View source for Residue of an analytic function

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r0815601.png" /> of one complex variable at a finite isolated [[Singular point|singular point]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r0815602.png" /> of unique character''

The coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r0815603.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r0815604.png" /> in the Laurent expansion of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r0815605.png" /> (cf. [[Laurent series|Laurent series]]) in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r0815606.png" />, or the integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r0815607.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r0815608.png" /> is a circle of sufficiently small radius with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r0815609.png" />, which is equal to it. The residue is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156010.png" />.

The theory of residues is based on the [[Cauchy integral theorem|Cauchy integral theorem]]. The residue theorem is fundamental in this theory. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156011.png" /> be a single-valued [[Analytic function|analytic function]] everywhere in a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156012.png" />, except for isolated singular points; then the integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156013.png" /> over any simple closed rectifiable curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156014.png" /> lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156015.png" /> and not passing through the singular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156016.png" /> can be computed by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156017.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156019.png" />, are the singular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156020.png" /> inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156021.png" />.

The residue of a function at the point at infinity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156022.png" />, for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156023.png" /> which is single-valued and analytic in a neighbourhood of that point, is defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156024.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156025.png" /> is a circle of sufficiently large radius, oriented clockwise, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156026.png" /> is the coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156027.png" /> in the Laurent expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156028.png" /> in a neighbourhood of the point at infinity. The residue theorem implies the theorem on the total sum of residues: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156029.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156030.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156031.png" /> be a pole of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156032.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156033.png" /> (cf. [[Pole (of a function)|Pole (of a function)]]); then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156034.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156035.png" /> (a simple pole), the formula becomes

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156036.png" /></td> </tr></table>

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156039.png" /> are regular in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156040.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156041.png" /> is a simple zero for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156042.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156043.png" /></td> </tr></table>

The application of the residue theorem to the logarithmic derivative yields the important theorem on logarithmic residues: If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156044.png" /> is meromorphic in a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156045.png" />, while the simple closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156046.png" /> lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156047.png" /> and does not pass through zeros or poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156048.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156049.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156050.png" /> is the number of zeros and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156051.png" /> is the number of poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156052.png" /> inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156053.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156054.png" /> (see also [[Argument, principle of the|Argument, principle of the]]).

Residues are employed in computing certain integrals of real-valued functions, such as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156055.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156056.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156057.png" /> is a rational function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156059.png" /> which is continuous if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156060.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156061.png" /> is a continuous function if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156062.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156063.png" /> is the imaginary part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156064.png" />, and is analytic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156065.png" /> except for a finite number of singular points. By substituting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156067.png" /> is reduced to the contour integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156068.png" /></td> </tr></table>

i.e. to the computation of the residues;

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156069.png" /></td> </tr></table>

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156070.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156073.png" />; and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156074.png" /></td> </tr></table>

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156075.png" /> satisfies the conditions of the [[Jordan lemma|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 [[#References|[1]]]–[[#References|[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 [[#References|[5]]] (cf. also [[Differential on a Riemann surface|Differential on a Riemann surface]]). The residue of an analytic differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156076.png" /> in a neighbourhood of (one of) its isolated singular points is defined as the coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156077.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156078.png" /> in the Laurent expansion of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156079.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156080.png" /> is a uniformizing parameter (cf. [[Uniformization|Uniformization]]) in a neighbourhood of this point. The integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156081.png" /> along any closed curve on the Riemann surface can be expressed in terms of the residues of the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156082.png" /> and its cyclic periods (the integrals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156083.png" /> along canonical cuts, cf. [[Canonical sections|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 [[#References|[8]]]–[[#References|[10]]], [[#References|[12]]], [[#References|[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 [[#References|[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|Abelian integral]]), and employed double residues as the basis of a two-dimensional analogue of [[Lagrange series|Lagrange series]].

