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	Taylor series - Wikipedia, the free encyclopedia
		Third, the (truncated) series can be used to compute function values approximately (often by recasting in the polynomial into the Chebyshev form and evaluating it with the Clenshaw algorithm).
		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.
		This theorem is an expansion on the Picard iteration.
	en.wikipedia.org /wiki/Taylor_series (782 words)

	Saf-T-Bar Series H Expansion Gaps \| Insulated Electrification Systems (Site not responding. Last check: 2007-10-16)
		Expansion Gap Assemblies are preassembled, ready to be installed between two adjacent sections of rail.
		The expansion mechanism is a telescoping interleaved unit, having a travel of 8", providing a constant sliding surface for the collector shoe for mechanical support only.
		Expansion gap assemblies are based on 500' conductor intervals with a 100°F temperature variation.
	www.keepitmoving.com /h-expansion.html (276 words)

	Resistoflex PTFE Expansion Joints
		Expansion joints are used to compensate for movement, misalignment, axial travel, angular deflection, and or vibration in piping systems.
		Resistoflex expansion joints are made of contour molded PTFE, providing exceptional corrosion resistance and flex-life twice that of our nearest competitor.
		The new E-Series expansion joints have limit cables instead of limit bolts, to allow for greater misalignment and angular deflection.
	www.ptfeflexjoints.com (378 words)

	Learn more about Taylor series in the online encyclopedia. (Site not responding. Last check: 2007-10-16)
		In mathematics, the Taylor series of an infinitely often differentiable real (or complex) function f defined on an open interval (a-r, a+r) is the power series
		If this series converges for every x in the interval (a-r, a+r) and the sum is equal to f(x), then the function f(x) is called analytic.
		The Parker-Sockacki theorem is a recent advance in finding Taylor series which are solutions to differential equations.
	www.onlineencyclopedia.org /t/ta/taylor_series.html (509 words)

	Maxima Manual - Series
		allows the user to define the Taylor series (about 0) of an arbitrary function of one variable as exp which may be a polynomial in that variable or which may be given implicitly as a power series using the SUM function.
		returns a list of all rational functions which have the given taylor-series expansion where the sum of the degrees of the numerator and the denominator is less than or equal to the truncation level of the power series, i.e.
		generates the general form of the power series expansion for exp in the variable var about the point pt (which may be INF for infinity).
	www.ma.utexas.edu /maxima/maxima_29.html (1303 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)

	The Mathematica Book Online: Advanced Mathematics in Mathematica \| Series, Limits and Residues
		Series can also generate some power series that involve fractional and negative powers, not directly covered by the standard Taylor series formula.
		The procedure that Series follows in constructing a power series is largely analogous to the procedure that N follows in constructing a real-number approximation.
		When you make a power series expansion in a variable x, Mathematica assumes that all objects that do not explicitly contain x are in fact independent of x.
	documents.wolfram.com /mathematica/book/section-3.6.1 (546 words)

	Making Power Series Expansions
		does not know the series expansion of a particular function, it writes the result symbolically in terms of derivatives.
		One way to understand what happens is to think of the analogy between power series taken to a certain order, and real numbers taken to a certain precision.
		Power series are "approximate formulas" in much the same sense as finite-precision real numbers are approximate numbers.
	documents.wolfram.com /v4/MainBook/3.6.1.html (413 words)

	Fourier Series
		And the notion of expansion in a set of elementary functions is more general than Fourier series.
		Expansion in a basis is a generalization of the Fourier series.
		An unknown wave function is to be expanded as a linear combination of well known wave functions and the problem is re-formulated to find the expansion coefficients.
	www.sci.wsu.edu /idea/quantum/fourier_series.htm (561 words)

	Power Series Expansions
		One method of approximating a function is by means of a power series expansion of the function about a given point.
		Note that the coefficients of this expansion are all in terms of the number e.
		We now wish to find a power series expansion of order 4 for the difference in values of this function at our two speeds.
	www.mathcs.emory.edu /~fox/NewCCS/ModuleIV/ModIVP8.html (1117 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-16)
		Date: 08/02/98 at 18:26:36 From: Doctor Pete Subject: Re: binomial expansion, roots Hi, The power series expansion of: (1 - x)^(-1) is: 1 + x + x^2 + x^3 + x^4 +....
		This power series expansion is usually considered as being derived from the theory of Taylor series (or more simply, from infinite geometric series), but not usually from the binomial theorem.
		It is easiest to see the above series expansion as a geometric series with common ratio x.
	mathforum.org /library/drmath/view/56423.html (471 words)

