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Topic: Laurent series

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Power series

Taylor expansion

Taylor series

Residue (complex analysis)

Holomorphic function

Isolated singularity

Examples of contour integration

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	Laurent series - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-07)
		In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree.
		Laurent series with complex coefficients are an important tool in complex analysis, especially to investigate the behavior of functions near singularities.
		Two such formal Laurent series may be added by adding the coefficients, and because of the finiteness of the negative-degree coefficients, they may also be multiplied using convolution of the coefficient sequences.
	en.wikipedia.org /wiki/Laurent_series (981 words)

	Introduction
		All elements in a series field of fixed precision have the same fixed relative precision; the only exception to this rule is that 0 in the quotient field is stored as zero with infinite absolute precision.
		All elements in a series ring of fixed precision have the same fixed absolute precision p, and the relative precision may be anything from 0 to p.
		Return the element of the series ring or field with a name attached to it, that is, return the `indeterminate' transcendental element generating R over its coefficient ring.
	www.math.ufl.edu /help/magma/text400.html (906 words)

	Element Operations
		If f is a series in x, the absolute precision of f is the exponent p such that x^p is the first term of f of which the coefficient is not known, that is, it is the least p such that f in O(x^p).
		Given a series f in R[[X]] or f in R((X)), return the smallest integer v (possibly negative for Laurent series) such that the coefficient of X^v in f is not known to be zero.
		Given a power series f (in x, say), this returns the inverse of f under composition, that is, an element g of the same power series ring such that its composition with f equals x to the best possible precision.
	www.math.uiuc.edu /Software/magma/text371.html (968 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-07)
		Usually these series are considered as formal objects; that is, no convergence properties have to be taken into account, although one does encounter convergence properties at some point when applying them to physical situations.
		The polynomial part of a Laurent series is the sum of all the terms with non-negative exponents of x.
		A Laurent series expansion of a function about c will not make sense unless there is an integer n (which is the degree) such that the limit as x grows without bound of f(x)/(x-c)^n equals a nonzero constant (which is just a(n)).
	mathforum.org /library/drmath/view/56944.html (565 words)

	Laurent Series, Mu-Ency at MROB
		When this was discussed in 1992, many believed that the limit of this series is greater than the limit of the pixel counting method.
		Ewing and Schober suggested either that the formula based on the Laurent series might be wrong (due most probably to poor understanding of the boundary of the Mandelbrot set and how the conformal map approaches it), or that perhaps the boundary has positive area.
		Given this, and the fact that the series converges so slowly, it is clear that the Ewing-Schober sequence has little practical value as a numerical algorithm for computing the area of the Mandelbrot set.
	www.mrob.com /pub/muency/laurentseries.html (1256 words)

	Creation Functions
		The easiest way to create power and Laurent series in a given ring is to use the angle bracket construction to attach names to the indeterminate, and to use these names to express the series (see the examples).
		Coerce s into the power series ring or Laurent series ring R. Here s is allowed to be a sequence of elements from (or coercible into) the coefficient ring of R, or just an element from (or coercible into) R. A sequence [a_1,..., a_d] is converted into the series a_1 + a_2X^1 +...
		Create the series 0 + O(x^n) where x is the `generator' of a (power or Laurent) series ring R and where n must be positive if R is a power series ring.
	www.math.ufl.edu /help/magma/text401.html (490 words)

	PlanetMath: formal power series (Site not responding. Last check: 2007-11-07)
		Formal power series allow one to employ much of the analytical machinery of power series in settings which don't have natural notions of convergence.
		Formal power series can also be interpreted as functions, but one has to be careful with the domain and codomain.
		This is version 8 of formal power series, born on 2002-07-01, modified 2005-02-13.
	planetmath.org /encyclopedia/GeneratingFunction.html (766 words)

	Laurent - Wikipedia, the free encyclopedia
		Laurent series in mathematics, named after the French mathematician Pierre Alphonse Laurent
		Laurent Véronnez A successful Belgium-based Trance DJ, whose DJ outfit is popularly known as "Airwave"
		Augustin Laurent, (1896-1990) a French minister and resistant
	en.wikipedia.org /wiki/Laurent (146 words)

	Laurent Series
		Laurent series for the inversion of perturbed linear operators on Hilbert space.
		A certain analogue of the Laurent series for functions of several complex variables that are holomorphic on strongly linearly convex sets.
		On A-convolutions of Laurent series in a circular annulus.
	math.fullerton.edu /mathews/c2003/LaurentSeriesBib/Links/LaurentSeriesBib_lnk_3.html (795 words)

