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Topic: Asymptotic series

	Series (mathematics) (Site not responding. Last check: 2007-10-08)
		In mathematics, a series is a sum of a sequence of termss.
		The harmonic series is the series 1 + 1/2 + 1/3 + 1/4 + 1/5...
		Asymptotic series, otherwise asymptotic expansions, are not typically convergent infinite series, but sequences of finite approximations each of which is a good asymptotic representation.
	www.brainyencyclopedia.com /encyclopedia/s/se/series__mathematics_.html (1534 words)

	Series (mathematics) - Wikipedia, the free encyclopedia
		Series may be finite, or infinite; in the first case they may be handled with elementary algebra, but infinite series require tools from mathematical analysis if they are to be applied in anything more than a tentative way.
		Series for the expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had been treated by Jakob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still earlier by Viète.
		Formal power series are also used in combinatorics to describe and study sequences that are otherwise difficult to handle; this is the method of generating functions.
	en.wikipedia.org /wiki/Series_(mathematics) (1747 words)

	Asymptotic expansion - Wikipedia, the free encyclopedia
		In mathematics an asymptotic expansion, asymptotic series or Poincaré expansion is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point.
		The most common type of asymptotic expansion is a power series in either positive or negative terms.
		While a convergent Taylor series fits the definition as given, a non-convergent series is what is usually intended by the phrase.
	en.wikipedia.org /wiki/Asymptotic_series (282 words)

	Integrating the Bell Curve (Asymptotic Series)
		One interesting approach (leading to the concept of asymptotic series) is to make use of the fact that, although G(t) cannot be integrated in closed form, there are functions that asymptotically approach G(t) in certain ranges and that DO have closed form integrals.
		This is a characteristic of asymptotic series: for a given argument the error becomes smaller as terms are added, until reaching a minimum, beyond which the error becomes larger.
		Thus, the series representation inf / / 1 1 13 135 1357 \
	www.mathpages.com /home/kmath045.htm (895 words)

	Infinite series (Site not responding. Last check: 2007-10-08)
		Series may be finite, or infinite; inthe first case they may be handled with elementary algebra, but infinite seriesrequire tools from mathematical analysis if they are tobe applied in anything more than a tentative way.
		The harmonic series isthe series 1 + 1/2 + 1/3 + 1/4 + 1/5...
		Asymptotic series, otherwise asymptotic expansions, are not typically convergent infiniteseries, but sequences of finite approximations each of which is a good asymptotic representation.
	www.therfcc.org /infinite-series-6287.html (1432 words)

	PlanetMath: asymptotic power series (Site not responding. Last check: 2007-10-08)
		This series may or may not converge when viewed as an infinite series, but a finite number of terms of this series,
		The series (1) is an asymptotic power series.
		This is version 6 of asymptotic power series, born on 2005-09-09, modified 2005-09-10.
	planetmath.org /encyclopedia/AsymptoticPowerSeries.html (217 words)

	Asymptotic series: A mathematical aside
		It is perfectly possible to have two different asymptotic series representations of the same function, as long as the difference between the two series is less than the intrinsic error associated with each series.
		The jumps in the coefficients of the subdominant series are chosen in such a manner that if we perform a complete circuit in the complex plane then the value of the asymptotic expansion is the same at the beginning and the end points.
		In other words, the asymptotic expansion is single-valued, despite the fact that it is built up out of two asymptotic series which are not single-valued.
	farside.ph.utexas.edu /teaching/jk1/lectures/node76.html (1487 words)

	StatLab Abstracts by Date
		Using this, we study the asymptotics ofa conditional empirical process indexed by classes of sets.Under assumptions on the richness of the indexing class in terms ofmetric entropy with bracketing, we have established uniform convergence, and asymptotic normality.
		Abstract: We derive the asymptotic distribution of a new backfitting procedure for estimating the closest additive approximation to a nonparametric regressionfunction.
		Abstract: A concept of asymptotically efficient estimation is presented whena misspecified parametric time series model is fitted to a stationary process.Efficiency of several minimum distance estimates is proved and the behavior ofthe Gaussian maximum likelihood estimate is studied.
	www.statlab.uni-heidelberg.de /reports (7977 words)