J. Leray [[#References|[7]]] (see also [[#References|[4]]], [[#References|[8]]]) developed the general theory of residues on a complex-analytic manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156084.png" />. Leray's residue theory describes, in particular, a method of computing integrals along certain cycles on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156085.png" /> of closed exterior differential forms with singularities on analytic submanifolds. He introduced the concept of a [[Residue form|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156086.png" /> with a first-order polar singularity on a complex-analytic submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156087.png" /> along a given cycle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156088.png" /> to the computation of an integral of the residue form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156089.png" /> along a cycle on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156090.png" /> of one dimension lower. In calculating integrals of closed forms with arbitrary singularities on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156091.png" />, the important concepts are those of a residue class (cf. [[Residue form|Residue form]]) and the Leray theorem, according to which any closed form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156092.png" /> has a corresponding cohomologous form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156093.png" /> with a first-order polar singularity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156094.png" />. For a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156095.png" /> with a singularity on several submanifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156096.png" /> one uses the composite residue form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156097.png" /></td> </tr></table>

the residue class

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156098.png" /></td> </tr></table>

and the residue formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r08156099.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560100.png" /> is the composite Leray coboundary operator associated to the Leray coboundary operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560102.png" /> is a cycle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560103.png" />.

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|Alexander duality]] [[#References|[8]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560105.png" />, be a holomorphic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560106.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560107.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560108.png" />-dimensional cycle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560109.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560110.png" /> is a basis of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560111.png" />-dimensional homology space of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560112.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560113.png" /></td> </tr></table>

is the expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560114.png" /> with respect to this basis, a generalization of the residue theorem has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560115.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560116.png" /></td> </tr></table>

is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560117.png" />-dimensional analogue of the residue and is called the residue of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560118.png" /> with respect to the basic cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560119.png" />. As distinct from the case of one variable, it is very difficult to find both a homology basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560120.png" /> and the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560121.png" />. In several cases (for example, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560122.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560123.png" /> is a polynomial) these problems may be solved with the aid of Alexander–Pontryagin duality. The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560124.png" /> are found as the linking coefficients of the cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560125.png" /> with the cycles on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560126.png" /> (compactified in a certain manner) which are dual to the cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560127.png" />. The residues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560128.png" /> can in some cases be found as the respective coefficients of the Laurent expansion of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560129.png" />.

Multi-dimensional analogues of logarithmic residues [[#References|[4]]], [[#References|[8]]]–[[#References|[9]]] express the number of common zeros (counted with multiplicities) of a system of holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560130.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560131.png" /> by means of the integrals

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560132.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560133.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560134.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560135.png" /> is some cycle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081560/r081560136.png" />. Residues of functions of several variables have found use in the study of Feynman integrals, in combinatorial analysis [[#References|[11]]] and in the theory of implicit functions [[#References|[8]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Evgrafov,   "Analytic functions" , Saunders , Philadelphia  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,   "Einführung in die Funktionentheorie" , '''1–3''' , Teubner  (1958–1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.V. Shabat,   "Introduction of complex analysis" , '''1–2''' , Moscow  (1985)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  G. Springer,   "Introduction to Riemann surfaces" , Addison-Wesley  (1957)  pp. Chapt.10</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  H. Poincaré,   "Sur les résidues des intégrales doubles"  ''Acta Math.'' , '''9'''  (1887)  pp. 321–380</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A.K. Tsikh,   "Multidimensional residues and its applications" , Amer. Math. Soc.  (Forthcoming)  (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  P.A. Griffiths,   "On the periods of certain rational integrals I"  ''Ann. of Math. (2)'' , '''90''' :  3  (1969)  pp. 460–495</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  G.P. Egorichev,   "Integral representation and the computation of combinatorial sums" , Amer. Math. Soc.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  P.A. Griffiths,   J.E. Harris,   "Principles of algebraic geometry" , Wiley (Interscience)  (1978)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  W.R. Coleff,   M.F. Herrera,   "Les courants residuals associés à une forme meromorphe" , ''Lect. notes in math.'' , '''633''' , Springer  (1978)</TD></TR></table>



====Comments====
See also the comments and references to [[Residue form|Residue form]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.S. Mitrinović,   J.D. Kečkić,   "The Cauchy method of residues: theory and applications" , Reidel  (1984)</TD></TR></table>