	Introduction
		Power series must have a non-negative integral valuation and integral exponents, Laurent series must have an integral valuation and integral exponents (possibly negative), while Puiseux series may have a general rational valuation and general rational exponents.
		Puiseux series in a variable x are often mathematically defined to be Laurent series in another variable y say, where y=x^(1/d), for a fixed positive integer d; this d is usually fixed for all the series under consideration.
		For each type of series ring, there are two sub-kinds: a free precision ring (the usual case), where elements of the ring have arbitrary precision, and a fixed precision ring, where all elements of the ring have a fixed precision.
	www.umich.edu /~gpcc/scs/magma/text721.htm (999 words)

	PlanetMath: Laurent series (Site not responding. Last check: 2007-11-07)
		Every Laurent series has an associated function, given by
		The coefficients of Laurent series for an analytic function can be determined using the Cauchy integral formula.
		This is version 7 of Laurent series, born on 2001-12-28, modified 2005-07-09.
	planetmath.org /encyclopedia/LaurentSeries.html (126 words)

	Calculus:Taylor series - Wikibooks, collection of open-content textbooks
		A function is analytic if and only if a power series converges to the function; the coefficients in that power series are then necessarily the ones given in the above Taylor series formula.
		Some functions cannot be written as Taylor series because they have a singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable x; see Laurent series.
		The Taylor series is named for mathematician Brook Taylor, who first published the power series formula in 1715.
	en.wikibooks.org /wiki/Calculus:Taylor_series (499 words)

	[No title]
		Laurent series ------------------------ diffop_laurent Global variables: x:=evaln(x) LL accuracy_laurent description_laurent set_laurents value_laurent LL.i are indeterminates which stand for Laurent series.
		The value of a Laurent series is given by: value_laurent[LL.i].
		Procedures for lifting Laurent series: These procedures are only to be called from the Laurent series procedures nmterms_laurent and nthterm_laurent lift_newton: Used for lifting coprime index 1 factorizations f=left*right.
	www.math.fsu.edu /~hoeij/compalg/diffop/beschrijving (1422 words)

	Laurent Series
		A finite laurent series is both a forward laurent series and a reverse laurent series.
		Thus the power series is a special case of the general laurent series, as computed by the contour integrals, when f happens to be analytic throughout the entire disk.
		Put this all together, and the laurent operator, from function to series, is indeed a transform from one vector space into another.
	www.mathreference.com /cx-pow,laur.html (1336 words)

	Unique Representation (Site not responding. Last check: 2007-11-07)
		However, if f is analytic on a disk, rather than an annulus, then the power series does exist, and is unique, and any attempt to compute the laurent series will produce the power series, as we discussed earlier.
		Suppose f is a function that is represented by a laurent series, yet the laurent operator produces infinitely many terms with negative exponents.
		When the laurent coefficients are computed, the resulting series is a shifted version of the original.
	www.mathreference.com /cx-pow,urep.html (493 words)

	Creation Functions
		By default, a global series ring will be returned; if the parameter Global is set to false, a non-global series ring will be returned (to which a separate name for the indeterminate can be assigned).
		Return the element of the series ring or field with a name attached to it, that is, return the `indeterminate' transcendental element generating S over its coefficient ring.
		Create the series O(x^v) where x is the generator of the parent of f and v is the valuation of f.
	www.math.wisc.edu /help/magma/text474.html (748 words)

	40: Sequences, series, summability
		Sequences and series are really just the most common examples of limiting processes; convergence criteria and rates of convergence are as important as finding "the answer".
		Taylor series of known functions) are of interest, as well as general methods for computing sums rapidly, or formally.
		Series can be estimated with integrals, their stability can be investigated with analysis.
	www.math.niu.edu /~rusin/known-math/index/40-XX.html (588 words)

	Uniform convergence.
		Corollary 7.3.6 (analyticity of the sum) Suppose that the series
		As a consequence, we have that power series converge uniformly on their open domain of convergence (v.i.
		Theorem 7.3.7 (term-by-term differentiation) Suppose that the series
	ndp.jct.ac.il /tutorials/complex/node48.html (110 words)

	Laurent Series.
		The convergence domain in the plane is the open unit disk centered at the origin; cf Fig 1(a).
		as a Laurent series convergent on an annulus.
		On this annulus, a Laurent series expansion of
	ndp.jct.ac.il /tutorials/complex/node51.html (142 words)