	Elliptic and Modular Functions
		Since the expansion is in terms of q rather than z we first must calculate the point q1 that corresponds to z1.
		Given a lattice L = [a, b] in the complex plane, this function returns the value of the elliptic j-invariant of L. This is the j-invariant of tau where tau = a/b or tau = b / a, whichever is in the upper half complex plane.
		Given a pair L = [a,b] of complex numbers generating a lattice in C, return the normalized q-series expansion of the discriminant Delta(q) evaluated at tau where tau = a/b or tau = b / a, whichever is in the upper half complex plane.
	www.math.wisc.edu /help/magma/text467.html (1285 words)

	sciforums.com - Infinite series expansion help needed
		Does anybody out there know of a general way to expand the product of two infinite series that are not identical (if they were identical I would just use multinomial expansion and be done with it).
		Your generating functions are two power series, ∑a(n)x^n and ∑b(n)x^n where n goes from 0 to infinity.
		If it were two x^n series I'd just put them into a more convenient form.
	www.sciforums.com /showthread.php?t=34126 (453 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-16)
		Date: 11/24/2001 at 15:14:59 From: Don Castaldini Subject: Taylor Series Expansion The distance from A to B is 1000 meters.
		Date: 11/25/2001 at 15:23:38 From: Doctor Pete Subject: Re: Taylor Series Expansion Hi, This is an interesting problem, one that depends on whether you consider it the distance discrete or continuous.
		If the problem is discrete (as implied by the wording, and your comments), then each time we step 1 meter forward, the length stretches 1000 meters under our feet.
	mathforum.org /library/drmath/view/52138.html (663 words)

	series -- compute a (generalized) series expansion (Site not responding. Last check: 2007-10-16)
		If no expansion exists that is valid in the complex plane, this argument can be used to request expansions that only need to be valid along the real line.
		Undirected, then an undirected series expansion around the complex infinity, i.e., the north pole of the Riemann sphere, is computed.
		In this case, the expansion is not a proper Puiseux series in the mathematical sense.
	www.sciface.com /STATIC/DOC25/eng/stdlib/series.shtml (2599 words)

	9.2 The Taylor series (Site not responding. Last check: 2007-10-16)
		The Taylor series expansion of a function f(x) is
		Differing approximations to the function are obtained from this series by truncation.
		The Taylor expansion must be used with some caution because the series does not converge for all values and sometimes converges very slowly.
	www.asp.ucar.edu /colloquium/1992/notes/part1/node71.html (106 words)

	Fourier Series
		The Fourier Series is a discrete-frequency "function," and values are obtained only for frequencies equal to n/T.
		T may be referred to as the time window for computing the series and 1/T is called the fundamental frequency.
		If x(t) is periodic in the window, the Fourier Series expansion (equations (1) and (4) above) may be used to approximate x(t) for all values of t.
	www.engr.trinity.edu /~paul/fourier/fourier/node7.html (523 words)

	HTX & SXHT Series - Hydronic Expansion Tanks
		Hydronic expansion tanks are the latest in a long line of quality products brought to you by Flexcon Industries.
		The in-line HTX Series are available from 2 to 15 gallons and utilize a welded system connection with built-in hex nut for easy installation.The brass shrader valve has an o-ring seal to ensure air side integrity.
		The SXHT Series are freestanding, and range in size from 15 to 81 gallons.
	www.flexconind.com /html/htx.html (272 words)

	Taylor series expansion of the selection differential
		A general expression for the within-generation change in trait mean involves a Taylor series expansion of relative fitness.
		The second summation evaluates to 1, which can be seen by writing the relative fitness function as an absolute fitness divided by the population mean fitness and performing a Taylor series expansion on the mean fitness.
		The Taylor series expansion relates fitness elasticities to the proportional selection gradient in equation (2).
	www.wsu.edu /~mmorgan/working/Elasticities/node8.html (556 words)

	[No title] (Site not responding. Last check: 2007-10-16)
		The "P" Series fire barrier have been Underwriters Laboratories tested for a two-hour rating.
		Please note that submissions or substitutions must be made within 10 working days of the general contract award to avoid project delay.
		C. Shop Drawing: Submit detailed drawings showing layout of expansion joint fire barrier and all special connections, jointing and accessories which are not completely shown in the manufacturer’s data.
	www.archart.com /spec/pserspec.txt (548 words)

