Difference between revisions of "Laurent series"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) pp. Chapt. 4 (Translated from Russian) {{MR|0444912}} {{ZBL|0357.30002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''1–2''' , Moscow (1976) pp. Vol. 1, Chapt. 2; Vol. 2, Chapt. 1 (In Russian) {{MR|}} {{ZBL|0799.32001}} {{ZBL|0732.32001}} {{ZBL|0732.30001}} {{ZBL|0578.32001}} {{ZBL|0574.30001}} </TD></TR></table> |
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− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H.C. Hutchins, "Examples of commutative rings" , Polygonal (1981) {{MR|0638720}} {{ZBL|0492.13001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.M. Cohn, "Skew field constructions" , Cambridge Univ. Press (1977) {{MR|0463237}} {{ZBL|0355.16009}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 241 {{MR|0510197}} {{MR|1535085}} {{MR|0188405}} {{MR|1570643}} {{MR|1528598}} {{MR|0054016}} {{ZBL|0395.30001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) {{MR|0344507}} {{ZBL|0271.32001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979) {{MR|0593142}} {{MR|0197687}} {{MR|1523319}} {{ZBL|0477.30001}} {{ZBL|0336.30001}} {{ZBL|0005.21004}} {{ZBL|65.0302.01}} {{ZBL|58.0297.01}} </TD></TR></table> |
Revision as of 16:59, 15 April 2012
A generalization of a power series in non-negative integral powers of the difference or in non-positive integral powers of
in the form
![]() | (1) |
The series (1) is understood as the sum of two series:
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the regular part of the Laurent series, and
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the principal part of the Laurent series. The series (1) is assumed to converge if and only if its regular and principal parts converge. Properties of Laurent series: 1) if the domain of convergence of a Laurent series contains interior points, then this domain is a circular annulus with centre at the point
; 2) at all interior points of the annulus of convergence
the series (1) converges absolutely; 3) as for power series, the behaviour of a Laurent series at points on the bounding circles
and
can be very diverse; 4) on any compact set
the series (1) converges uniformly; 5) the sum of the series (1) in
is an analytic function
; 6) the series (1) can be differentiated and integrated in
term-by-term; 7) the coefficients
of a Laurent series are defined in terms of its sum
by the formulas
![]() | (2) |
where is any circle with centre
situated in
; and 8) expansion in a Laurent series is unique, that is, if
in
, then all the coefficients of their Laurent series in powers of
coincide.
For the case of a centre at the point at infinity, , the Laurent series takes the form
![]() | (3) |
and now the regular part is
![]() |
while the principal part is
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The domain of convergence of (3) has the form
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and formulas (2) go into
![]() |
where . Otherwise all the properties are the same as in the case of a finite centre
.
The application of Laurent series is based mainly on Laurent's theorem (1843): Any single-valued analytic function in an annulus
can be represented in
by a convergent Laurent series (1). In particular, in a punctured neighbourhood
of an isolated singular point
of single-valued character an analytic function
can be represented by a Laurent series, which serves as the main instrument for investigating its behaviour in a neighbourhood of an isolated singular point.
For holomorphic functions of several complex variables
the following proposition can be regarded as the analogue of Laurent's theorem: Any function
, holomorphic in the product
of annuli
, can be represented in
as a convergent multiple Laurent series
![]() | (4) |
is which the summation extends over all integral multi-indices
![]() |
![]() |
![]() |
where is the product of the circles
,
. The domain of convergence of the series (4) is logarithmically convex and is a relatively-complete Reinhardt domain. However, the use of multiple Laurent series (4) is limited, since for
holomorphic functions
cannot have isolated singularities.
References
[1] | A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) pp. Chapt. 4 (Translated from Russian) MR0444912 Zbl 0357.30002 |
[2] | B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) pp. Vol. 1, Chapt. 2; Vol. 2, Chapt. 1 (In Russian) Zbl 0799.32001 Zbl 0732.32001 Zbl 0732.30001 Zbl 0578.32001 Zbl 0574.30001 |
Comments
Let be any field. The term Laurent series is also often used to denote a formal expansion of the form
![]() |
The expressions are added termwise and multiplied as follows:
![]() |
where
![]() |
(note that this sum is finite). The result is a field, denoted by . It is the quotient field of the ring of formal power series
, and is called the field of formal Laurent series. A valuation is defined by
if
. This makes
a discretely valued complete field; the ring of integers is
, the maximal ideal is
and the residue field is
. (Cf. also Valuation.)
A Laurent polynomial over is an expression
,
,
.
More generally one also defines (formal) Laurent series in several variables and non-commutative Laurent series, cf. [a1].
References
[a1] | H.C. Hutchins, "Examples of commutative rings" , Polygonal (1981) MR0638720 Zbl 0492.13001 |
[a2] | P.M. Cohn, "Skew field constructions" , Cambridge Univ. Press (1977) MR0463237 Zbl 0355.16009 |
[a3] | L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 241 MR0510197 MR1535085 MR0188405 MR1570643 MR1528598 MR0054016 Zbl 0395.30001 |
[a4] | L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) MR0344507 Zbl 0271.32001 |
[a5] | E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979) MR0593142 MR0197687 MR1523319 Zbl 0477.30001 Zbl 0336.30001 Zbl 0005.21004 Zbl 65.0302.01 Zbl 58.0297.01 |
Laurent series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Laurent_series&oldid=17288