	Asymptotic expansion -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics an asymptotic expansion, (additional info and facts about asymptotic series) asymptotic series or (additional info and facts about Poincaré expansion) Poincaré expansion is a formal series of functions
		See (additional info and facts about asymptotic analysis) asymptotic analysis and (additional info and facts about big O notation) big O notation for the notation.
		given, a non-convergent series is what is usually intended by the phrase.
	www.absoluteastronomy.com /encyclopedia/a/as/asymptotic_expansion.htm (419 words)

	Asymptotic Series (Site not responding. Last check: 2007-10-08)
		In a general sense, a series is a related set of things that occur one after the other or are otherwise connected oneafter the other.
		Seriation is a method of dating objects in the field of archaeology.
		Serialism is a rigorous system of writing music in which various elements ofthe piece are ordered according to a pre-determined ordered set or sets, and variations on them,
	www.bodawg.com /point/35444-asymptotic-series.html (170 words)

	Asymptotic Series for Singularly Perturbed Kolmogorov--Fokker--Planck Equations (Site not responding. Last check: 2007-10-08)
		We derive limit theorems for the transition densities of diffusion processes and develop asymptotic expansions for solutions of a class of singularly perturbed Kolmogorov--Fokker--Planck equations.
		The study is motivated by a wide range of applications involving singularly perturbed Markov processes in manufacturing systems, reliability analysis, queueing networks, statistical physics, population biology, financial economics, and many other related fields.
		It is shown that the initial layer terms in the expansion decay at an exponential rate.
	epubs.siam.org /sam-bin/dbq/article/27008 (161 words)

	Worksheet 29
		By the alternating series test the # sum of the exact values of the terms A(k,10), k=0..20, should approximate Si(10) with an # error no more than the absolute value of the next term in the sequence, that is # abs(A(21,10)).
		The exact value of the sum of the first 21 terms in the series is: -------------------------------------------------------------------------------- > convert([seq(A(k,10),k=0..20)],`+`);k:='k': 9733366047998806860644040501494565352596170 ------------------------------------------- 5869315987738947410279534073197070730772067 -------------------------------------------------------------------------------- > evalf(""); 1.658347594 -------------------------------------------------------------------------------- # This is in perfect agreement with the alternating series test.
		It is not surprising that adding # up an alternating series with such large terms is not accurate, unless one does the # calculations to a great many places.
	www.ugrad.math.ubc.ca /coursedoc/m210/worksh29.html (1836 words)

	Stieltjes Series
		For instance, in [29] it was shown that the asymptotic series for the modified Bessel function of the second kind,
		The classic example of a Stieltjes function with a strongly divergent Stieltjes series is the Euler integral (2.6) and its associated asymptotic series, the Euler series (2.7).
		Stieltjes functions and their associated Stieltjes series are very important in the theory of divergent series, since they possess a highly developed representation and convergence theory [25,71,26,72,73,74].
	www.apmaths.uwo.ca /~rcorless/AM563/NOTES/report/node6.html (644 words)

	Numerical Algorithms for Uniform Airy-type Asymptotic Expansions - Temme (ResearchIndex)
		Abstract: Airy-type asymptotic representations of a class of special functions are considered from a numerical point of view.
		It is well known that the evaluation of the coefficients of the asymptotic series near the transition point is a difficult problem.
		One method is based on expanding the coefficients of the asymptotic series in Maclaurin series.
	citeseer.ist.psu.edu /temme97numerical.html (395 words)

