==
* {{springer|title=Argument, principle of the|id=p/a013250}}

{{DEFAULTSORT:Argument Principle}}
[[Category:Theorems in complex analysis]]