41. K10.4-Fourier.html Even expansions. Let call S(n,x), the Fourier series approximations withn coefficients S=(m,x) 1/2+sum(Ak*cos(k*Pi/2*x),k=1..m);. http://www.mapleapps.com/categories/engineering/engmathematics/html/K10.4-Fourie

Even expansions ge:=x*(Heaviside(x)-Heaviside(x-1))+(2-x)*(Heaviside(x-1)-Heaviside(x-2))+(-x)*(Heaviside(x+1)-Heaviside(x))+(2+x)*(Heaviside(x+2)-Heaviside(x+1)); plot(ge,x=-2.01..2.01); The coefficient a[0]:=1/4*Int('f',x=-2..2)=2*(1/4*(int(f1,x=0..1)+int(f2,x=1..2))); The coefficients a[n]:=2*1/2*Int('f'*cos(n*Pi/2*x),x=-2..2)=simplify(2*1/2*(int(f1*cos(n*Pi/2*x),x=0..1)+int(f2*cos(n*Pi/2*x),x=1..2))); The coefficient are all zero A[k]:=subs(n=k,simplify(2*1/2*(int(f1*cos(n*Pi/2*x),x=0..1)+int(f2*cos(n*Pi/2*x),x=1..2)))); Let call S(n,x), the Fourier series approximations with n coefficients: Now, we look at different approximations of S as m increases: plot([ge,S(3,x),S(7,x)],x=-2..2,title="Higher approximations: n=3 (Green),n=7 (Blue)",color=[red,green,blue],numpoints=100); The approximation gets better., Let's try some higher order approximations plot([S(100,x)],x=-6..6,title="Higher approximations: n=100 (Blue)",color=[red,blue],numpoints=150);

42. K10.4-Fourier.html plot(S(100,x),x=6..6,title= Higher approximations n=100 (Blue) ,color=red,blue,numpoints=150);.Comparison between odd and even expansions. http://www.mapleapps.com/categories/engineering/engmathematics/html/K10.4-Fourie

Section 1: Even vs Odd expansions Consider the triangle: g:=x*(Heaviside(x)-Heaviside(x-1))+(2-x)*(Heaviside(x-1)-Heaviside(x-2)); plot(g,x=0..2,title="The triangle"); f1:=x;f2:=(2-x); Even expansions Odd expansions go:=x*(Heaviside(x)-Heaviside(x-1))+(2-x)*(Heaviside(x-1)-Heaviside(x-2))+(x)*(Heaviside(x+1)-Heaviside(x))+(-2-x)*(Heaviside(x+2)-Heaviside(x+1)); plot(go,x=-2.01..2.01); The coefficients are all zero (including n=0) b[n]:=2*1/2*Int('f'*sin(n*Pi/2*x),x=-2..2)=simplify(2*1/2*(int(f1*sin(n*Pi/2*x),x=0..1)+int(f2*sin(n*Pi/2*x),x=1..2))); The coefficient are all zero B[k]:=subs(n=k,simplify(2*1/2*(int(f1*sin(n*Pi/2*x),x=0..1)+int(f2*sin(n*Pi/2*x),x=1..2)))); Let call S(n,x), the Fourier series approximations with n coefficients: Now, we look at different approximations of S as m increases: plot([go,S(3,x),S(7,x)],x=-2..2,title="Higher approximations: n=3 (Green),n=7 (Blue)",color=[red,green,blue],numpoints=100); The approximation gets better., Let's try some higher order approximations plot([S(100,x)],x=-6..6,title="Higher approximations: n=100 (Blue)",color=[red,blue],numpoints=150); Comparison between odd and even expansions plot([S(j,1),Seven(j,1)],j=2..51,title="Convergence of the even expansion (blue) and odd expansion (red) at one point x=1",color=[red,blue],numpoints=50);

43. Calculus&Mathematica: 153 Course Description person knows 1/(1 x); e x; sinx; cosx. expansions for approximations.Science and math experience. Experiments geared toward http://socrates.math.ohio-state.edu/about/coursedesc/describe153.php3

Go to.... About our Program For Students For Schools The People Contact Us Links Mathematica Bill Davis, Horacio Porta and Jerry Uhl Technical Crew: Don Brown, Gary Binyamin, Alan DeGuzman, Justin Gallivan, Corey Mutter, David Taubenheim, Jennifer Welch and David Wiltz The rights to all modifications are assigned to Addison-Wesley Publishing Company, Inc. Developed with support from the National Science Foundation at the University of Illinois at Urbana-Champaign and the Ohio State University. Mathematica Approximations Book 3 3.01 Splines Mathematics Remarkable plots explained by order of contact. Splining for smoothness at the knots. Science and math experience. Experiments geared at discovering that the smoother the transition from one curve to another at a knot, the better both curves approximate each other near the knot. Splining functions and polynomials. Splines in road design. Landing an airplane. The natural cubic spline. Order of contact for derivatives and integrals. 3.02 Expansions Mathematics The expansion of a function f[x] in powers of x as a file of polynomials with higher and higher orders of contact with f[x] at x = 0. The expansions every literate calculus person knows:

44. Contents Page Contents List. Module M1.7 Series expansions and approximations. 1Opening items 2 Finite series 2.1 Sequences 2.2 An introduction http://physics.open.ac.uk/flap/schools/M1_7cl.html

Contents List Module M1.7 Series expansions and approximations 1 Opening items 2 Finite series 2.1 Sequences 2.2 An introduction to series 2.3 Properties of the summation symbol 2.4 Sums of powers of positive integers 2.5 Arithmetic progressions 2.6 Geometric progressions 2.7 The binomial expansion 3 Infinite series 3.1 Some standard infinite series 3.3 Convergence and divergence: the ratio test 3.4 Approximations to infinite series 4 Closing items 5 Answers and comments Return to Previous page (c) 1998, The Open University

45. R. Wong Asymptotic approximations of Integrals contains the distributional method, whichis not in the general theory of asymptotic expansions, including smoothing http://ec-securehost.com/SIAM/CL34.html

new books author index subject index series index Purchase options are located at the bottom of the page. The catalog and shopping cart are hosted for SIAM by EasyCart. Your transaction is secure. If you have any questions about your order, contact harris@siam.org Asymptotic Approximations of Integrals R. Wong Classics in Applied Mathematics 34 Asymptotic methods are frequently used in many branches of both pure and applied mathematics, and this classic text remains the most up-to-date book dealing with one important aspect of this area, namely, asymptotic approximations of integrals. In Asymptotic Approximations of Integrals , all results are proved rigorously, and many of the approximation formulas are accompanied by error bounds. A thorough discussion on multidimensional integrals is given, and references are provided. Asymptotic Approximations of Integrals contains the "distributional method," which is not available elsewhere. Most of the examples in this text come from concrete applications. Since its publication twelve years ago, significant developments have occurred in the general theory of asymptotic expansions, including smoothing of the Stokes phenomenon, uniform exponentially improved asymptotic expansions, and hyperasymptotics. These new concepts belong to the area now known as "exponential asymptotics." Expositions of these new theories are available in papers published in various journals, but not yet in book form.

46. First Steps In Numerical Analysis f(0.5), using a calculator and the first four terms of the Taylor expansions. foundin 1.3 to give linear, quadratic, and cubic polynomial approximations for f http://kr.cs.ait.ac.th/~radok/math/mat7/step5.htm

47. Robert Harlander Aymptotic Expansions Padé approximations (for singlet diagrams). (For a review on Padé approximationssee here.) The results obtained through asymptotic expansions can be combined http://www.robert-harlander.de/research/lmp.html

48. Paper Info Special functions, orthogonal polynomials, harmonic analysis, ODE s, differentialrelations, calculus of variations, approximations, expansions, asymptotics; http://math.ucr.edu/home/baez/paper.info.html

Paper Info John Baez You can get a bunch of math and physics papers electronically by going to the xxx e-print archives . There is a lot of information to read if you're having trouble figuring out how it works. Once you know what you're doing, it's easy to download papers using the xxx `form interface'. From there, you can easily get ahold of papers in the following physics archives: astro-ph Astrophysics cond-mat Condensed Matter gr-qc hep-th High Energy Physics - Experiment hep-lat High Energy Physics - Lattice hep-ph High Energy Physics - Phenomenology hep-th High Energy Physics - Theory math-ph Mathematical Physics nucl-ex Nuclear Experiment nucl-th Nuclear Theory quant-ph Quantum Physics physics Physics (various topics not covered above) as well as those in the math archive: math Another nice interface to the math archive is the U.C. Davis front end Math papers are divided into the following subject classifications. (Some of these subject classifications subsume the old math archives alg-geom, dg-ga, funct-an, and q-alg.) AG - Algebraic Geometry subsumes alg-geom Real and complex algebraic geometry, schemes, stacks, commutative rings, local rings

49. PROBLEMS OF THE NUMERICAL ANALYSIS OF ITO STOCHASTIC DIFFERENTIAL Chapter 4 ends with a discussion of various expansions and approximations of multiplestochastic Stratonovich integrals on polynomial and trigonometric systems http://www.neva.ru/journal/eng/ref/1998/vol1/e_kulbk.htm