	J1(P) - Online Information article about J1(P)
		From the series (15), it may be at once proved that Ps(cos 0) = - (n+!) n (sin 2) 2+..
		The Asymptotic Series for Bessel's Functions.—It may be shown, by means of definite integral expressions for the Bessel's functions, that j()} Jm(P) = P P cos (211+5—0 + Q sin (2~+I-P) Ym(P) =.
		function is less than the next term; thus in using the series for calculation, we must stop at a term which is small.
	encyclopedia.jrank.org /INV_JED/J1_P_.html (1353 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		A while ago somebody posted a question on Stirling's Formula, to which I replied, and you then asked me if I knew what the exact formula for the nth term of the Asymptotic series was.
		This is the difference between asymptotic series, and the kind of convergence which we think of for normal series.
		Now, while the asymptotic series is an approxiamation (no asymptotic series serves to be exact, as they diverge for the infinite partial sum), I believe that the formula:
	www.math2.org /math/integrals/more/stirling.htm (364 words)

	Symbolic Asymptotics:Multiseries of Inverse Functions
		Thus multiseries are asymptotic series in the scale element of fastest decrease, with coefficients which are non-zero functions having multiseries in the remaining scale elements.
		Similarly we need to be sure that the series we have for these coefficient functions are indeed asymptotic series for them, and that we have suitable expressions for their coefficients, and so on.
		The computations alternate between asymptotic and exact representations, replacing n by the left-hand side of (3.1) to get expressions in terms of R only, on which the exp-log machinery can be used.
	algo.inria.fr /papers/html/SaSh99/inv.html (4856 words)

	MUG: asymptotics of ln(ln(x)+a) (15.4.96)
		Apparently: in general the asymptotic gauge functions are x^p, and sometimes exp(p*x).
		Mixed expressions of x and exp(x) can be handled under a logarithm, but not under another power, and ln(x) is ignored as a function of x, and treated as a constant.
		Otherwise, the name "asymptotic expansion" is a bit misleading, and should be something like "Laurent series expansion around infinity".
	www.math.rwth-aachen.de /mapleAnswers/html/137.html (432 words)

	Asymptotic analysis (Site not responding. Last check: 2007-10-08)
		Approximations using Taylor's theorem; order notation; approximate solution of algebraic and transcendental equations; asymptotic series; evaluation of integrals with a large or small parameter; Laplace's method; Watson's lemma; method of stationary phase; method of steepest descents.
		Asymptotic approximations of the roots of a quadratic function
		Asymptotic approximations of the roots of a cubic function
	www.ma.hw.ac.uk /~simonm/ae (683 words)

	All about series (Site not responding. Last check: 2007-10-08)
		The notion of series is closely related to the sum of numbers.
		In fact, whenever one hears the word series, the first thing to come to mind is the sum of numbers.
		This is the basic difference between series and sequences.
	about-serials.org (331 words)

	Table of Contents
		The foundations of the study of asymptotic series in the theory of differential equations were laid by Poincaré in the late 19th century, but it was not until the middle of this century that it became apparent how essential asymptotic series are to understanding the solutions of ordinary differential equations.
		In this outstanding text, the first book devoted exclusively to the subject, the author concentrates on the mathematical ideas underlying the various asymptotic methods; however, asymptotic methods for differential equations are included only if they lead to full, infinite expansions.
		Asymptotic Expansions with Respect to a Parameter 24.
	www.doverpublications.com /cgi-bin/toc.pl/0486495183 (405 words)

	Amazon.com: Asymptotics and Mellin-Barnes Integrals (Encyclopedia of Mathematics and its Applications): Books: R. B. ... (Site not responding. Last check: 2007-10-08)
		Asymptotics and Mellin-Barnes Integrals provides an account of the use and properties of a type of complex integral representation that arises frequently in the study of special functions typically of interest in classical analysis and mathematical physics.
		After developing the properties of these integrals, their use in determining the asymptotic behavior of special functions is detailed.
		The book also fills a gap in the literature on asymptotic analysis and special functions by providing a thorough account of the use of Mellin-Barnes integrals that is otherwise not available in standard references on asymptotics.
	www.amazon.com /exec/obidos/tg/detail/-/0521790018?v=glance (712 words)

