In
complex analysis, the argument principle (or Cauchy's argument principle) relates the difference between the number of
zeros and poles
In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. In some sense, it is the simplest type of singularity. Technically, a point is a pole of a function if ...
of a
meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
to a
contour integral
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.
Contour integration is closely related to the calculus of residues, a method of complex analysis.
...
of the function's
logarithmic derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function ''f'' is defined by the formula
\frac
where f' is the derivative of ''f''. Intuitively, this is the infinitesimal relative change in ''f'' ...
.
Specifically, if ''f''(''z'') is a meromorphic function inside and on some closed contour ''C'', and ''f'' has no zeros or poles on ''C'', then
:
where ''Z'' 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
Multiplicity may refer to: In science and the humanities
* Multiplicity (mathematics), the number of times an element is repeated in a multiset
* Multiplicity (philosophy), a philosophical concept
* Multiplicity (psychology), having or using mult ...
and
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''(''z'') is a meromorphic function on an
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
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
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...
to a point inside Ω. For each point ''z'' ∈ Ω, let ''n''(''C'',''z'') be the
winding number
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of t ...
of ''C'' around ''z''. Then
:
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
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.
Contour integration is closely related to the calculus of residues, a method of complex analysis.
...
can be interpreted as 2π''i'' times the winding number of the path ''f''(''C'') around the origin, using the substitution ''w'' = ''f''(''z''):
:
That is, it is ''i'' times the total change in the
argument of ''f''(''z'') as ''z'' travels around ''C'', explaining the name of the theorem; this follows from
:
and the relation between arguments and logarithms.
Proof of the argument principle
Let ''z''
''Z'' be a zero of ''f''. We can write ''f''(''z'') = (''z'' − ''z''
''Z'')
''k''''g''(''z'') where ''k'' is the multiplicity of the zero, and thus ''g''(''z''
''Z'') ≠ 0. We get
:
and
:
Since ''g''(''z''
''Z'') ≠ 0, it follows that ''g' ''(''z'')/''g''(''z'') has no singularities at ''z''
''Z'', and thus is analytic at ''z''
Z, which implies that the
residue
Residue may refer to:
Chemistry and biology
* An amino acid, within a peptide chain
* Crop residue, materials left after agricultural processes
* Pesticide residue, refers to the pesticides that may remain on or in food after they are applied ...
of ''f''′(''z'')/''f''(''z'') at ''z''
''Z'' 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
''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''
''Z'' 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
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as wel ...
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''
''Z'' 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
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's function inside a rectangle intersecting the critical line.
The proof of
Rouché's theorem
Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions and holomorphic inside some region K with closed contour \partial K, if on \partial K, then and have the same number of zeros inside K, wher ...
uses the argument principle.
Modern books on feedback control theory quite frequently use the argument principle to serve as the theoretical basis of the
Nyquist stability criterion
In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer at Siemens in 1930 and the Swedish-American electrical engineer Harry ...
.
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
:
For example, if ''f'' is a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
having zeros ''z''
1, ..., ''z''
p inside a simple contour ''C'', and ''g''(''z'') = ''z''
k, then
:
is
power sum symmetric polynomial In mathematics, specifically in commutative algebra, the power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients can be expressed as a su ...
of the roots of ''f''.
Another consequence is if we compute the complex integral:
:
for an appropriate choice of ''g'' and ''f'' we have the
Abel–Plana formula
In mathematics, the Abel–Plana formula is a summation formula discovered independently by and . It states that
:\sum_^\infty f(n)=\frac 1 2 f(0)+ \int_0^\infty f(x) \, dx+ i \int_0^\infty \frac \, dt.
It holds for functions ''f'' that are hol ...
:
:
which expresses the relationship between a discrete sum and its integral.
Generalized argument principle
There is an immediate generalization of the argument principle. Suppose that g is analytic in the region
. Then
:
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
Frank Smithies FRSE (1912–2002) was a British mathematician who worked on integral equations, functional analysis, and the history of mathematics.
He was elected as a fellow of the Royal Society of Edinburgh in 1961.
He was an alumnus and an ...
(''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 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
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function ''f'' is defined by the formula
\frac
where f' is the derivative of ''f''. Intuitively, this is the infinitesimal relative change in ''f'' ...
*
Nyquist stability criterion
In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer at Siemens in 1930 and the Swedish-American electrical engineer Harry ...
References
*
*
*
* Backlund, R.-J. (1914) Sur les zéros de la fonction zeta(s) de Riemann, C. R. Acad. Sci. Paris 158, 1979–1982.
External links
*
{{DEFAULTSORT:Argument Principle
Theorems in complex analysis