Pole (of a function)
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 |
Pole (of a function). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pole_(of_a_function)&oldid=15756