Argument principle

In complex analysis, the Argument principle (or Cauchy's argument principle) states that if "f"("z") is a
meromorphic function inside and on some closed contour "C", with "f" having no zeros or poles on "C", then the following formula holds: $oint\_\{C\}\; \{f\text{'}(z)\; over\; f(z)\},\; dz=2pi\; i\; (N-P)$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 and order respectively. 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 "C" is a curve, oriented counter-clockwise, which is contractible to a point inside an open set Ω in the complex plane. For each point "z" ∈ Ω, let "n"("C","z") be the winding number of "C" around the point "z". Then:$oint\_\{C\}\; frac\{f\text{'}(z)\}\{f(z)\},\; dz\; =\; 2pi\; i\; left(sum\_a\; n(C,a)\; -\; sum\_b\; n(C,b)\; ight)$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".

Proof

Let "z"_N be a zero of "f". We can write "f"("z") = ("z" −"z"_N)^"k""g"("z") where "k" is the multiplicity of the zero, and thus"g"("z"_N) ≠ 0. We get: $f\text{'}(z)=k(z-z\_N)^\{k-1\}g(z)+(z-z\_N)^kg\text{'}(z),!$and: $\{f\text{'}(z)over\; f(z)\}=\{k\; over\; z-z\_N\}+\{g\text{'}(z)over\; g(z)\}.$Since "g"("z"_N) ≠ 0, it follows that "g"′("z")/"g"("z") has no singularities at"z"_N, and thus is analytic at "z"_N, which implies that the residue of"f"′("z")/"f"("z") at "z"_N is "k".

Let "z"_P be a pole of "f". We can write "f"("z") = ("z" −"z"_P)^−"m""h"("z") where "m" is the order of the pole, and thus"h"("z"_P) ≠ 0. Then,: $f\text{'}(z)=-m(z-z\_P)^\{-m-1\}h(z)+(z-z\_P)^\{-m\}h\text{'}(z),!.$and: $\{f\text{'}(z)over\; f(z)\}=\{-m\; over\; z-z\_P\}+\{h\text{'}(z)over\; h(z)\}$similarly as above. It follows that "h"′("z")/"h"("z") has no singularities at "z"_P since"h"("z"_P) ≠ 0 and thus it is analytic at "z"_P. We find that the residue of"f"′("z")/"f"("z") at "z"_P is −"m".

Putting these together, each zero "z"_N of multiplicity "k" of "f" creates a simple pole for"f"′("z")/"f"("z") with the residue being "k", and each pole "z"_P of order "m" of"f" creates a simple pole for "f"′("z")/"f"("z") with the residue being −"m". (Here, by a simple pole wemean a pole of order one.) In addition, it can be shown that "f"′("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 2πi and the sum of the residues.Together, the sum of the "k" 's for each zero "z"_N is the number of zeros counting multiplicities of the zeros,and likewise for the poles, and so we have our result.

Consequences

This has consequences in considering the winding number of "f"("z") about the origin, say, if "C" is a closed contour centered on the origin. We see that the integral of "f"′("z")/"f"("z") about "C" is the change in values of log "f"("z"). Since "C" is closed we only need consider the change in $i$arg "f"("z") over "C" − which will be some multiple of 2π$i$ since "C" is closed (but may wind more than once about the origin). But since by the argument principle : $oint\_C\; \{f\text{'}(z)\; over\; f(z)\},\; dz=2pi\; i\; (N-P)$the factors of 2π$i$ cancel and so we are left with : $N-P\; =\; I(C,\; 0),!$where "I"("C",0) denotes the winding number of "f" over "C" about 0.

A consequence of the more general theorem is that, under the same hypothesis, if "g" is an analytic function in Ω, then

:$frac\{1\}\{2pi\; i\}\; oint\_C\; g(z)frac\{f\text{'}(z)\}\{f(z)\},\; dz\; =\; sum\_a\; n(C,a)g(a)\; -\; sum\_b\; n(C,b)g(b).$

For example, if "f" is a polynomial having zeros "z"₁, ..., "z"_p inside a simple contour "C", and "g"("z") = "z"^k, then:$frac\{1\}\{2pi\; i\}\; oint\_C\; z^kfrac\{f\text{'}(z)\}\{f(z)\},\; dz\; =\; z\_1^k+z\_2^k+dots+z\_p^k,$is power sum symmetric function of the roots of "f".

Another consequence is if we compute the complex integral:

: $oint\_C\; f(z)\{g\text{'}(z)\; over\; g(z)\},\; dz$

for an appropriate election for "g" and "f" we have the Abel-Plana formula:

: $sum\_\{n=0\}^\{infty\}f(n)-int\_\{0\}^\{infty\}f(x),dx=\; f(0)/2+iint\_\{0\}^\{infty\}frac\{f(it)-f(-it)\}\{e^\{2pi\; t\}-1\},\; dt$

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

History

According to the book by Frank Smithies ("Cauchy and the Creation of Complex Function Theory", Cambridge University Press, 1997), Augustin Louis Cauchy presented a theory similar to the above on 27 November 1831, during his self-imposed exile in Turin (capital of the Kingdom of Piedmont-Sardinia in 1831 but now just a city in northern Italy after the unification of Italy) away from France. (Please see page 177.) However, according to this book, only zeroes were mentioned, not poles. This theory by Cauchy was only published many years later in 1974 in a hand-written form and so is quite difficult to read. It can be found that Cauchy published a paper with a discussion on both "zeroes" and "poles" in 1855, two years before his death. Theorem 1 involved only "zeroes". Theorem 2 of Cauchy's 1855 paper stated that the "compteurs logarithmiques" (the logarithmic residue according to modern textbooks) of a function Z of a complex variable is equal to the difference of the number of the roots of Z and the roots of 1/Z (zeroes and poles of the function Z according to modern textbooks). Thus the modern "Argument Principle" can be found as a theorem in an 1855 paper by Augustin Louis Cauchy.

Applications

Modern books on feedback control theory quite frequently use the "Argument Principle" to serve as the theoretical basis of Nyquist stability criterion. The original 1932 paper by Harry Nyquist (H. Nyquist, "Regeneration theory", Bell System Technical Journal, vol. 11, pp. 126-147, 1932) used a rather clumsy and primitive approach to derive the Nyquist stability criterion. In his 1932 paper, Harry Nyquist did not mention Cauchy's name at all. Subsequently, both Leroy MacColl (Fundamental theory of servomechanisms, 1945) and Hendrik Bode (Network analysis and feedback amplifier design, 1945) started from the "Argument Principle" to derive the Nyquist stability criterion. MacColl (Bell Laboratories) mentioned the "Argument Principle" as Cauchy's theorem. Thus the "Argument Principle" has strong impact both on pure mathematics and control engineering. Nowadays, the "Argument Principle" can be found in many modern textbooks on complex analysis or control engineering.

References

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External links

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* [http://math.fullerton.edu/mathews/c2003/RoucheTheoremMod.html Module for the Argument Principle by John H. Mathews]
* [http://demonstrations.wolfram.com/AnIllustrationOfTheArgumentPrinciple/ An Illustration of the Argument Principle] by Keith Schneider, The Wolfram Demonstrations Project.

See also

* Rouché's theorem
* Contour integral
* Nyquist stability criterion
* Augustin Louis Cauchy