Divided Differences, Falling Factorials, and Discrete Splines Another Look at Trend Filtering and Related Problems
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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. -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
Euler and His Work on Infinite Series
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 44, Number 4, October 2007, Pages 515–539 S 0273-0979(07)01175-5 Article electronically published on June 26, 2007 EULER AND HIS WORK ON INFINITE SERIES V. S. VARADARAJAN For the 300th anniversary of Leonhard Euler’s birth Table of contents 1. Introduction 2. Zeta values 3. Divergent series 4. Summation formula 5. Concluding remarks 1. Introduction Leonhard Euler is one of the greatest and most astounding icons in the history of science. His work, dating back to the early eighteenth century, is still with us, very much alive and generating intense interest. Like Shakespeare and Mozart, he has remained fresh and captivating because of his personality as well as his ideas and achievements in mathematics. The reasons for this phenomenon lie in his universality, his uniqueness, and the immense output he left behind in papers, correspondence, diaries, and other memorabilia. Opera Omnia [E], his collected works and correspondence, is still in the process of completion, close to eighty volumes and 31,000+ pages and counting. A volume of brief summaries of his letters runs to several hundred pages. It is hard to comprehend the prodigious energy and creativity of this man who fueled such a monumental output. Even more remarkable, and in stark contrast to men like Newton and Gauss, is the sunny and equable temperament that informed all of his work, his correspondence, and his interactions with other people, both common and scientific. It was often said of him that he did mathematics as other people breathed, effortlessly and continuously. -
On the Euler Integral for the Positive and Negative Factorial
On the Euler Integral for the positive and negative Factorial Tai-Choon Yoon ∗ and Yina Yoon (Dated: Dec. 13th., 2020) Abstract We reviewed the Euler integral for the factorial, Gauss’ Pi function, Legendre’s gamma function and beta function, and found that gamma function is defective in Γ(0) and Γ( x) − because they are undefined or indefinable. And we came to a conclusion that the definition of a negative factorial, that covers the domain of the negative space, is needed to the Euler integral for the factorial, as well as the Euler Y function and the Euler Z function, that supersede Legendre’s gamma function and beta function. (Subject Class: 05A10, 11S80) A. The positive factorial and the Euler Y function Leonhard Euler (1707–1783) developed a transcendental progression in 1730[1] 1, which is read xedx (1 x)n. (1) Z − f From this, Euler transformed the above by changing e to g for generalization into f x g dx (1 x)n. (2) Z − Whence, Euler set f = 1 and g = 0, and got an integral for the factorial (!) 2, dx ( lx )n, (3) Z − where l represents logarithm . This is called the Euler integral of the second kind 3, and the equation (1) is called the Euler integral of the first kind. 4, 5 Rewriting the formula (3) as follows with limitation of domain for a positive half space, 1 1 n ln dx, n 0. (4) Z0 x ≥ ∗ Electronic address: [email protected] 1 “On Transcendental progressions that is, those whose general terms cannot be given algebraically” by Leonhard Euler p.3 2 ibid. -
Coalgebraic Foundations of the Method of Divided Differences
ADVANCES IN MATHEMATICS 91, 75~135 (1992) Coalgebraic Foundations of the Method of Divided Differences PHILIP S. HIRSCHHORN AND LOUISE ARAKELIAN RAPHAEL* Department qf Muthenmtics, Howard Unhrersity, Washington. D.C. 20059 1. INTRODUCTION 1.1. Zntroductiorz The analogy between properties of the function e’ and the properties of the function l/( 1 -x) has led, in the past two centuries, to much new mathematics. At the most elementary level, it was noticed long ago that the function l/( 1 - x) is (roughly) the Laplace transform of the function e-‘;, and the theory of Laplace transforms, in its heyday, was concerned with drafting a correspondence table between properties of functions and properties of their Laplace transforms. In the development of functional analysis in the forties