Residue of an analytic function
of one complex variable at a finite isolated singular point
of unique character
The coefficient of
in the Laurent expansion of the function
(cf. Laurent series) in a neighbourhood of
, or the integral
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where is a circle of sufficiently small radius with centre at
, which is equal to it. The residue is denoted by
.
The theory of residues is based on the Cauchy integral theorem. The residue theorem is fundamental in this theory. Let be a single-valued analytic function everywhere in a simply-connected domain
, except for isolated singular points; then the integral of
over any simple closed rectifiable curve
lying in
and not passing through the singular points of
can be computed by the formula
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where ,
, are the singular points of
inside
.
The residue of a function at the point at infinity , for a function
which is single-valued and analytic in a neighbourhood of that point, is defined by the formula
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where is a circle of sufficiently large radius, oriented clockwise, while
is the coefficient of
in the Laurent expansion of
in a neighbourhood of the point at infinity. The residue theorem implies the theorem on the total sum of residues: If
is a single-valued analytic function in the extended complex plane, except for a finite number of singular points, then the sum of all residues of
, including the residue at the point at infinity, is zero.
Thus, the computation of integrals of analytic functions along closed curves (contour integrals) is reduced to the computation of residues, which is particularly simple in the case of finite poles. Let be a pole of order
of the function
(cf. Pole (of a function)); then
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If (a simple pole), the formula becomes
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if , where
and
are regular in a neighbourhood of
, and if
is a simple zero for
, then
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The application of the residue theorem to the logarithmic derivative yields the important theorem on logarithmic residues: If a function is meromorphic in a simply-connected domain
, while the simple closed curve
lies in
and does not pass through zeros or poles of
, then
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where is the number of zeros and
is the number of poles of
inside
counted with multiplicities. The expression on the left-hand side of the formula is called the logarithmic residue of the function with respect to the curve
(see also Argument, principle of the).
Residues are employed in computing certain integrals of real-valued functions, such as
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where is a rational function of
,
which is continuous if
, and
is a continuous function if
, where
is the imaginary part of
, and is analytic if
except for a finite number of singular points. By substituting
,
is reduced to the contour integral
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i.e. to the computation of the residues;
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if as
,
,
; and
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if satisfies the conditions of the Jordan lemma.
Residues have found numerous important applications in problems of analytic continuation, decomposition of meromorphic functions into partial fractions, summation of power series, asymptotic estimation, and many other problems of theoretical and applied analysis [1]–[4].
The theory of residues in one variable was mostly developed by A.L. Cauchy in 1825–1829. A number of results concerning the generalizations of the theory were obtained by Ch. Hermite (a theorem on the sum of the residues of doubly-periodic functions), P. Laurent, Yu.V. Sokhotskii, E. Lindelöf, and others.
Residues of analytic differentials rather than residues of analytic functions are studied on Riemann surfaces [5] (cf. also Differential on a Riemann surface). The residue of an analytic differential in a neighbourhood of (one of) its isolated singular points is defined as the coefficient
of
in the Laurent expansion of the function
, where
is a uniformizing parameter (cf. Uniformization) in a neighbourhood of this point. The integral of
along any closed curve on the Riemann surface can be expressed in terms of the residues of the differential
and its cyclic periods (the integrals of
along canonical cuts, cf. Canonical sections). The theorem on the total sum of residues is applicable to Riemann surfaces: The sum of all residues of a meromorphic differential on a compact Riemann surface is zero.
The theory of residues of analytic functions of several complex variables.
See [8]–[10], [12], [13]. This theory is based on the integral theorems of Stokes and Cauchy–Poincaré, which make it possible to replace the integral of a closed form along one cycle by an integral of this form along another cycle which is homologous to the former. The foundations of the theory of residues of functions of several variables were laid by H. Poincaré [6], who was the first (1887) to generalize Cauchy's integral theorem and the concept of a residue to functions of two complex variables; he showed, in particular, that the integral of a rational function of two complex variables along a two-dimensional cycle which does not pass through the singularities of the integrand can be reduced to the periods of Abelian integrals (cf. Abelian integral), and employed double residues as the basis of a two-dimensional analogue of Lagrange series.
