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Topic: Laurent expansion

	PlanetMath: Laurent series
		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 (127 words)

	Laurent series - Wikipedia, the free encyclopedia
		Laurent series with complex coefficients are an important tool in complex analysis, especially to investigate the behavior of functions near singularities.
		of the Laurent expansion of such a function is called the residue of f(z) at the singularity c; it plays a prominent role in the residue theorem.
		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 (961 words)

	Laurent series (Site not responding. Last check: 2007-10-22)
		Laurent series with complex coefficients are an tool in complex analysis especially to investigate the behavior of near singularities.
		More generally Laurent series can be used express holomorphic functions defined on an annulus much as power series are used to express holomorphic functions on a disc.
		Two such formal Laurent series may added by adding the coefficients and because the finiteness of the negative-degree coefficients they also be multiplied using convolution of the coefficient sequences.
	www.freeglossary.com /Laurent_expansion (1062 words)

	Laurent series - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-22)
		In mathematics, a Laurent series is an infinite series.
		The coefficients of the Laurent series can be determined with an integral formula which generalizes Cauchy's integral formula:
		Pick any rectifiable path γ in A which is closed (has the same beginning and ending points), does not have any self-intersections, and moves around the annulus counterclockwise.
	encyclopedia.worldsearch.com /laurent_series.htm (1132 words)

	Laurent expansion (Site not responding. Last check: 2007-10-22)
		[math/9409208] Laurent coefficients and Ext of finite graded modules...
		Orthogonal Rational Expansions of Functions using Laurent Gaussian Quadrature Ru...
		Engel series expansions of Laurent series and Hausdorff dimensions...
	www.scienceoxygen.com /math/472.html (101 words)

	Hypergeometric functions
		It is based on the analytical calculation of series arising after expansion of the hypergeometric function via systems of difference/differential equations.
		The result of expansion are expressible in terms of polylogarithms, Nielsen polylogarithms and new functcion, related with harmonic polylogarithm.
		An efficient algorithm for the calculation of the higher-order coefficients of Laurent expansion is constructed [4].
	theor.jinr.ru /~kalmykov/hypergeom/hyper.html (495 words)

	The Laurent Expansion of Pencils that are Singular at the Origin - Schweitzer, Stewart (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		The Laurent Expansion of Pencils that are Singular at the Origin (1991)
		Laurent series for the inversion of perturbation operators..
		P.J. Schweitzer and G.W. Stewart, The Laurent expansion of pencils that are singular at the origin, Linear Algebra Appl., 183 (1993), pp.
	citeseer.ist.psu.edu /schweitzer91laurent.html (597 words)

	[No title]
		This kind of logarithmic expansions for solutions of systems of ordinary differential equations were called psi-series by Hille (see, for instance, {[Hil76]}), although they had already been studied in the beginning of the century in systems of the form $t \d x/\d t=f(t,x)$ by Horn~{[Hor96]}.
		The convergence region of the expansions of the singularities of b) type is a Riemann surface with a finite number of sheets whereas the expansions of the singularities of c) type is a Riemann surface with an infinite number of sheets.
		We note that the expansion of $x(t)$ begins with a term of ``$\log\tau$'' type instead of a term of pole type, and therefore it is very difficult to find such singularities with the standard methods.
	www.ma.utexas.edu /mp_arc/html/papers/97-136 (4314 words)

	Askaroo : Laurent Series (Site not responding. Last check: 2007-10-22)
		Laurent series with complex coefficients an and a complex center c.
		The coefficient a-1 of the Laurent expansion of such a function is called the
		Formal Laurent series are Laurent series that are used without regard for their
	www.askaroo.com /info/Laurent_series.html (904 words)

	PlanetMath: Laurent series
		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.
	www.planetmath.org /encyclopedia/LaurentSeries.html (126 words)