and fifties, properties of semigroups of linear operators on Banach spaces were thrown back onto properties of their resolvents; the functional equation satisfied by the resolvent of an operator was clearly perceived at that time as an analog-though still a mysterious one-of the Cauchy functional equation satisfied by the exponential function. A far more ancient, though, it must be admitted, still little known instance of this analogy is the Newton expansion of the calculus of divided difference (see, e.g., [lo] ). Although Taylor’s formula can be formally seen as a special case of a Newton expansion (obtained by taking equal arguments), nonetheless, the algebraic properties of the two expansions were radically different. Suitably reinterpreted, Taylor’s formula led in this * Partially supported by National Science Foundation Grant ECS8918050. 75 OOOl-8708/92 $7.50 Copyright ( 1992 by Academic Press. -
Interpolation
Interpolation 1 What is interpolation? For a certain function f (x) we know only the values y1 = f (x1),. ,yn = f (xn). For a pointx ˜ different from x1;:::;xn we would then like to approximate f (x˜) using the given data x1;:::;xn and y1;:::;yn. This means we are constructing a function p(x) which passes through the given points and hopefully is close to the function f (x). It turns out that it is a good idea to use polynomials as interpolating functions (later we will also consider piecewise polynomial functions). 2 Why are we interested in this? • Efficient evaluation of functions: For functions like f (x) = sin(x) it is possible to find values using a series expansion (e.g. Taylor series), but this takes a lot of operations. If we need to compute f (x) for many values x in an interval [a;b] we can do the following: – pick points x1;:::;xn in the interval – find the interpolating polynomial p(x) – Then: for any given x 2 [a;b] just evaluate the polynomial p(x) (which is cheap) to obtain an approximation for f (x) Before the age of computers and calculators, values of functions like sin(x) were listed in tables for values x j with a certain spacing. Then function values everywhere in between could be obtained by interpolation. A computer or calculator uses the same method to find values of e.g. sin(x): First an interpolating polynomial p(x) for the interval [0;p=2] was constructed and the coefficients are stored in the computer. -
List of Mathematical Symbols by Subject from Wikipedia, the Free Encyclopedia
List of mathematical symbols by subject From Wikipedia, the free encyclopedia This list of mathematical symbols by subject shows a selection of the most common symbols that are used in modern mathematical notation within formulas, grouped by mathematical topic. As it is virtually impossible to list all the symbols ever used in mathematics, only those symbols which occur often in mathematics or mathematics education are included. Many of the characters are standardized, for example in DIN 1302 General mathematical symbols or DIN EN ISO 80000-2 Quantities and units – Part 2: Mathematical signs for science and technology. The following list is largely limited to non-alphanumeric characters. It is divided by areas of mathematics and grouped within sub-regions. Some symbols have a different meaning depending on the context and appear accordingly several times in the list. Further information on the symbols and their meaning can be found in the respective linked articles. Contents 1 Guide 2 Set theory 2.1 Definition symbols 2.2 Set construction 2.3 Set operations 2.4 Set relations 2.5 Number sets 2.6 Cardinality 3 Arithmetic 3.1 Arithmetic operators 3.2 Equality signs 3.3 Comparison 3.4 Divisibility 3.5 Intervals 3.6 Elementary functions 3.7 Complex numbers 3.8 Mathematical constants 4 Calculus 4.1 Sequences and series 4.2 Functions 4.3 Limits 4.4 Asymptotic behaviour 4.5 Differential calculus 4.6 Integral calculus 4.7 Vector calculus 4.8 Topology 4.9 Functional analysis 5 Linear algebra and geometry 5.1 Elementary geometry 5.2 Vectors and matrices 5.3 Vector calculus 5.4 Matrix calculus 5.5 Vector spaces 6 Algebra 6.1 Relations 6.2 Group theory 6.3 Field theory 6.4 Ring theory 7 Combinatorics 8 Stochastics 8.1 Probability theory 8.2 Statistics 9 Logic 9.1 Operators 9.2 Quantifiers 9.3 Deduction symbols 10 See also 11 References 12 External links Guide The following information is provided for each mathematical symbol: Symbol: The symbol as it is represented by LaTeX. -