PROBLEMS OF THE NUMERICAL ANALYSIS OF ITO STOCHASTIC DIFFERENTIAL EQUATIONS The monograth D.F.Kuznetsov Department of Mathematics St.-Petersburg State Technical University St.-Petersburg, Russia, e-mail: control1@citadel.stu.neva.ru Abstract. The book is devoted to the problem of numerical analysis of Ito stochastic differential equations. The book consists of seven chapters. Chapter 1 is an introduction and begins with an exposition of general facts from the elementary theory of probability. Some problems formulated in terms of stochastic differential equations are presented. Chapter 2 deals with the problem of integration order replacement for multiple stochastic Ito integrals. For one class of multiple stochastic Ito integrals we give proofs of the integration order replacement theorems. Stochastic (Taylor-Ito, Taylor-Stratonovich, and unified Taylor-Ito) expansions of Ito processes are considered in Chapter 3. The unified Taylor-Ito expansions are constructed via integration order replacement theorems for multiple stochastic Ito integrals obtained in Chapter 2. Examples of the unified Taylor-Ito expansions for solutions of certain scalar and vector stochastic differential Ito equations are given. Chapter 4 provides methods of expansion and approximation of multiple stochastic Stratonovich and Ito's integrals. We give a new method of multiple Stratonovich stochastic integral approximation based on multiple Fourier series on full orthonormal systems of functions. The comparison of this method with the Milstein method of expansion and approximation of multiple stochastic Stratonovich integrals is given. General formulas for expansion, approximation, and mean-square error of approximation of multiple stochastic Stratonovich integral of a multiplicity k are obtained. We suggest a new method of multiple Ito stochastic integral approximation based on multiple integral sums. Chapter 4 ends with a discussion of various expansions and approximations of multiple stochastic Stratonovich integrals on polynomial and trigonometric systems of functions.

50. KOPS-Datenbank 41XX approximations and expansions, {For all approximation theory in the complexdomain, See {30Exx} {30E05 and 30E10}; for all trigonometric approximation http://www.ub.uni-konstanz.de/kops/msc_ebene2.php?zahl=41&anzahl=0

51. EPrint Series Of Department Of Mathematics, Hokkaido University - Subject: 41-xx Subject 41xx approximations AND expansions. MSC2000 (137) 41-xx approximationsAND expansions. This list was generated on Tue May 18 061009 JST 2004. http://eprints.math.sci.hokudai.ac.jp/view/subjects/41-xx.html

EPrint Series of Department of Mathematics, Hokkaido University Home About Browse Search ... Help Subject: 41-xx APPROXIMATIONS AND EXPANSIONS 41-xx APPROXIMATIONS AND EXPANSIONS This list was generated on Sun Jun 6 06:10:09 JST 2004 Site Administrator: nami@math.sci.hokudai.ac.jp

52. ICMS-Mathematics Preprints Server-Listing approximations and expansions {For all approximation theory in the complex domain,see 30Exx, 30E05 and 30E10; for all trigonometric approximation and http://mps.math-net.info/msc.html?code=41-XX&d_class=41

53. Citations Adaptive Nonlinear Approximations - Davis the problem of finding M vector optimal approximations is NP hard. Because of thenumerical intractability of computing optimal expansions, in section 3 we http://citeseer.ist.psu.edu/context/21384/185946

54. SSRN-Edgeworth Approximations For Semiparametric Instrumental Variable Estimator Edgeworth approximations for Semiparametric Instrumental Variable Estimators andTest Statistics, the validity of higher order asymptotic expansions to the http://papers.ssrn.com/sol3/papers.cfm?abstract_id=241351

55. 4 Extrapolation Methods For Orthogonal Expansions For extrapolation of expansions in orthogonal polynomials of the form. simple formula,but applying it is expected to yield a sequence of approximations that is http://www.uni-regensburg.de/Fakultaeten/nat_Fak_IV/Theoretische_Chemie/Homeier/

Next: 5 A Simple Example Up: On Convergence Acceleration of Previous: 3 Three-center Nuclear Attraction 4 Extrapolation Methods for Orthogonal Expansions For extrapolation of expansions in orthogonal polynomials of the form with partial sums not many methods are known to work well. Besides the well-known algorithm of Wynn and the transformations [ ], there are methods based on the acceleration of related complex power series [ ]. If one does not like to work with complex arithmetics, one may use also the so-called transformation [ ]. The transformation may be regarded as a generalization of the transformation that has proved to be useful for trigonometric Fourier series [ ]. Both transformations are obtained by iteration of a simple transformation. We sketch the main ideas that lead to the transformation. Consider the sequence of the partial sums to be extrapolated as given in Eq. ( ), and write it in terms of the limit s and a tail as By using the usual three-term recurrence relation of the orthogonal polynomials repeatedly, one may express the tail as