	Laurent series - TheBestLinks.com - Laurent expansion theorem, Complex analysis, Complex number, Convolution, ... (Site not responding. Last check: 2007-10-22)
		Laurent series - TheBestLinks.com - Laurent expansion theorem, Complex analysis, Complex number, Convolution,...
		Laurent expansion theorem, Laurent series, Complex analysis, Complex number...
		The Laurent series is named after French mathematician Pierre Laurent (1813-1854).
	www.thebestlinks.com /Laurent_expansion_theorem.html (1048 words)

	WP 3.1
		Solution: The form of the Laurent expansion, as always, is (3.23a); our task is merely to find a
		The pole/zero plot is as shown in Fig.
		< 2, the Laurent expansion of f(z) is just the Taylor series found in Example 3.7.
	www.wiley.com /college/engin/cavicchi124729/resources2/wp3.htm (779 words)

	Laurent series for the inversion of perturbation operators on Hilbert space (ResearchIndex)
		Then, the generalization of an efficient algorithm of Schweitzer and Stewart can be applied to obtain a Laurent expansion for the inverse of perturbed operators.
		6 The Laurent expansion of pencils that are singular at the or..
		5 The Laurent expansion of a generalized resolvent with some a..
	citeseer.ist.psu.edu /howlett99laurent.html (378 words)

	Analysis, Convergence, Series, Complex Analysis - Numericana
		Laurent series of a function about one of its poles.
		That's the Laurent expansion of f about the pole z=1.
		The so-called "residue" of f for the pole z=1 is the coefficient of 1/h in the Laurent expansion (here, that's -1); that's the only thing that comes into play when integrating over a closed contour.
	home.att.net /~numericana/answer/analysis.htm (4108 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		However, if the expansion point is a branch point, then the returned expansion may not approximate the function f for values of x close to the branch cut.
		In this case, the expansion is not a proper Puiseux series in the mathematical sense.
		The second argument is the series variable; the expansion point is always the origin 0; other expansion points are internally moved to the origin by a change of variables.
	www.sciface.com /STATIC/DOC30/eng/stdlib_series.html (3315 words)

	Laurent, Auguste - Hutchinson encyclopedia article about Laurent, Auguste
		French chemist who, in collaboration with Charles Gerhardt, made important contributions to the classification of elements.
		Laurent was also one of the first to recognize a clear distinction between atoms and molecules.
		Laurent was the son of a mining engineer from Langres, and himself qualified in mining engineering in Paris.
	encyclopedia.farlex.com /Laurent%2c+Auguste (155 words)

	Singular invariant hyperfunctions on the square matrix space and the alternating matrix space, Masakazu Muro
		By expanding the complex powers of the determinant function or the Pfaffian function into the Laurent series with respect to the complex parameter, we can construct singular invariant hyperfunctions as their Laurent expansion coefficients.
		The author presents here the exact orders of the poles of the complex powers and determines the exact supports of the Laurent expansion coefficients.
		By applying these results, we prove that every quasi-relatively invariant hyperfunction can be expressed as a linear combination of the Laurent expansion coefficients of the complex powers and that every singular quasi-relatively invariant hyperfunction is in fact relatively invariant on the generic points of its support.
	projecteuclid.org /getRecord?id=euclid.nmj/1114631809 (490 words)

	Math Forum Discussions
		> A function has a Laurent Expansion near an isolated singularity.
		Yes, but the point is not exactly the "isolated" caracter of the
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /kb/thread.jspa?messageID=104913&tstart=0 (269 words)

	MATH 3410 - Assignment no 5
		Hence the principal part of the Laurent expansion is [e
		Thus the principal part of the Laurent expansion is [((-1)
		The third factor is analytic at n (with value 1 there) and the second factor has a removable singularity (with limiting value 1) there.
	www.math.yorku.ca /Who/Faculty/Muldoon/cv/cvas5s.htm (400 words)

