Euler and His Work on Infinite Series
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7 Numerical Integration
7 Numerical Integration 7.1 Elementary Algorithms Let us suppose we are confronted with a function f(x) tabulated at the points x1, x2, x3...xn, not necessarily equally spaced. We require the integral of f(x) from x1 to xn. How do we proceed? The simplest algorithm we can use, valid both for equally and unequally spaced points, is the trapezoidal rule. The trapezoidal rule assumes that the function is linear between the tabulated points. With this assumption, it can be seen that the integral from x1 to x2 is given by x2 1 f(x)dx h(f1 + f2) Zx1 ≈ 2 where h = x x . 2 − 1 ¨¨u ¨¨ ¨¨ ¨¨ u¨ f2 f1 u u x1 x2 This equation can be extended to n points for equally spaced points xn 1 1 f(x)dx h( f1 + f2 + f3 ... + fn) (1) Zx1 ≈ 2 2 and xn 1 1 1 f(x)dx (x2 x1)(f1+f2)+ (x3 x2)(f2+f3) ...+ (xn xn−1)(fn−1+fn) Zx1 ≈ 2 − 2 − 2 − for unevenly spaced points. Exercise 7.1: Use the trapezoidal rule to evaluate the integral 2 x sin xdx Z1 1 with 10 points. Double the number of points and re-evaluate the integral. The “exact” value of this integral is 1.440422. Rewrite your program to evaluate the integral at an arbitrary number of points input by the user. Are 1000 points better than 100 points? For equally spaced points the trapezoidal rule has an error O(h3), which means that the error for each interval in the rule is proportional to the cube of the spacing. -
Generalizing the Trapezoidal Rule in the Complex Plane
Generalizing the trapezoidal rule in the complex plane Bengt Fornberg ∗ Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA May 22, 2020 Abstract In computational contexts, analytic functions are often best represented by grid-based func- tion values in the complex plane. For integrating periodic functions, the spectrally accurate trapezoidal rule (TR) then becomes a natural choice, due to both accuracy and simplicity. The two key present observations are (i) The accuracy of TR in the periodic case can be greatly increased (doubling or tripling the number of correct digits) by using function values also along grid lines adjacent to the line of integration, and (ii) A recently developed end correction strategy for finite interval integrations applies just as well when using these enhanced TR schemes. Keywords: Euler-Maclaurin, trapezoidal rule, contour integrals, equispaced grids, hexago- nal grid. AMS classification codes: Primary: 65B15, 65E05; Secondary: 30-04. 1 Introduction There are two main scenarios when approximating contour integrals in the complex plane: i. The function to integrate can be obtained at roughly equal cost for arbitrary spatial locations, and ii. Function values are available primarily on an equispaced grid in the complex plane.1 In case (i), a commonly used option is Gaussian quadrature2, either applied directly over the whole interval of interest, or combined with adaptive interval partitioning. Conformal (and other) mappings may be applied both for straightening curved integration paths (such as Hankel contours) and to improve the resolution in critical places. ∗Email: [email protected] 1If an analytic functions is somewhat costly to evaluate, it may be desirable to pay a one-time cost of evaluating over a grid and then re-use this data for multiple purposes, such as for graphical displays (e.g., [7, 12]) and, as focused on here, for highly accurate contour integrations (assuming some vicinity of the integration paths to be free of singularities). -
A Trapezoidal Rule Error Bound Unifying
Downloaded from http://rspa.royalsocietypublishing.org/ on November 10, 2016 A trapezoidal rule error bound unifying the Euler–Maclaurin formula and geometric rspa.royalsocietypublishing.org convergence for periodic functions Mohsin Javed and Lloyd N. Trefethen Research Mathematical Institute, University of Oxford, Oxford, UK Cite this article: Javed M, Trefethen LN. 2014 A trapezoidal rule error bound unifying the The error in the trapezoidal rule quadrature formula Euler–Maclaurin formula and geometric can be attributed to discretization in the interior and convergence for periodic functions. Proc. R. non-periodicity at the boundary. Using a contour Soc. A 470: 20130571. integral, we derive a unified bound for the combined http://dx.doi.org/10.1098/rspa.2013.0571 error from both sources for analytic integrands. The bound gives the Euler–Maclaurin formula in one limit and the geometric convergence of the trapezoidal Received: 27 August 2013 rule for periodic analytic functions in another. Similar Accepted:15October2013 results are also given for the midpoint rule. Subject Areas: 1. Introduction computational mathematics Let f be continuous on [0, 1] and let n be a positive Keywords: integer. The (composite) trapezoidal rule approximates the integral trapezoidal rule, midpoint rule, 1 = Euler–Maclaurin formula I f (x)dx (1.1) 0 by the sum Author for correspondence: n −1 k In = n f , (1.2) Lloyd N. Trefethen n k=0 e-mail: [email protected] where the prime indicates that the terms k = 0andk = n are multiplied by 1/2. Throughout this paper, f may be real or complex, and ‘periodic’ means periodic with period 1. -