	ClusterExpansion (Site not responding. Last check: 2007-10-16)
		We have implemented a completely object oriented program for generating series expansions for quantum lattice models.
		The algorithm was designed to optimize memory allocation in generating ground state expansions, operator expansions and effective hamiltonians, thus allowing to calculate high order expansions.
		A primary goal of the implementation was to keep the general applicability of series expansions in the studies of quantum lattice problems.
	brahms.th.physik.uni-bonn.de /ClusterExpansion/DOC/Intro.html (123 words)

	Recurrence relations and a series expansion for ArcSin (Site not responding. Last check: 2007-10-16)
		This shows that the series is a representation for the arcsin of x.
		With this relationship, all the terms in the series can be generated once one is given.
		This method makes it possible to compute the series expansion to arbitrary order, and produces a significant savings in time.
	www.utexas.edu /its/rc/answers/math/mathematica/faq/ArcSin_series_expansion.html (564 words)

	A series expansion approach to the inverse problem, J. M. Angulo, M. D. Ruiz-Medina
		The purpose of this paper is to present an approach to this problem using a Riesz-based or orthonormal-based series expansion of the input random field with uncorrelated random coefficients.
		We establish conditions under which the input series expansion induces (via the integral equation) a Riesz-based or orthonormal-based series expansion for the output random field.
		Finally, we discuss this approach in the case of transmissivity estimation from piezometric head data, which was the original motivation of this work.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.jap/1032192853 (244 words)

	Taylor Series Expansion (Site not responding. Last check: 2007-10-16)
		As seen in Section 5.5, Euler's method can be derived from a two-term Taylor series expansion.
		It's only natural to consider the consequence of adding the third term of the Taylor series into the formula for the numerical approximation.
		The ``degree n'' Taylor series approximation to the differential equation is obtained by omitting the remainder term,
	www.mathcs.emory.edu /ccs/ccs315/ccs315/node31.html (312 words)

	Saf-T-Bar Series T Expansion Gaps \| Insulated Electrification Systems (Site not responding. Last check: 2007-10-16)
		Expansion Gaps may be required every 150 feet for steel conductors and every 100 feet for copper conductors, this length expanding 1" in 100°F temperature change.
		The support bolt on the guide frame must go to a conductor mounting bracket.
		(For low amperage applications, if there are no runway expansion gaps to match, expansion gaps may be omitted.
	www.keepitmoving.com /tgaps.html (159 words)

	Series expansion for a stochastic sandpile
		Using operator algebra, we extend the series for the activity density in a one-dimensional stochastic sandpile with fixed particle density p, the first terms of which were obtained via perturbation theory (Dickman and Vidigal 2002 J.
		The expansion is in powers of the time; the coefficients are polynomials in p.
		Constructing Padé approximants to a suitably transformed series, we obtain predictions for the activity that compare well against simulations, in the supercritical regime.
	stacks.iop.org /0305-4470/37/1145 (243 words)

	taylor -- compute a Taylor series expansion (Site not responding. Last check: 2007-10-16)
		The number of terms is counted from the lowest degree term on for finite expansion points, and from the highest degree term on for expansions around infinity, i.e., ``
		Note, however, that the actual number of terms in the resulting series expansion may differ from the requested number of terms.
		If a Taylor series expansion cannot be computed, then the function call with evaluated arguments is returned symbolically together with a warning:
	www.sciface.com /STATIC/DOC25/eng/stdlib/taylor.shtml (613 words)

	The Motor Sports Forum - PR-USAC Silver Crown Series Expansion (Site not responding. Last check: 2007-10-16)
		Homestead-Miami and Kansas Speedways are among tracks expected to be a part of the expanded 2006 racing calendar, with discussions in progress for new events at Kentucky and Nashville Speedways, in addition to already-existing Silver Crown venues.
		The series' presence at these high-profile facilities and its potential inclusion with major traditional racing weekends is expected to generate increased exposure for the series through major media outlets and television.
		Specifications will be distributed to traditional Silver Crown Series car builders who are anticipating construction of their own unique racing machines in the coming months in preparation for the 2006 season.
	www.motorsportsforum.com /contents/showthread.php?threadid=17938 (518 words)

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