Residue of an analytic function

$ f( z) $ of one complex variable at a finite isolated singular point $ a $ of unique character

The coefficient $ c _ {-1} $ of $ {( z - a) } ^ {- 1} $ in the Laurent expansion of the function $ f( z) $ (cf. Laurent series) in a neighbourhood of $ a $, or the integral

$$ \frac{1}{2 \pi i } \int\limits _ \gamma f ( z) dz, $$

where $ \gamma $ is a circle of sufficiently small radius with centre at $ a $, which is equal to it. The residue is denoted by $ \mathop{\rm res} [ f ( z) ; a ] $.

The theory of residues is based on the Cauchy integral theorem. The residue theorem is fundamental in this theory. Let $ f( z) $ be a single-valued analytic function everywhere in a simply-connected domain $ G $, except for isolated singular points; then the integral of $ f( z) $ over any simple closed rectifiable curve $ \gamma $ lying in $ G $ and not passing through the singular points of $ f ( z) $ can be computed by the formula

$$ \int\limits _ \gamma f ( z) dz = \ 2 \pi i \sum _ {k = 1 } ^ { N } \mathop{\rm res} [ f ( z) ; a _ {k} ], $$

where $ a _ {k} $, $ k= 1 \dots N $, are the singular points of $ f( z) $ inside $ \gamma $.

The residue of a function at the point at infinity $ a = \infty $, for a function $ f( z) $ which is single-valued and analytic in a neighbourhood of that point, is defined by the formula

$$ \mathop{\rm res} [ f ( z) ; \infty ] = \ \frac{1}{2 \pi i } \int\limits _ {\gamma ^ {-} } f ( z) dz = - c _ {- 1} , $$

where $ \gamma ^ {-} $ is a circle of sufficiently large radius, oriented clockwise, while $ c _ {- 1} $ is the coefficient of $ z ^ {- 1} $ in the Laurent expansion of $ f ( z) $ in a neighbourhood of the point at infinity. The residue theorem implies the theorem on the total sum of residues: If $ f( z) $ 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 $ f( z) $, 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 $ a \neq \infty $ be a pole of order $ m $ of the function $ f( z) $ (cf. Pole (of a function)); then

$$ \mathop{\rm res} [ f ( z) ; a ] = \ \frac{1}{( m - 1)! } \lim\limits _ {z \rightarrow a } \left \{ \frac{d ^ {m - 1 } }{dz ^ {m - 1 } } [( z - a) ^ {m} f ( z)] \right \} . $$

If $ m = 1 $ (a simple pole), the formula becomes

$$ \mathop{\rm res} [ f ( z) ; a ] = \ \lim\limits _ {z \rightarrow a } [( z - a) f ( z)]; $$

if $ f( z) = \phi ( z)/ \psi ( z) $, where $ \phi ( z) $ and $ \psi ( z) $ are regular in a neighbourhood of $ a $, and if $ a $ is a simple zero for $ \psi ( z) $, then

$$ \mathop{\rm res} \left [ \frac{\phi ( z) }{\psi ( z) } ; a \right ] = \ \frac{\phi ( a) }{\psi ^ \prime ( a) } . $$

The application of the residue theorem to the logarithmic derivative yields the important theorem on logarithmic residues: If a function $ f( z) $ is meromorphic in a simply-connected domain $ G $, while the simple closed curve $ \gamma $ lies in $ G $ and does not pass through zeros or poles of $ f( z) $, then

$$ \frac{1}{2 \pi i } \int\limits _ \gamma \frac{f ^ { \prime } ( z) }{f ( z) } dz = N - P, $$

where $ N $ is the number of zeros and $ P $ is the number of poles of $ f( z) $ inside $ \gamma $ 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 $ \gamma $ (see also Argument, principle of the).