Notes on the Convergence of Trapezoidal-Rule Quadrature
Notes on the convergence of trapezoidal-rule quadrature Steven G. Johnson Created Fall 2007; last updated March 10, 2010 1 Introduction Numerical quadrature is another name for numerical integration, which refers to the approximation of an integral f (x)dx of some function f (x) by a discrete summation ∑wi f (xi) over points xi with´ some weights wi. There are many methods of numerical quadrature corresponding to different choices of points xi and weights wi, from Euler integration to sophisticated methods such as Gaussian quadrature, with varying degrees of accuracy for various types of functions f (x). In this note, we examine the accuracy of one of the simplest methods: the trapezoidal rule with uniformly spaced points. In particular, we discuss how the convergence rate of this method is determined by the smoothness properties of f (x)—and, in practice, usually by the smoothness at the end- points. (This behavior is the basis of a more sophisticated method, Clenshaw-Curtis quadrature, which is essentially trapezoidal integration plus a coordinate transforma- tion to remove the endpoint problem.) For simplicity, without loss of generality, we can take the integral to be for x 2 [0;2p], i.e. the integral 2p I = f (x)dx; ˆ0 which is approximated in the trapezoidal rule1 by the summation: f (0)Dx N−1 f (2p)Dx IN = + ∑ f (nDx)Dx + ; 2 n=1 2 2p where Dx = N . We now want to analyze how fast the error EN = jI − INj decreases with N. Many books estimate the error as being O(Dx2) = O(N−2), assuming f (x) is twice differ- entiable on (0;2p)—this estimate is correct, but only as an upper bound. -
Ch. 15 Power Series, Taylor Series
Ch. 15 Power Series, Taylor Series 서울대학교 조선해양공학과 서유택 2017.12 ※ 본 강의 자료는 이규열, 장범선, 노명일 교수님께서 만드신 자료를 바탕으로 일부 편집한 것입니다. Seoul National 1 Univ. 15.1 Sequences (수열), Series (급수), Convergence Tests (수렴판정) Sequences: Obtained by assigning to each positive integer n a number zn z . Term: zn z1, z 2, or z 1, z 2 , or briefly zn N . Real sequence (실수열): Sequence whose terms are real Convergence . Convergent sequence (수렴수열): Sequence that has a limit c limznn c or simply z c n . For every ε > 0, we can find N such that Convergent complex sequence |zn c | for all n N → all terms zn with n > N lie in the open disk of radius ε and center c. Divergent sequence (발산수열): Sequence that does not converge. Seoul National 2 Univ. 15.1 Sequences, Series, Convergence Tests Convergence . Convergent sequence: Sequence that has a limit c Ex. 1 Convergent and Divergent Sequences iin 11 Sequence i , , , , is convergent with limit 0. n 2 3 4 limznn c or simply z c n Sequence i n i , 1, i, 1, is divergent. n Sequence {zn} with zn = (1 + i ) is divergent. Seoul National 3 Univ. 15.1 Sequences, Series, Convergence Tests Theorem 1 Sequences of the Real and the Imaginary Parts . A sequence z1, z2, z3, … of complex numbers zn = xn + iyn converges to c = a + ib . if and only if the sequence of the real parts x1, x2, … converges to a . and the sequence of the imaginary parts y1, y2, … converges to b. Ex. -
[Math.CA] 1 Jul 1992 Napiain.Freape Ecnlet Can We Example, for Applications
Convolution Polynomials Donald E. Knuth Computer Science Department Stanford, California 94305–2140 Abstract. The polynomials that arise as coefficients when a power series is raised to the power x include many important special cases, which have surprising properties that are not widely known. This paper explains how to recognize and use such properties, and it closes with a general result about approximating such polynomials asymptotically. A family of polynomials F (x), F (x), F (x),... forms a convolution family if F (x) has degree n 0 1 2 n ≤ and if the convolution condition F (x + y)= F (x)F (y)+ F − (x)F (y)+ + F (x)F − (y)+ F (x)F (y) n n 0 n 1 1 · · · 1 n 1 0 n holds for all x and y and for all n 0. Many such families are known, and they appear frequently ≥ n in applications. For example, we can let Fn(x)= x /n!; the condition (x + y)n n xk yn−k = n! k! (n k)! kX=0 − is equivalent to the binomial theorem for integer exponents. Or we can let Fn(x) be the binomial x coefficient n ; the corresponding identity n x + y x y = n k n k Xk=0 − is commonly