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	Laurent Series Representations
		In this case, therefore, there are no negative powers involved, and the Laurent series reduces to the Taylor series.
		Theorem 7.9 delineates two important aspects of the Laurent series.
		The uniqueness of the Laurent series is an important property because the coefficients in the Laurent expansion of a function are seldom found by using Equation
	math.fullerton.edu /mathews/c2003/LaurentSeriesMod.html (332 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)

	University Of Ghana (Site not responding. Last check: 2007-11-06)
		Proof of the fundamental theorem of calculus and of the major basic results involved in its proof: mean-value theorem, Rolle’s theorem, maximum value theorem, intermediate value theorem.
		Kelvin’s circulation theorem; complex potential for two-dimensional incompressible irrotational motion; three-dimensional irrotational flow, the circle theorem and Blasius’s theorem.
		Weingarten equations, curvatures; fundamental theorems of surfaces, surfaces of constant curvature.
	www.ug.edu.gh /deptdetails.php?recordID=13 (1669 words)

	Omnipelagos.com ~ article "Laurent series"
		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.
		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.
	www.omnipelagos.com /entry?n=laurent_expansion_theorem (757 words)

	Bambooweb: Laurent expansion theorem
		Finally, a Laurent series with no terms of negative degree and with only a finite number of non-zero terms is a polynomial.
		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.
		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.
	www.bambooweb.com /articles/l/a/Laurent_expansion_theorem.html (977 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-06)
		Hermite (a theorem on the sum of the residues of doubly-periodic functions), P.
		The theorem on the total sum of residues is applicable to Riemann surfaces: The sum of all residues of a meromorphic differential on a compact Riemann surface is zero.
		This theory is based on the integral theorems of Stokes and Cauchy–Poincaré, which make it possible to replace the integral of a closed form along one cycle by an integral of this form along another cycle which is homologous to the former.
	eom.springer.de /r/r081560.htm (1045 words)

	Taylor series Summary
		A series expansion is a representation of a function as a sum of powers in one of its variables or a sum of powers of another function.
		He considered these expansions so important to calculus that he maintained that in order to understand a continuous function one needed to know only the derivatives of a function at a given number of points.
		The Taylor series, power series, and infinite series expansions of functions were first discovered in India by Madhava in the 14th century.
	www.bookrags.com /Taylor_series (1855 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.
		Use the residue theorem to evaluate certain real integrals, to evaluate the sum of certain real series and to assist in certain partial fraction decompositions.
	www.mcs.le.ac.uk /Modules/MA-02-03/MA3121.html (400 words)

	Complex Variables
		The elegant fundamental theorem of Cauchy's theory is ∫ f(z)dz = 0, where the integral is taken about a closed curve C surrounding a region where f'(z) exists, as shown in the diagram.
		Laurent extended Taylor's theorem to the case where the domain of analyticity was the region between two concentric circles, a large one and a small one surrounding a point (the origin, for purposes of argument) which could be a singularity.
		The coefficient b of 1/z (or 1/(z-a)) in the Laurent expansion is special because in integrating the series along a circle surrounding a pole, it is the only term that contributes to the integral, in an amount 2πib.
	www.du.edu /~jcalvert/math/complex.htm (4434 words)

	First year curriculum
		Differentiation: mean value theorem, Taylor's theorem and Taylor's series, partial differentiation and total differentiability of functions of several variables.
		Laurent expansion, poles and essential singularities, Residue theorem, contour integration.
		Theory of manifolds: differentiable manifolds, charts, tangent bundles, transversality, Sard's theorem, vector and tensor fields and differential forms: Frobenius' theorem, integration on manifolds, Stokes' theorem in n dimensions, de Rham cohomology.
	www.math.upenn.edu /grad/1stYearGrad.html (895 words)

	PlanetMath: Bessel's equation
		Let's first look the the solution of (1) with
		Let us prove (8) by using the general expression
		Cross-references: sine, cosine, addition formula, residue, origin, paths, coefficients of Laurent series, point, Laurent expansion, analytic, essential singularity, function, generating function, linearly dependent, gamma function, ratio, linearly independent, converge, power series, positive, particular solution, integer, solution, roots, equations, vanish, powers, derivatives, indefinite, parameter, series, general solution, real, linear differential equation
	planetmath.org /encyclopedia/BesselFunction.html (457 words)

