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| [[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>\oint_{C} {f'(z) \over f(z)}\, dz=2\pi i (4-5)</math>.]]
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| In [[complex analysis]], the '''argument principle''' (or '''Cauchy's argument principle''') relates the difference between the number of [[Zero (complex analysis)|zeros]] and [[Pole (complex analysis)|poles]] of a [[meromorphic function]] to a [[contour integral]] of the function's [[logarithmic derivative]].
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| Specifically, if ''f''(''z'') is a meromorphic function inside and on some closed contour ''C'', and ''f'' has no [[Zero (complex analysis)|zeros]] or [[Pole (complex analysis)|poles]] on ''C'', then
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| : <math>\oint_{C} {f'(z) \over f(z)}\, dz=2\pi i (N-P)</math>
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| where ''N'' and ''P'' denote respectively the number of zeros and poles of ''f''(''z'') 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.
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| More generally, suppose that ''f''(''z'') 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
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| :<math>\oint_{C} \frac{f'(z)}{f(z)}\, dz = 2\pi i \left(\sum_a n(C,a) - \sum_b n(C,b)\right)</math>
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| 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.
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| ==Interpretation of the contour integral==
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| The contour integral <math>\oint_{C} \frac{f'(z)}{f(z)}\, dz</math> can be interpreted in two ways:
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| * as 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
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| :<math>\frac{d}{dz}\log(f(z))=\frac{f'(z)}{f(z)}</math>
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| and the relation between arguments and logarithms.
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| * as 2π''i'' times the winding number of the path ''f''(''C'') around the origin, using the substitution ''w'' = ''f''(''z''):
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| :<math>\oint_{C} \frac{f'(z)}{f(z)}\, dz = \oint_{f(C)} \frac{1}{w}\, dw</math>
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| ==Proof of the argument principle==
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| Let ''z''<sub>''N''</sub> be a zero of ''f''. We can write ''f''(''z'') = (''z'' − ''z''<sub>''N''</sub>)<sup>''k''</sup>''g''(''z'') where ''k'' is the multiplicity of the zero, and thus ''g''(''z''<sub>''N''</sub>) ≠ 0. We get
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| : <math>f'(z)=k(z-z_N)^{k-1}g(z)+(z-z_N)^kg'(z)\,\!</math>
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| and
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| : <math>{f'(z)\over f(z)}={k \over z-z_N}+{g'(z)\over g(z)}.</math>
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| Since ''g''(''z''<sub>''N''</sub>) ≠ 0, it follows that ''g' ''(''z'')/''g''(''z'') has no singularities at ''z''<sub>''N''</sub>, and thus is analytic at ''z''<sub>N</sub>, which implies that the [[Residue (complex analysis)|residue]] of ''f''′(''z'')/''f''(''z'') at ''z''<sub>''N''</sub> is ''k''.
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| Let ''z''<sub>P</sub> be a pole of ''f''. We can write ''f''(''z'') = (''z'' − ''z''<sub>P</sub>)<sup>−''m''</sup>''h''(''z'') where ''m'' is the order of the pole, and thus
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| ''h''(''z''<sub>P</sub>) ≠ 0. Then,
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| : <math>f'(z)=-m(z-z_P)^{-m-1}h(z)+(z-z_P)^{-m}h'(z)\,\!.</math>
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| and
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| : <math>{f'(z)\over f(z)}={-m \over z-z_P}+{h'(z)\over h(z)}</math>
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| similarly as above. It follows that ''h''′(''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
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| ''f''′(''z'')/''f''(''z'') at ''z''<sub>P</sub> is −''m''.
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| Putting these together, each zero ''z''<sub>''N''</sub> of multiplicity ''k'' of ''f'' creates a simple pole for
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| ''f''′(''z'')/''f''(''z'') with the residue being ''k'', and each pole ''z''<sub>P</sub> of order ''m'' of
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| ''f'' creates a simple pole for ''f''′(''z'')/''f''(''z'') with the residue being −''m''. (Here, by a simple pole we
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| mean a pole of order one.) In addition, it can be shown that ''f''′(''z'')/''f''(''z'') has no other poles,
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| and so no other residues.
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| 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>''N''</sub> is the number of zeros counting multiplicities of the zeros, and likewise for the poles, and so we have our result.
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| ==Applications and consequences==
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| 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.
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| The proof of [[Rouché's theorem]] uses the argument principle.
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| Modern books on feedback control theory quite frequently use the argument principle to serve as the theoretical basis of the [[Nyquist stability criterion]].
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| 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
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| :<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> | |
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| 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
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| :<math> \frac{1}{2\pi i} \oint_C z^k\frac{f'(z)}{f(z)}\, dz = z_1^k+z_2^k+\dots+z_p^k,</math>
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| is [[power sum symmetric polynomial]] of the roots of ''f''.
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| Another consequence is if we compute the complex integral:
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| : <math>\oint_C f(z){g'(z) \over g(z)}\, dz</math>
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| for an appropriate choice of ''g'' and ''f'' we have the [[Abel–Plana formula]]:
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| : <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>
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| which expresses the relationship between a discrete sum and its integral.
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| ==Generalized argument principle==
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| There is an immediate generalization of the argument principle. The integral
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| : <math>\oint_{C} {f'(z) \over f(z)} g(z) \, dz</math>
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| is equal to g evaluated at the zeroes, minues g evaluated at the poles.
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| ==History==
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| 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. However, according to this book, only zeroes were mentioned, not poles. This theorem by Cauchy was only published many years later in 1974 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.
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| == See also ==
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| * [[Logarithmic derivative]]
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| ==References==
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| * {{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}}
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| * {{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}}
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| * {{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}}
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| * Backlund, R.-J. (1914) Sur les zéros de la fonction zeta(s) de Riemann, C. R. Acad. Sci. Paris 158, 1979-1982.
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| {{DEFAULTSORT:Argument Principle}}
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| [[Category:Complex analysis]]
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| [[Category:Theorems in complex analysis]]
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