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Pole (of a function)

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An isolated singular point of single-valued character of an analytic function of the complex variable for which increases without bound when approaches : . In a sufficiently small punctured neighbourhood of the point , or in the case of the point at infinity , the function can be written as a Laurent series of special form:

(1)

or, respectively,

(2)

with finitely many negative exponents if , or, respectively, finitely many positive exponents if . The natural number in these expressions is called the order, or multiplicity, of the pole ; when the pole is called simple. The expressions (1) and (2) show that the function if , or if , can be analytically continued (cf. Analytic continuation) to a full neighbourhood of the pole , and, moreover, . Alternatively, a pole of order can also be characterized by the fact that the function has a zero of multiplicity at .

A point of the complex space , , is called a pole of the analytic function of several complex variables if the following conditions are satisfied: 1) is holomorphic everywhere in some neighbourhood of except at a set , ; 2) cannot be analytically continued to any point of ; and 3) there exists a function , holomorphic in , such that the function , which is holomorphic in , can be holomorphically continued to the full neighbourhood , and, moreover, . Here also

however, for , poles, as with singular points in general, cannot be isolated.

References

[1] B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian)


Comments

For see [a1]. For see [a2][a3].

For the use of poles in the representation of analytic functions see Integral representation of an analytic function; Cauchy integral.

References

[a1] L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. Chapt. 8
[a2] H. Grauert, K. Fritzsche, "Several complex variables" , Springer (1976) (Translated from German)
[a3] R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. 1, Sect. 3
How to Cite This Entry:
Pole (of a function). Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Pole_(of_a_function)&oldid=15756
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article