called Vandermonde’s convolution. How special is the convolution condition? Mathematica will readily find all sequences of polynomials that work for, say, 0 n 4: ≤ ≤ F[n_,x_]:=Sum[f[n,j]x^j,{j,0,n}]/n! arXiv:math/9207221v1 [math.CA] 1 Jul 1992 conv[n_]:=LogicalExpand[Series[F[n,x+y],{x,0,n},{y,0,n}] ==Series[Sum[F[k,x]F[n-k,y],{k,0,n}],{x,0,n},{y,0,n}]] Solve[Table[conv[n],{n,0,4}], [Flatten[Table[f[i,j],{i,0,4},{j,0,4}]]]] Mathematica replies that the F ’s are either identically zero or the coefficients of Fn(x) = fn0 + f x + f x2 + + f xn /n! satisfy n1 n2 · · · nn f00 = 1 , f10 = f20 = f30 = f40 = 0 , 2 3 f22 = f11 , f32 = 3f11f21 , f33 = f11 , 2 2 4 f42 = 4f11f31 + 3f21 , f43 = 6f11f21 , f44 = f11 . -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
Rearrangement of Divergent Fourier Series
The Australian Journal of Mathematical Analysis and Applications AJMAA Volume 14, Issue 1, Article 3, pp. 1-9, 2017 A NOTE ON DIVERGENT FOURIER SERIES AND λ-PERMUTATIONS ANGEL CASTILLO, JOSE CHAVEZ, AND HYEJIN KIM Received 20 September, 2016; accepted 4 February, 2017; published 20 February, 2017. TUFTS UNIVERSITY,DEPARTMENT OF MATHEMATICS,MEDFORD, MA 02155, USA [email protected] TEXAS TECH UNIVERSITY,DEPARTMENT OF MATHEMATICS AND STATISTICS,LUBBOCK, TX 79409, USA [email protected] UNIVERSITY OF MICHIGAN-DEARBORN,DEPARTMENT OF MATHEMATICS AND STATISTICS,DEARBORN, MI 48128, USA [email protected] ABSTRACT. We present a continuous function on [−π, π] whose Fourier series diverges and it cannot be rearranged to converge by a λ-permutation. Key words and phrases: Fourier series, Rearrangements, λ-permutations. 2000 Mathematics Subject Classification. Primary 43A50. ISSN (electronic): 1449-5910 c 2017 Austral Internet Publishing. All rights reserved. This research was conducted during the NREUP at University of Michigan-Dearborn and it was sponsored by NSF-Grant DMS-1359016 and by NSA-Grant H98230-15-1-0020. We would like to thank Y. E. Zeytuncu for valuable discussion. We also thank the CASL and the Department of Mathematics and Statistics at the University of Michigan-Dearborn for providing a welcoming atmosphere during the summer REU program. 2 A. CASTILLO AND J. CHAVEZ AND H. KIM 1.1. Fourier series. The Fourier series associated with a continuous function f on [−π, π] is defined by ∞ X inθ fe(θ) ∼ ane , n=−∞ where Z π 1 −inθ an = f(θ)e dθ . 2π −π Here an’s are called the Fourier coefficients of f and we denote by fethe Fourier series associ- ated with f. -
The Summation of Power Series and Fourier Series
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 12&13 (1985) 447-457 447 North-Holland The summation of power series and Fourier . series I.M. LONGMAN Department of Geophysics and Planetary Sciences, Tel Aviv University, Ramat Aviv, Israel Received 27 July 1984 Abstract: The well-known correspondence of a power series with a certain Stieltjes integral is exploited for summation of the series by numerical integration. The emphasis in this paper is thus on actual summation of series. rather than mere acceleration of convergence. It is assumed that the coefficients of the series are given analytically, and then the numerator of the integrand is determined by the aid of the inverse of the two-sided Laplace transform, while the denominator is standard (and known) for all power series. Since Fourier series can be expressed in terms of power series, the method is applicable also to them. The treatment is extended to divergent series, and a fair number of numerical examples are given, in order to illustrate various techniques for the numerical evaluation of the resulting integrals. Keywork Summation of series. 1. Introduction We start by considering a power series, which it is convenient to write in the form s(x)=/.Q-~2x+&x2..., (1) for reasons which will become apparent presently. Here there is no intention to limit ourselves to alternating series since, even if the pLkare all of one sign, x may take negative and even complex values. It will be assumed in this paper that the pLk are real. -
3.3 the Trapezoidal Rule