	Residue Theorem - Residue Calculus
		You will learn how Laurent expansions can give useful information concerning seemingly unrelated properties of complex functions.
		in the Laurent expansion, we seek a method to calculate the residue from special information about the nature of the singularity at
		The theory of residues can be used to expand the quotient of two polynomials into its partial fraction representation.
	math.fullerton.edu /mathews/c2003/ResidueCalcMod.html (424 words)

	Math 132 Applet 6 (Site not responding. Last check: 2007-10-22)
		This applet displays the various Taylor and Laurent expansions of the function
		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)

	Laurent Series.
		Theorem 8.2.2 (Laurent expansion of a function) Suppose that the function f is analytic on the annulus
		Then f has an expansion as a Laurent series in D.
		This series expansion of f is unique and its coefficients are given by:
	ndp.jct.ac.il /tutorials/complexwin/node48.htm (231 words)

	Solutions (Site not responding. Last check: 2007-10-22)
		We determine the residue there by expanding the integrand in a Laurent series.
		We substitute this Laurent series into the Residue Formula to verify it.
		We could also calculate the residue at infinity by recalling that the sum of all residues of this function in the extended complex plane is zero.
	www.acm.caltech.edu /~seanm/applied_math_html/node139.html (1097 words)

	Type::Series -- a type representing truncated Puiseux, Laurent, and Taylor series expansions (Site not responding. Last check: 2007-10-22)
		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.sciface.com /STATIC/DOC25/eng/Type/Series.shtml (730 words)

	Footnotes
		No further use will be made of it in this course.
		Laurent expansion follows from the fact that the generating function is a product of an
		The desired Laurent expansion may be obtained, in principle, by multiplying one-sided series for the exponential and inverted exponential together.
	ccrma.stanford.edu /~jos/rbeats/footnode.html (167 words)

	MA3121 Complex Analysis
		Prove Taylor's theorem and appreciate that a function which is differentiable in a neighbourhood of a point has a Taylor series expansion about that point.
		Prove Laurent's theorem and appreciate that a function which is differentiable in a punctured neighbourhood of a point has a Laurent expansion about that point.
		Determine whether a function has a singularity at a given point, and the nature and order of singularities.
	www.mcs.le.ac.uk /Modules/MA-02-03/MA3121.html (400 words)

	Exercises (Site not responding. Last check: 2007-10-22)
		Determine the radius of convergence with the ratio test.
		Find the first three terms in the following Taylor series and state the convergence properties for the following.
		Without determining the series, specify the region of convergence for a Laurent series representing
	www.acm.caltech.edu /~seanm/applied_math_html/node123.html (362 words)

	Session 2 \\{\bf Series Solutions of ODE's}
		When converging series are used to represent functions the operations of analysis (differentiation and integration) can be carried out on both sides of the equation and the equality is maintained.
		Laurent series generalize Taylor series by allowing negative powers.
		By introducing a crosscut Cauchy's integral formula can be applied on a single contour and the Laurent expansion of f(z) becomes
	www.rh.edu /~ernesto/C_S2000/mes/Notes/mes02.html (836 words)

	Laurent Series.
		Theorem 8.2.2 (Laurent expansion of a function) Suppose that the function
		The convergence domain in the plane is the open unit disk centered at the origin; cf Fig 1(a).
		The intersection of these convergence domains is the annulus displayed on Fig 1(b).
	ndp.jct.ac.il /tutorials/complex/node51.html (142 words)

	BSHM: Abstracts -- U
		At least five mathematicians tried to prove denumerability of the value set, to which the names of two of them are attached.
		Ullrich, Peter, ‘The proof of the Laurent expansion by Weierstrass’, Eberhard Knobloch and David E. Rowe (editors), The history of modern mathematics iii: images, ideas and communities, Academic Press 1994, 139-153
		In 1841 Weierstrass proved the existence of the Laurent expansion for functions on an annulus, using complex integration in a decisive way, independently of Laurent and at most indirectly influenced by Cauchy.
	www.dcs.warwick.ac.uk /bshm/abstracts/U.html (328 words)

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