	Creation of Local Field Elements (Site not responding. Last check: 2007-11-07)
		Given a power series ring R[[X]] or Laurent series ring R((X)), integers e and p >= e, and a sequence a=[a_1,..., a_d] of elements of R, return the series a_1X^e +...
		Coerce s into the power series ring or Laurent series ring R. Here s is allowed to be a sequence of elements from (or coercable into) the coefficient ring of R, or just an element from (or coercable into) R. A sequence [a_1,..., a_d] is converted into the series a_1 + a_2X^1 +...
		Given a (power or Laurent) series f, this returns the precision that is stored with f.
	www.math.uiuc.edu /Software/magma/text381.html (274 words)

	Kimberlyscott.com
		Laurent's crisp lines and solid laminate finished surfaces reflect top quality and superb practicality.
		Whether you need a functional, task oriented work area or a stylish executive setting, Laurent offers the options necessary to create a perfect fit.
		The Laurent collection was designed to satisfy all your needs, even incorporating the warmth of a 2" radius genuine wood trim.
	www.kimberlyscott.com /desk-laurent.htm (59 words)

	Type::Series -- a type representing truncated Puiseux, Laurent, and Taylor series expansions (Site not responding. Last check: 2007-11-07)
		A Laurent series is a Puiseux series with integral exponents.
		If the expansion point is finite, then a Taylor series is a Puiseux series with non-negative integral exponents.
		A Laurent series that is not a Taylor series:
	www.mupad.de /doc/25/eng/Type/Series.shtml (730 words)

	Travel Series
		Dion Laurent's Travel Series began in 1990 in Nashville with his first bulldozer painting, and during travels through Europe that same year, the concept started taking shape.
		In 1991 with a trip to Jamaica at the beginning of the Gulf War the series began in earnest.
		Though he was unable to complete each of these three works for the 25 countries he has thus far visited, the series has blossomed and is the backbone of his efforts and installations since 1990.
	www.dionlaurent.com /TravelSeries.htm (234 words)

	"Laurent" buy "Laurent" at InstaOffice (Site not responding. Last check: 2007-11-07)
		Laurent Series Left Return for Workstation Desk, 48 x 24, Mahogany
		Laurent Series Left Return for Workstation Desk, 48 x 24, Oak
		Laurent Series Workstation Desk for Left Return, 66 x 29-1/2, Mahogany
	www.instaoffice.com /Laurent.0.3.0.htm (419 words)

	Math 132 Applet 6 (Site not responding. Last check: 2007-11-07)
		Note that there are some white circles for which none of the three series converge; these include the singularities of
		To read off all the displayed terms of a Taylor or Laurent series, one may have to use the cursor keys to navigate the text windows.
		Note that the co-efficients of the Taylor series become much larger as one approaches a singularity.
	www.math.ucla.edu /~tao/java/Laurent.html (114 words)

	Inversion of (Taylor, Laurent) series (Site not responding. Last check: 2007-11-07)
		I wonder whether someone could give me some hints to what extent this can be generalized to Laurent series.
		I understand that a function defined by a Taylor series f(z) = sum a(i) x^i i = 0,...
		If I have now a function defined as a Laurent series of the form g(z) = sum a(i) x^i i = -n, -n+1,...
	www.uni-koeln.de /REDUCE/reduce-forum/91/msg106.html (267 words)

	Power, Laurent and Puiseux Series Rings
		Provided this is kept in mind, calculations with elements of a power series ring (though not field) are always precise.
		Since elements of such a field are infinite series, calculation is necessarily approximate.
		Puiseux series with arbitrary fractional exponents are also supported (since V2.4).
	magma.maths.usyd.edu.au /magma/Features/node129.html (117 words)

	Fabian Waleffe -- Math 321 -- U. of Wisconsin
		Laurent vs Taylor: Expand 1/z into a Laurent series about z=0 (hint: this is really easy...).
		Hence the Taylor and Laurent series are complementary.
		Find the first 3 terms in the Laurent series expansion of 1/sin(z) about z=0.
	www.math.wisc.edu /~waleffe/M321/Spring04/HWK.html (628 words)

	Engel series expansions of Laurent series and Hausdorff dimensions
		Engel series expansions of Laurent series and Hausdorff dimensions
		be the field of all formal Laurent series
		be the Engel expansion of Laurent series of
	www.austms.org.au /Publ/JAustMS/V75P1/c74.html (85 words)

	Laurent Shop For Laurent at Instawares (Site not responding. Last check: 2007-11-07)
		Laurent Series Two-Drawer Lateral File, 36w x 20d...
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	www.instawares.com /Laurent.0.3.0.htm (429 words)

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