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

$$ J _ {1} = \ \int\limits _ { 0 } ^ { {2 } \pi } R ( \sin t, \cos t) dt, $$

$$ J _ {2} = \int\limits _ {- \infty } ^ \infty f ( x) dx,\ J _ {3} = \int\limits _ {- \infty } ^ \infty e ^ {ix} f( x) dx, $$

where $ R ( \sin t, \cos t ) $ is a rational function of $ \sin t $, $ \cos t $ which is continuous if $ 0 \leq t \leq 2 \pi $, and $ f( z) $ is a continuous function if $ \mathop{\rm Im} z \geq 0 $, where $ \mathop{\rm Im} z $ is the imaginary part of $ z $, and is analytic if $ \mathop{\rm Im} z > 0 $ except for a finite number of singular points. By substituting $ e ^ {it} = z $, $ J _ {1} $ is reduced to the contour integral

$$ \int\limits _ {| z | = 1 } R \left ( \frac{z ^ {2} - 1 }{2 iz } ,\ \frac{z ^ {2} + 1 }{2z } \right ) \ \frac{dz }{iz } , $$

i.e. to the computation of the residues;

$$ J _ {2} = 2 \pi i \sum _ { \mathop{\rm Im} a > 0 } \mathop{\rm res} [ f ( z) ; a ], $$

if $ f( z) z ^ {r} \rightarrow 0 $ as $ z \rightarrow \infty $, $ \mathop{\rm Im} z \geq 0 $, $ r > 1 $; and

$$ J _ {3} = 2 \pi i \sum _ { \mathop{\rm Im} a > 0 } \mathop{\rm res} [ e ^ {iz} f ( z) ; a ], $$

if $ f ( z) $ 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 $ d Z $ in a neighbourhood of (one of) its isolated singular points is defined as the coefficient $ c _ {- 1} $ of $ z ^ {- 1} $ in the Laurent expansion of the function $ g( z) = d Z/ d z $, where $ z $ is a uniformizing parameter (cf. Uniformization) in a neighbourhood of this point. The integral of $ d Z $ along any closed curve on the Riemann surface can be expressed in terms of the residues of the differential $ d Z $ and its cyclic periods (the integrals of $ d Z $ 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 $ X $. Leray's residue theory describes, in particular, a method of computing integrals along certain cycles on $ X $ 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 $ \omega $ with a first-order polar singularity on a complex-analytic submanifold $ S $ along a given cycle in $ X \setminus S $ to the computation of an integral of the residue form $ \mathop{\rm res} [ \omega ] $ along a cycle on $ S $ of one dimension lower. In calculating integrals of closed forms with arbitrary singularities on $ S $, the important concepts are those of a residue class (cf. Residue form) and the Leray theorem, according to which any closed form $ \omega \in C ^ \infty ( X \setminus S) $ has a corresponding cohomologous form $ \omega _ {0} $ with a first-order polar singularity on $ S $. For a form $ \omega $ with a singularity on several submanifolds $ ( S _ {1} \cup \dots \cup S _ {m} ) $ one uses the composite residue form

$$ {\rm res } ^ {m} [ \omega ] \in C ^ \infty ( S _ {1} \cap \dots \cap S _ {m} ), $$

the residue class

$$ {\rm Res} ^ {m} [ \omega ] \in H ^ {*} ( S _ {1} \cap \dots \cap S _ {m} ) $$

and the residue formula

$$ \int\limits _ {\delta ^ {m} \gamma } \omega = ( 2 \pi i) ^ {m} \int\limits _ \gamma {\rm Res} ^ {m} [ \omega ], $$

where $ \delta ^ {m} $ is the composite Leray coboundary operator associated to the Leray coboundary operator $ \delta $ and $ \gamma $ is a cycle in $ S _ {1} \cap \dots \cap S _ {m} $.