	Residue Theorem - Residue Calculus
		You will learn how Laurent expansions can give useful information concerning seemingly unrelated properties of complex functions.
		We begin with a theorem relating residues to the evaluation of complex integrals.
		Theorem 8.2 gives methods for evaluating residues at poles.
	math.fullerton.edu /mathews/c2003/ResidueCalcMod.html (424 words)

	Analysis, Convergence, Series, Complex Analysis - Numericana
		Laurent series of a function about one of its poles.
		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.
		[the fundamental theorem of complex analysis] to multivalued functions (like the square root function involved here), it is important to specify a so-called "cut" in the complex plane were the function is allowed to be discontinuous, so that it is everywhere else continuous and single-valued.
	home.att.net /~numericana/answer/analysis.htm (4397 words)

	1 Introduction
		These sets are the indices of the coefficients in a Laurent series that share a common value.
		In these cases, the tools used to prove the transcendence were automata theoretic or language theoretic in nature.
		The purpose of this paper is to explain the connection between definability of sets of integers and the algebraicity of Laurent series in
	www.u.arizona.edu /~miller/webthesis/mthesis/node5.html (274 words)

	Timeline of mathematics - Encyclopedia, History, Geography and Biography
		1100s - Bhaskara Acharya conceives differential calculus, and also develops Rolle's theorem, Pell's equation, a proof for the Pythagorean Theorem, proves that division by zero is infinity, computes π to 5 decimal places, and calculates the time taken for the earth to orbit the sun to 9 decimal places
		1300s - Parameshvara, a Kerala school mathematician, presents a series form of the sine function that is equivalent to its Taylor series expansion, states the mean value theorem of differential calculus, and is also the first mathematician to give the radius of circle with inscribed cyclic quadrilateral
		1400 - Madhava discovers the series expansion for the inverse-tangent function, the infinite series for arctan and sin, and many methods for calculating the circumference of the circle, and uses them to compute π correct to 11 decimal places
	www.arikah.com /encyclopedia/Timeline_of_mathematics (4486 words)

	Laurent series \| MaplePrimes
		The local ring (k[x,y]/(f))_(x,y) can be viewe inside the Laurent series field k((x)).
		the Laurent series are the quotient field (as your local ring is integer).
		No series expansion needed I think, it should be all in the context
	beta.mapleprimes.com /forum/laurent-series (394 words)

	Timeline of mathematics - Gurupedia
		Euclidean algorithm; he states the law of reflection in Catoptrics, and he proves the fundamental theorem of arithmetic
		Cauchy integral theorem for integration around the boundary of a rectangle in the complex plane,
		Cauchy integral theorem for general integration paths -- he assumes the function being integrated has a continuous derivative, and he introduces the theory of
	www.gurupedia.com /t/ti/timeline_of_mathematics.htm (2575 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)

	Texts by Konrad Knopp
		The foundations of the theory are therefore presented with special care, while the developmental aspects are limited by the scope and purpose of the book.
		All definitions are clearly stated; all theorems are proved with enough detail to make them readily comprehensible.
		Chapters four and five explain power series and the development of the theory of convergence, while chapter six treats expansion of the elementary functions.
	www-groups.dcs.st-and.ac.uk /~history/Extras/Knopp_texts.html (478 words)

	MATH 502 Spring 1997, Lecture Schedule (Site not responding. Last check: 2007-11-06)
		Path integrals, Cauchy's formula and theorem for a disc, analytic implies power series at each point,
		Cauchy's estimate, Maximum Modulus Principle, Liouville's Theorem, Fundamental Theorem of Algebra, isolation and finite multiplicity of roots
		Morera's Theorem, theorem on existence of primitives, Goursat's Theorem, isolated singularities
	www.ima.umn.edu /~arnold/502.s97/complexlectures.html (160 words)