Leonhard Euler English Version
LEONHARD EULER (April 15, 1707 – September 18, 1783) by HEINZ KLAUS STRICK , Germany Without a doubt, LEONHARD EULER was the most productive mathematician of all time. He wrote numerous books and countless articles covering a vast range of topics in pure and applied mathematics, physics, astronomy, geodesy, cartography, music, and shipbuilding – in Latin, French, Russian, and German. It is not only that he produced an enormous body of work; with unbelievable creativity, he brought innovative ideas to every topic on which he wrote and indeed opened up several new areas of mathematics. Pictured on the Swiss postage stamp of 2007 next to the polyhedron from DÜRER ’s Melencolia and EULER ’s polyhedral formula is a portrait of EULER from the year 1753, in which one can see that he was already suffering from eye problems at a relatively young age. EULER was born in Basel, the son of a pastor in the Reformed Church. His mother also came from a family of pastors. Since the local school was unable to provide an education commensurate with his son’s abilities, EULER ’s father took over the boy’s education. At the age of 14, EULER entered the University of Basel, where he studied philosophy. He completed his studies with a thesis comparing the philosophies of DESCARTES and NEWTON . At 16, at his father’s wish, he took up theological studies, but he switched to mathematics after JOHANN BERNOULLI , a friend of his father’s, convinced the latter that LEONHARD possessed an extraordinary mathematical talent. At 19, EULER won second prize in a competition sponsored by the French Academy of Sciences with a contribution on the question of the optimal placement of a ship’s masts (first prize was awarded to PIERRE BOUGUER , participant in an expedition of LA CONDAMINE to South America). -
Math 63: Winter 2021 Lecture 17
Math 63: Winter 2021 Lecture 17 Dana P. Williams Dartmouth College Monday, February 15, 2021 Dana P. Williams Math 63: Winter 2021 Lecture 17 Getting Started 1 We should be recording. 2 Remember it is better for me if you have your video on so that I don't feel I'm just talking to myself. 3 Our midterm will be available Friday after class and due Sunday by 10pm. It will cover through Today's lecture. More details soon. 4 Time for some questions! Dana P. Williams Math 63: Winter 2021 Lecture 17 Rules To Differentiate With The following are routine consequences of the definition. Lemma If f (x) = c for all x 2 R, then f 0(x) = 0 for all x 2 R. If f (x) = x for all x 2 R, then f 0(x) = 1 for all x 2 R. Theorem Suppose that f and g are differentiable at x0. Then so are f ± g, f fg, and if g(x0) 6= 0, g . Futhermore, 0 0 0 1 (f ± g) (x0) = f (x0) ± g (x0), 0 0 0 2 (fg) (x0) = f (x0)g(x0) + f (x0)g (x0), and 0 f 0(x )g(x )−f (x )g 0(x ) 3 f 0 0 0 0 (x0) = 2 . g g(x0) Dana P. Williams Math 63: Winter 2021 Lecture 17 More Formulas Corollary Suppose that f is differentiable at x0 and c 2 R. Then cf is 0 0 differentiable at x0 and (cf ) (x0) = cf (x0). Corollary Suppose that n 2 Z and f (x) = xn. -
Summation of Divergent Power Series by Means of Factorial Series
Summation of divergent power series by means of factorial series Ernst Joachim Weniger Institut f¨ur Physikalische und Theoretische Chemie, Universit¨at Regensburg, D-93040 Regensburg, Germany Abstract Factorial series played a major role in Stirling’s classic book Methodus Differentialis (1730), but now only a few specialists still use them. This article wants to show that this neglect is unjustified, and that factorial series are useful numerical tools for the summation of divergent (inverse) power series. This is documented by summing the divergent asymptotic expansion for the exponential integral E1(z) and the factorially divergent Rayleigh-Schr¨odinger perturba- tion expansion for the quartic anharmonic oscillator. Stirling numbers play a key role since they occur as coefficients in expansions of an inverse power in terms of inverse Pochhammer symbols and vice versa. It is shown that the relationships involving Stirling numbers are special cases of more general orthogonal and triangular transformations. Keywords: Factorial series; Divergent asymptotic (inverse) power series; Stieltjes series; Quartic anharmonic oscillator; Stirling numbers; General orthogonal and triangular transformations; 2010 MSC: 11B73; 40A05; 40G99; 81Q15; 1. Introduction Power series are extremely important analytical tools not only in mathematics, but also in the mathematical treat- ment of scientific and engineering problems. Unfortunately, a power series representation for a given function is from a numerical point of view a mixed blessing. A power series converges within its circle of convergence and diverges outside. Circles of convergence normally have finite radii, but there are many series expansions of consider- able practical relevance, for example asymptotic expansions for special functions or quantum mechanical perturbation expansions, whose circles of convergence shrink to a single point. -
Confluent Vandermonde Matrices, Divided Differences, and Lagrange
CONFLUENT VANDERMONDE MATRICES, DIVIDED DIFFERENCES, AND LAGRANGE-HERMITE INTERPOLATION OVER QUATERNIONS VLADIMIR BOLOTNIKOV Abstract. We introduce the notion of a confluent Vandermonde matrix with quater- nion entries and discuss its connection with Lagrange-Hermite interpolation over quaternions. Further results include the formula for the rank of a confluent Van- dermonde matrix, the representation formula for divided differences of quaternion polynomials and their extensions to the formal power series setting. 1. Introduction The notion of the Vandermonde matrix arises naturally in the context of the La- grange interpolation problem when one seeks a complex polynomial taking prescribed values at given points. Confluent Vandermonde matrices come up once interpolation conditions on the derivatives of an unknown interpolant are also imposed; we refer to the survey [10] for confluent Vandermonde matrices and their applications. The study of Vandermonde matrices over division rings (with further specialization to the ring of quaternions) was initiated in [11]. In the follow-up paper [12], the Vandermonde ma- trices were studied in the setting of a division ring K endowed with an endomorphism s and an s-derivation D, along with their interactions with skew polynomials from the Ore domain K[z,s,D]. The objective of this paper is to extend the results from [11] in a different direction: to introduce a meaningful notion of a confluent Vandermonde matrix (over quaternions only, for the sake of simplicity) and to discuss its connections with Lagrange-Hermite interpolation problem for quaternion polynomials. Let H denote the skew field of quaternions α = x0+ix1+jx2+kx3 where x0,x1,x2,x3 ∈ arXiv:1505.03574v1 [math.RA] 13 May 2015 R and where i, j, k are the imaginary units commuting with R and satisfying i2 = j2 = k2 = ijk = −1. -
Polynomials, Power Series and Interpolation*
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MATHEMATlCAL ANALYSIS AND APPLICATIONS 80, 333-371 (1981) Polynomials, Power Series and Interpolation* STEVEN ROMAN Department of Mathematics, University of South Florida. Tampa, Florida 33520 Submitted bJ>Gian-Carlo Rota Contents. I. Introduction. 2. Linear f’unctionals. 3. Linear operators. 4. Associated sequences. 5. The Expansion Theorem. 6. Decentralized geometric sequences. 7. Formal power series. 1. INTRODUCTION In this paper we study the relationship between formal power series, sequences of polynomials and polynomial interpolation. We do this by extending and applying the techniques of the umbra1 calculus. Since our approach is a completely fresh one, no previous knowledge of the umbra1 calculus is required. Our main goals are threefold. Let us begin from the point of view of inter- polation. Let P be the algebra of polynomials in a single variable and let P* be the dual vector space of all linear functionals on P. If L E P* and P(X) E P, we denote the application of L on p(x) by (L I P(X)). If M,, M,, M *,... is a sequence in P* we are interested in studying the polynomials p,(x) satisfying (Mk I P,(X)) = h,,, if indeed they exist, where a,,, is the Kronecker delta function. Let us call the sequencep,(x) an associated sequencefor M,. For a linear functional L we define the order o(L) of L to be the smallest integer k such that (L 1x”) # 0. A sequence M, in P* is triangular if o(M,) = 0 and if o(MJ is a strictly increasing sequenceof integers.