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 $ f( z) $, $ z = ( z _ {1} \dots z _ {n} ) $, be a holomorphic function in a domain $ G \subset \mathbf C ^ {n} $, and let $ \sigma $ be an $ n $-dimensional cycle in $ G $. If $ \{ \sigma _ {1} \dots \sigma _ {p} \} $ is a basis of the $ n $-dimensional homology space of the domain $ G $ and

$$ \sigma \sim \ \sum _ {v = 1 } ^ { p } k _ {v} \sigma _ {v} $$

is the expansion of $ \sigma $ with respect to this basis, a generalization of the residue theorem has the form

$$ \int\limits _ \sigma f ( z) dz = \ ( 2 \pi i) ^ {n} \sum _ {v = 1 } ^ { p } k _ {v} R _ {v} ,\ \ dz = dz _ {1} \wedge \dots \wedge dz _ {n} , $$

where

$$ R _ {v} = \ \frac{1}{( 2 \pi i) ^ {n} } \int\limits _ {\sigma _ {v} } f ( z) dz $$

is an $ n $-dimensional analogue of the residue and is called the residue of the function $ f( z) $ with respect to the basic cycle $ \sigma _ {v} $. As distinct from the case of one variable, it is very difficult to find both a homology basis $ \{ \sigma _ {v} \} $ and the coefficients $ \{ k _ {v} \} $. In several cases (for example, when $ G = \mathbf C ^ {2} \setminus \{ P ( z _ {1} , z _ {2} ) = 0 \} $, where $ P $ is a polynomial) these problems may be solved with the aid of Alexander–Pontryagin duality. The coefficients $ k _ {v} $ are found as the linking coefficients of the cycle $ \sigma $ with the cycles on the set $ \mathbf C ^ {n} \setminus G $ (compactified in a certain manner) which are dual to the cycles $ \sigma _ {v} $. The residues $ R _ {v} $ can in some cases be found as the respective coefficients of the Laurent expansion of the function $ f( z) $.

Multi-dimensional analogues of logarithmic residues [4], [8]–[9] express the number of common zeros (counted with multiplicities) of a system of holomorphic functions $ f = ( f _ {1} \dots f _ {n} ) $ in a domain $ D \subset \subset G \subset \mathbf C ^ {n} $ by means of the integrals

$$ N ( f, D) = \ \frac{( n - 1)! }{( 2 \pi i) ^ {n} } \int\limits _ {\partial D } \frac{1}{| f | ^ {2n} } \times $$

$$ \times \sum _ {v = 1 } ^ { n } {\overline{f}\; } _ {v} df _ {v} \wedge d {\overline{f}\; } _ {1} \wedge df _ {1} \wedge \dots [ v] \dots \wedge d {\overline{f}\; } _ {n} \wedge df _ {n} , $$

$$ N ( f, D) = \frac{1}{( 2 \pi i) ^ {n} } \int\limits _ \gamma \frac{df _ {1} }{f _ {1} } \wedge \dots \wedge \frac{df _ {n} }{f _ {n} } , $$

where $ \gamma $ is some cycle in $ \partial D \setminus \cup _ {j= 1} ^ {n} \{ f _ {j} ( z) = 0 \} $. 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) MR0444912 Zbl 0357.30002
[2]	M.A. Evgrafov, "Analytic functions" , Saunders , Philadelphia (1966) (Translated from Russian) MR0197686 Zbl 0147.32605
[3]	I.I. [I.I. Privalov] Priwalow, "Einführung in die Funktionentheorie" , 1–3 , Teubner (1958–1959) (Translated from Russian) MR0342680 MR0264037 MR0264036 MR0264038 MR0123686 MR0123685 MR0098843 Zbl 0177.33401 Zbl 0141.26003 Zbl 0141.26002 Zbl 0082.28802
[4]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1985) (In Russian) Zbl 0578.32001 Zbl 0574.30001
[5]	G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 MR0092855 Zbl 0078.06602
[6]	H. Poincaré, "Sur les résidues des intégrales doubles" Acta Math. , 9 (1887) pp. 321–380 Zbl 19.0275.01
[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) MR0735793
[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 MR0260733 Zbl 0215.08103
[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) MR0507725 Zbl 0408.14001
[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) MR754560 Zbl 0546.30004