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 Image File history File links No higher resolution available. ... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. ... In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0. ...

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. In mathematics, the multiplicity of a member of a multiset is how many memberships in the multiset it has. ... In complex analysis, a pole of a holomorphic function is a certain type of simple singularity that behaves like the singularity 1/zn at z = 0. ...

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 This is a glossary of some terms used in the branch of mathematics known as topology. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... A point z0 and a curve C In mathematics, the winding number is a topological invariant playing a leading role in complex analysis. ...

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)^kg(z) where k is the multiplicity of the zero, and thus g(z_N) ≠ 0. We get

and

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. In complex analysis, the residue is a complex number which describes the behavior of line integrals of a meromorphic function around a singularity. ...

Let z_P be a pole of f. We can write f(z) = (z − z_P)^−mh(z) where m is the order of the pole, and thus h(z_P) ≠ 0. Then,

and

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 we mean 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. The residue theorem in complex analysis is a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. ...

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 2πi since C is closed (but may wind more than once about the origin). But since by the argument principle A point z0 and a curve C In mathematics, the winding number is a topological invariant playing a leading role in complex analysis. ...

the factors of 2πi cancel and so we are left with

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

For example, if f is a polynomial having zeros z₁, ..., z_p inside a simple contour C, and g(z) = z^k, then In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...

is power sum symmetric function of the roots of f. In mathematics, Newtons identities relate two different ways of describing the roots of a polynomial. ... In mathematics, the theory of symmetric functions is part of the theory of polynomial equations, and also a substantial chapter of combinatorics. ...

Another consequence is if we compute the complex integral:

for an appropriate election for g and f we have the Abel-Plana formula: Niels Henrik Abel (August 5, 1802–April 6, 1829), Norwegian mathematician, was born in Nedstrand, near Finnøy where his father acted as rector. ... Giovanni Antonio Amedeo Plana (November 6, 1781–January 20, 1864) was an Italian astronomer and mathematician. ...

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 27th 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 many years later in 1974 in a hand-written form and so is quite difficult to read. (This happened probably because of Cauchy's self-imposed exile in 1831.) However, according to this paper presented in 1831, only zeroes were mentioned, not poles. It can be found by literature survey 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 a 1855 paper by Augustin Louis Cauchy, who was one of the greatest French mathematicians. Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ...

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. The Nyquist plot for . ... Harry Nyquist (February 7, 1889 - April 4, 1976) was an important contributor to information theory. ... The Nyquist plot for . ... Harry Nyquist (February 7, 1889 - April 4, 1976) was an important contributor to information theory. ... Dr. Hendrik Wade Bode Hendrik Wade Bode (pronounced Boh-dee in English, Boh-dah in Dutch),[1] (24 December 1905 Madison, Wisconsin – 21 June 1982 Cambridge, Massachusetts) was a gifted researcher, prolific inventor and eloquent and nuanced engineer, author and scientist, an American of Dutch ancestry. ... The Nyquist plot for . ... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... Control engineering is the engineering discipline that focuses on the mathematical modelling systems of a diverse nature, analysing their dynamic behaviour, and using control theory to make a controller that will cause the systems to behave in a desired manner. ...

References

• Ahlfors, Lars (1979). Complex Analysis. McGraw Hill. 

External links

• Eric W. Weisstein, Argument Principle at MathWorld.
• Module for the Argument Principle by John H. Mathews

Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

See also

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