J. Leray [7] (see also [4], [8]) developed the general theory of residues on a complex-analytic manifold . Leray's residue theory describes, in particular, a method of computing integrals along certain cycles on
of closed exterior differential forms with singularities on analytic submanifolds. He introduced the concept of a residue form, which generalizes the concept of a residue of an analytic function of a single variable; the residue formula thus obtained makes it possible to reduce the computation of the integral of a form
with a first-order polar singularity on a complex-analytic submanifold
along a given cycle in
to the computation of an integral of the residue form
along a cycle on
of one dimension lower. In calculating integrals of closed forms with arbitrary singularities on
, the important concepts are those of a residue class (cf. Residue form) and the Leray theorem, according to which any closed form
has a corresponding cohomologous form
with a first-order polar singularity on
. For a form
with a singularity on several submanifolds
one uses the composite residue form
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the residue class
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and the residue formula
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where is the composite Leray coboundary operator associated to the Leray coboundary operator
and
is a cycle in
.
There exists another approach to the theory of residues of functions of several complex variables — the method of distinguishing a homology basis, based on an idea of E. Martinelli and involving the use of Alexander duality [8]. Let ,
, be a holomorphic function in a domain
, and let
be an
-dimensional cycle in
. If
is a basis of the
-dimensional homology space of the domain
and
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is the expansion of with respect to this basis, a generalization of the residue theorem has the form
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where
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is an -dimensional analogue of the residue and is called the residue of the function
with respect to the basic cycle
. As distinct from the case of one variable, it is very difficult to find both a homology basis
and the coefficients
. In several cases (for example, when
, where
is a polynomial) these problems may be solved with the aid of Alexander–Pontryagin duality. The coefficients
are found as the linking coefficients of the cycle
with the cycles on the set
(compactified in a certain manner) which are dual to the cycles
. The residues
can in some cases be found as the respective coefficients of the Laurent expansion of the function
.
Multi-dimensional analogues of logarithmic residues [4], [8]–[9] express the number of common zeros (counted with multiplicities) of a system of holomorphic functions in a domain
by means of the integrals
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where is some cycle in
. Residues of functions of several variables have found use in the study of Feynman integrals, in combinatorial analysis [11] and in the theory of implicit functions [8].
References
[1] | A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian) |
[2] | M.A. Evgrafov, "Analytic functions" , Saunders , Philadelphia (1966) (Translated from Russian) |
[3] | I.I. [I.I. Privalov] Priwalow, "Einführung in die Funktionentheorie" , 1–3 , Teubner (1958–1959) (Translated from Russian) |
[4] | B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1985) (In Russian) |
[5] | G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 |
[6] | H. Poincaré, "Sur les résidues des intégrales doubles" Acta Math. , 9 (1887) pp. 321–380 |
[7] | J. Leray, "Le calcule différentiel et intégral sur une variété analytique complexe (Problème de Cauchy, III)" Bull. Soc. Math. France , 87 (1959) pp. 81–180 |
[8] | L.A. Aizenberg, A.P. Yuzhakov, "Integral representations and residues in multidimensional complex analysis" , Transl. Math. Monogr. , 58 , Amer. Math. Soc. (1983) (Translated from Russian) |
[9] | A.K. Tsikh, "Multidimensional residues and its applications" , Amer. Math. Soc. (Forthcoming) (Translated from Russian) |
[10] | P.A. Griffiths, "On the periods of certain rational integrals I" Ann. of Math. (2) , 90 : 3 (1969) pp. 460–495 |
[11] | G.P. Egorichev, "Integral representation and the computation of combinatorial sums" , Amer. Math. Soc. (1984) (Translated from Russian) |
[12] | P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) |
[13] | W.R. Coleff, M.F. Herrera, "Les courants residuals associés à une forme meromorphe" , Lect. notes in math. , 633 , Springer (1978) |
Comments
See also the comments and references to Residue form.
References
[a1] | D.S. Mitrinović, J.D. Kečkić, "The Cauchy method of residues: theory and applications" , Reidel (1984) |
Residue of an analytic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Residue_of_an_analytic_function&oldid=23957