	SFU Department of Mathematics: MATH 254-3
		Vector functions of a single variable, space curves, scalar and vector fields, conservative fields, surface and volume integrals, and theorems of Gauss, Green and Stokes.
		Taylor and Laurent expansion, method of residues, integral transform and conformal mapping.
		Residue Calculus: Poles and residues, the residue theorem, contour integration, evaluation of definite integrals, the integral transform.
	www.math.sfu.ca /math/courses/math254.shtml (317 words)

	[No title]
		Can you write down the Laurent expansion of that (I messed up, then sorta corrected mistake and he said it was fine...used expansion of 1/(1-z)).
		I stated the one that the fourier transform is continuous & goes to zero for L1 functions.
		He called this "Lebesgue's Integral Theorem", which confused me for a bit) - L2 is a Hilbert space - what do we know about Hilbert spaces in general (he wanted me to say they're all equivalent to L2(mu) for some mu, which I did -I gave the example of Bergman space).
	www.math.princeton.edu /generals/dowling_blair (528 words)

	YE OLDE TWIN PARADOX
		In Lorentz reference frames, the spacetime coordinates of events are given by (x,y,z,t), where x,y, and z are space coordinates, and t is the time coordinate.
		As with the length expansion argument above, one may ask what happens if the observer in T tries to make exactly the same observation as the observer in S does.
		The idea here is to give an analysis of each problem from the point of view of an observer in each of the different reference frames, and to show that the various analyses give the same results.
	chabin.laurent.free.fr /twinzz.htm (7206 words)

	Wilkes Math Courses
		Topics include systems of linear differential equations; nonlinear differential equations; qualitative, numerical, and finite difference methods; theorems of Green and Stokes and the Divergence Theorem.
		A study of group theory (including the Sylow Theorems and solvable groups); ring theory (including the Noetherian rings and UFDs); modules, tensor algebra, and semi-simple rings.
		Prerequisites: MTH 331 and a course in linear algebra or consent of instructor.
	www.wilkes.edu /pages/1170.asp (500 words)

	Complex-Analysis
		When the complex derivative is defined "everywhere," the function is said to be analytic.
		Let O be a bounded domain in R^N with C^2 boundary, and suppose that O is geometrically convex.
		Let O in R^{N} be a geometrically convex domain, p in ∂O, and q in O. Show that for all 0
	www.adwan.net /Complex-Analysis.html (480 words)

	Course Descriptions -- Ch10: 2000-2001 UVa Graduate Record
		Topics include the solution of the heat, potential, and wave equation in rectangular and polar coordinates; separation of variables and eigenfunction expansion techniques for nonhomogeneous boundary-value problems.
		Analyzes methods of solving linear partial differential equations; Green's functions; transform methods; asymptotic expansions; and a variety of linear boundary/initial value problems in various coordinate systems.
		Analyzes the role of statistics in science; hypothesis tests of significance; confidence intervals; design of experiments; regression; correlation analysis; analysis of variance; and introduction to statistical computing with statistical software libraries.
	www.virginia.edu /registrar/records/00gradrec/chapter10/gchap10-4.2.html (883 words)

	Mathematics (M.S.)
		Classification of second-order partial differential equations, background on eigenfunction expansions and Fourier series, solution of the wave equation, reflection of waves, solution of the heat equation in bounded and unbounded media, Laplace's equation, Dirichlet and Neumann problems.
		The algebraic and topological structure of the complex plane, analytic functions, Cauchy's integral theorem and integral formula, power series, elementary functions and their Riemann surfaces, isolated singularities, residues, the Laurent expansion, the Riemann mapping.
		Local and global theory of curves and surfaces: Fenchel’s theorem; the first and second fundamental forms; surface area; Bernstein’s theorem; Gauss theorema egregium; local intrinsic geometry of surfaces; Riemannian surfaces; Lie derivatives; covariant differentiation; geodesics; the Riemann curvature tensor; the second variation of arclength; selected topics in the global theory of surfaces.
	main.uab.edu /show.asp?durki=24908 (1691 words)

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