Analysis and Implementation of an Age Structured Model of the Cell Cycle Anna Broms [email protected] principal supervisor: Sara Maad Sasane assistant supervisor: Gustaf S¨oderlind Master's thesis Lund University September 2017 Abstract In an age-structured model originating from cancer research, the cell cycle is divided into two phases: Phase 1 of variable length, consisting of the biologically so called G1 phase, and Phase 2 of fixed length, consisting of the so called S, G2 and M phases. A system of nonlinear PDEs along with initial and boundary data describes the number densities of cells in the two phases, depending on time and age (where age is the time spent in a phase). It has been shown that the initial and boundary value problem can be rewritten as a coupled system of integral equations, which in this M.Sc. thesis is implemented in matlab using the trapezoidal and Simpson rule. In the special case where the cells are allowed to grow without restrictions, the system is uncoupled and possible to study analytically, whereas otherwise, a nonlinearity has to be solved in every step of the iterative equation solving. The qualitative behaviour is investigated numerically and analytically for a wide range of model components. This includes investigations of the notions of crowding, i.e. that cell division is restricted for large population sizes, and quorum sensing, i.e. that a small enough tumour can eliminate itself through cell signalling. In simulations, we also study under what conditions an almost eliminated tumour relapses after completed therapy. Moreover, upper bounds for the number of dividing cells at the end of Phase 2 at time t are determined for specific cases, where the bounds are found to depend on the existence of so called cancer stem cells. -
Divergent and Conditionally Convergent Series Whose Product Is Absolutely Convergent
DIVERGENTAND CONDITIONALLY CONVERGENT SERIES WHOSEPRODUCT IS ABSOLUTELYCONVERGENT* BY FLORIAN CAJOKI § 1. Introduction. It has been shown by Abel that, if the product : 71—« I(¥» + V«-i + ■•• + M„«o)> 71=0 of two conditionally convergent series : 71—0 71— 0 is convergent, it converges to the product of their sums. Tests of the conver- gence of the product of conditionally convergent series have been worked out by A. Pringsheim,| A. Voss,J and myself.§ There exist certain conditionally- convergent series which yield a convergent result when they are raised to a cer- tain positive integral power, but which yield a divergent result when they are raised to a higher power. Thus, 71 = «3 Z(-l)"+13, 7>=i n where r = 7/9, is a conditionally convergent series whose fourth power is con- vergent, but whose fifth power is divergent. || These instances of conditionally * Presented to the Society April 28, 1900. Received for publication April 28, 1900. fMathematische Annalen, vol. 21 (1883), p. 327 ; vol. 2(5 (1886), p. 157. %Mathematische Annalen, vol. 24 (1884), p. 42. § American Journal of Mathematics, vol. 15 (1893), p. 339 ; vol. 18 (1896), p. 195 ; Bulletin of the American Mathematical Society, (2) vol. 1 (1895), p. 180. || This may be a convenient place to point out a slight and obvious extension of the results which I have published in the American Journal of Mathematics, vol. 18, p. 201. It was proved there that the conditionally convergent series : V(_l)M-lI (0<r5|>). Sí n when raised by Cauchy's multiplication rule to a positive integral power g , is convergent whenever (î — l)/ï <C r ! out the power of the series is divergent, if (q— 1 )¡q > r. -
Notes on Euler's Work on Divergent Factorial Series and Their Associated
Indian J. Pure Appl. Math., 41(1): 39-66, February 2010 °c Indian National Science Academy NOTES ON EULER’S WORK ON DIVERGENT FACTORIAL SERIES AND THEIR ASSOCIATED CONTINUED FRACTIONS Trond Digernes¤ and V. S. Varadarajan¤¤ ¤University of Trondheim, Trondheim, Norway e-mail: [email protected] ¤¤University of California, Los Angeles, CA, USA e-mail: [email protected] Abstract Factorial series which diverge everywhere were first considered by Euler from the point of view of summing divergent series. He discovered a way to sum such series and was led to certain integrals and continued fractions. His method of summation was essentialy what we call Borel summation now. In this paper, we discuss these aspects of Euler’s work from the modern perspective. Key words Divergent series, factorial series, continued fractions, hypergeometric continued fractions, Sturmian sequences. 1. Introductory Remarks Euler was the first mathematician to develop a systematic theory of divergent se- ries. In his great 1760 paper De seriebus divergentibus [1, 2] and in his letters to Bernoulli he championed the view, which was truly revolutionary for his epoch, that one should be able to assign a numerical value to any divergent series, thus allowing the possibility of working systematically with them (see [3]). He antic- ipated by over a century the methods of summation of divergent series which are known today as the summation methods of Cesaro, Holder,¨ Abel, Euler, Borel, and so on. Eventually his views would find their proper place in the modern theory of divergent series [4]. But from the beginning Euler realized that almost none of his methods could be applied to the series X1 1 ¡ 1!x + 2!x2 ¡ 3!x3 + ::: = (¡1)nn!xn (1) n=0 40 TROND DIGERNES AND V.