56. Analytical Approximations Of Fractional Delays: Lagrange Interpolators And Allpa by approximating the ideal Fourier transform of the fractional delay according totwo different Padé approximations series expansions and continued fraction http://www.ircam.fr/equipes/analyse-synthese/tassart/these/icassp97/article.html

Next: Introduction S. Tassart, Ph. Depalle IRCAM, Analysis/Synthesis Team 1 place Igor-Stravinsky, 75004 PARIS, FRANCE tassartircam.fr, phdircam.fr Analytical Approximations of Fractional Delays: Lagrange Interpolators and Allpass Filters Abstract: Introduction Ideal Fractional Delay From Continuous to Discrete An Ideal Transfer Function ... About this document ... Stephan Tassart Wed May 21 17:49:28 MET DST 1997

57. T.M. Dunster: Publication List Publication list. 25 TM Dunster, Convergent expansions for solutions of linear 24 TM Dunster, Uniform asymptotic approximations for the Whittaker functions http://www-rohan.sdsu.edu/~dunster/publications

Dowloads of recent ones (pdf format): Manuscript Manuscript Manuscript Manuscript ... Manuscript and figures . (Note: In the manuscript [] should read +/ Publication list [25] T. M. Dunster , Convergent expansions for solutions of linear ordinary differential equations having a simple pole, with an application to associated Legendre functions, (Submitted). [24] T. M. Dunster , Uniform asymptotic approximations for the Whittaker functions M k , i m and W k , i m , Analysis and Applications, 1 (2003) pp. 199-212. [23] T. M. Dunster , Uniform asymptotic expansions for associated Legendre functions of large order, Proc. Roy. Soc. Edinburgh Sec. A, 133A, (2003) pp. 807-827. [22] T. M. Dunster , Uniform asymptotic expansions for Charlier polynomials, J. Approx. Theory, 112 (2001) pp. 93-133. [21] T. M. Dunster , Convergent expansions for linear ordinary differential equations having a simple turning point, with an application to Bessel functions, Studies in Applied Math., 107 (2001) pp. 293-323. [20] T. M. Dunster , Uniform asymptotic expansions for the reverse generalised Bessel polynomials, and related functions, SIAM J. Math. Anal., 32 (5) (2001) pp. 987-1013. [19] T. M. Dunster

58. RR-3427 : Analytic Expansions Of (max,+) Lyapunov Exponents Translate this page order, together with an error estimate for finite order Taylor approximations. Severalextensions of this are discussed, including expansions of multinomial http://www.inria.fr/rrrt/rr-3427.html

RR-3427 - Analytic Expansions of (max,+) Lyapunov Exponents Hong, Dohy Rapport de recherche de l'INRIA- Sophia Antipolis Fichier PostScript / PostScript file (382 Ko) Fichier PDF / PDF file (595 Ko) Equipe : MISTRAL 51 pages - Mai 1998 - Document en anglais KEY-WORDS : TAYLOR SERIES / LYAPUNOV EXPONENTS / (MAX / +) SEMIRING / STRONG COUPLING / RENOVATING EVENTS / STATIONARY STATE VARIABLES / ANALYTICITY / VECTORIAL RECURRENCE RELATION / NETWORK MODELING / STOCHASTIC PETRI NETS

59. MSC91 41XX approximations and expansions, {For all approximation theory in the complexdomain, See {30Exx} {30E05 and 30E10}; for all trigonometric approximation http://scidok.sulb.uni-saarland.de/msc_ebene2.php?zahl=41&anzahl=0

60. Asymptotic Methods In Statistical Inference Download (PostScript format),; Chapter 3 Edgeworth expansions. First sectioncontaining the Edgeworth approximations. Download (PostScript format). http://www.dina.dk/~ib/asymp/plan.html

Asymptotic methods in statistical inference NB - new time and place Mondays 12.15 - 14.00 in Aud. 3-07. Course plan The course plan below shows the tentative content of each of the mondays. No doubt this will change as the course progresses, and the final part of the course will depend on the participants. First time is August 30. October 18 is cancelled (vacation) and so is November 8. 30/8-99. Introduction. From the likelihood equation to probability statements. Mathematical foundation (1): Asymptotic expansions, the order symbols, multilinear algebra. 13/9-99. Mathematical foundation (2): Differentials. 20/9-99. Mathematical foundation (3): Moments and cumulants. The saddlepoint approximation (via exponential tilting). 27/9-99. Laplace integration (1): Watson's lemma and extensions. 4/10-99. Laplace integration methods (ctd), uniform expansions, tail integrals. 11/10-99. Transformations of densities between surfaces, (the p-star formula). 18/10-99. VACATION 25/10-00. Inversion integrals and Edgeworth expansions for densities (and cdf's). 1/11-99. The delta method, cumulant computations, (marginal distributions).

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