	On the Choice of the Remainder Estimates
		Instead, it is at least in principle possible to find for a given sequence an infinite variety of different remainder estimates which all satisfy the asymptotic condition (5.2), and which may perform quite differently in convergence acceleration and summation processes [24].
		On the basis of heuristic and asymptotic arguments, Levin [38] suggested for sequences of partial sums (1.1) several simple explicit remainder estimates.
		A more detailed discussion of the properties of these remainder estimates, some generalizations, additional heuristic motivation, and a description of the types of sequences, for which these estimates should be effective, can be found in section 7 of [23].
	www.apmaths.uwo.ca /~rcorless/AM563/NOTES/report/node5.html (1063 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		From: israel@math.ubc.ca (Robert Israel) Subject: Re: Complex analysis Date: 17 Oct 1999 20:58:05 GMT Newsgroups: sci.math Keywords: series of analytic functions [deletia --djr] The answer to Eric Brunet's question about asymptotic series is no. That is, there are functions F(z,c) which are entire functions of z (i.e.
		analytic on the complex plane C) for all c > 0, have asymptotic series sum_n a_n(z) c^n as c -> 0+, but a_n are not all entire.
		Thus F(z,c) has the asymptotic series sum_{n=0}^infinity a_n(z) c^n as c -> 0+, where a_0(0) = g(0), a_n(z) = 0 otherwise.
	www.math.niu.edu /~rusin/known-math/99/nonentire (204 words)

	AN ASYMPTOTIC SERIES APPROACH TO QUALOCATION METHODS (Site not responding. Last check: 2007-10-08)
		In this paper we develop an asymptotic analysis of qualocation methods when applied to a particular class of pseudodifferential equations.
		As by-products of this expansion we obtain some estimations of pointwise convergence and an asymptotic expansion between the exact and the numerical solution under the action of regularizing operators.
		In addition to this, using the error expansion we deduce sufficient conditions to obtain qualocation methods of higher order for some particular equations.
	math.la.asu.edu /~rmmc/jie/Vol15-2/DOM/DOM.html (104 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		The good news is that if a is very close to zero, you can add a lot of terms before the series start to go berserk, and it will get very close to the right answer before it goes bad.
		The point is that while all convergent series are automatically asymptotic to the functions they converge to, not all asymptotic series are convergent to the functions they are asymptotic to.
		They have a bunch of general theorems about asymptotic series, and also a bunch of tables illustrating what happens for the anharmonic oscillator.
	www.math.niu.edu /~rusin/known-math/00_incoming/pathint (1800 words)

	Asymptotic Normality of Series Estimators for Nonparametric and Semiparametric Regression Models
		This paper establishes the asymptotic normality of series estimators for nonparametric regression models.
		Gallant's Fourier flexible form estimators, trigonometric series estimators, and polynomial series estimators are prime examples of the estimators covered by the results.
		The paper also considers series estimators for additive interactive regression, semiparametric regression, and semiparametric index regression models, and shows them to be consistent and asymptotically normal.
	www.ideas.uqam.ca /ideas/data/Articles/ecmemetrpv:59:y:1991:i:2:p:307-45.html (578 words)

	Numerical algorithms III: special functions
		The terms of the series are however not monotonic: first the terms grow and then they start to decrease, like in the Taylor series for the exponential function evaluated at a large argument.
		It consists of estimating the error of the approximation of gamma by an asymptotic series.
		The error of a truncated asymptotic series is not larger than the first discarded term if the number of terms is larger than n-1/2.
	homepage.mac.com /yacas/manual/Algochapter6.html (6721 words)

	The W.K.B. solutions as asymptotic series
		As the number of terms in the series is increased, the intrinsic error decreases, and the value of
		As we shall discover, the reflection is directly associated with the fact that the expansion (4.261) exhibits a Stokes phenomenon.
		It is, of course, impossible for a convergent power series expansion to exhibit a Stokes phenomenon.
	farside.ph.utexas.edu /teaching/jk1/lectures/node77.html (459 words)

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