An Arithmetical Theory of the Bernoulli Numbers
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An Appreciation of Euler's Formula
Rose-Hulman Undergraduate Mathematics Journal Volume 18 Issue 1 Article 17 An Appreciation of Euler's Formula Caleb Larson North Dakota State University Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Larson, Caleb (2017) "An Appreciation of Euler's Formula," Rose-Hulman Undergraduate Mathematics Journal: Vol. 18 : Iss. 1 , Article 17. Available at: https://scholar.rose-hulman.edu/rhumj/vol18/iss1/17 Rose- Hulman Undergraduate Mathematics Journal an appreciation of euler's formula Caleb Larson a Volume 18, No. 1, Spring 2017 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 [email protected] a scholar.rose-hulman.edu/rhumj North Dakota State University Rose-Hulman Undergraduate Mathematics Journal Volume 18, No. 1, Spring 2017 an appreciation of euler's formula Caleb Larson Abstract. For many mathematicians, a certain characteristic about an area of mathematics will lure him/her to study that area further. That characteristic might be an interesting conclusion, an intricate implication, or an appreciation of the impact that the area has upon mathematics. The particular area that we will be exploring is Euler's Formula, eix = cos x + i sin x, and as a result, Euler's Identity, eiπ + 1 = 0. Throughout this paper, we will develop an appreciation for Euler's Formula as it combines the seemingly unrelated exponential functions, imaginary numbers, and trigonometric functions into a single formula. To appreciate and further understand Euler's Formula, we will give attention to the individual aspects of the formula, and develop the necessary tools to prove it. -
Newton and Leibniz: the Development of Calculus Isaac Newton (1642-1727)
Newton and Leibniz: The development of calculus Isaac Newton (1642-1727) Isaac Newton was born on Christmas day in 1642, the same year that Galileo died. This coincidence seemed to be symbolic and in many ways, Newton developed both mathematics and physics from where Galileo had left off. A few months before his birth, his father died and his mother had remarried and Isaac was raised by his grandmother. His uncle recognized Newton’s mathematical abilities and suggested he enroll in Trinity College in Cambridge. Newton at Trinity College At Trinity, Newton keenly studied Euclid, Descartes, Kepler, Galileo, Viete and Wallis. He wrote later to Robert Hooke, “If I have seen farther, it is because I have stood on the shoulders of giants.” Shortly after he received his Bachelor’s degree in 1665, Cambridge University was closed due to the bubonic plague and so he went to his grandmother’s house where he dived deep into his mathematics and physics without interruption. During this time, he made four major discoveries: (a) the binomial theorem; (b) calculus ; (c) the law of universal gravitation and (d) the nature of light. The binomial theorem, as we discussed, was of course known to the Chinese, the Indians, and was re-discovered by Blaise Pascal. But Newton’s innovation is to discuss it for fractional powers. The binomial theorem Newton’s notation in many places is a bit clumsy and he would write his version of the binomial theorem as: In modern notation, the left hand side is (P+PQ)m/n and the first term on the right hand side is Pm/n and the other terms are: The binomial theorem as a Taylor series What we see here is the Taylor series expansion of the function (1+Q)m/n. -
Accessing Bernoulli-Numbers by Matrix-Operations Gottfried Helms 3'2006 Version 2.3
Accessing Bernoulli-Numbers by Matrix-Operations Gottfried Helms 3'2006 Version 2.3 Accessing Bernoulli-Numbers by Matrixoperations ........................................................................ 2 1. Introduction....................................................................................................................................................................... 2 2. A common equation of recursion (containing a significant error)..................................................................................... 3 3. Two versions B m and B p of bernoulli-numbers? ............................................................................................................... 4 4. Computation of bernoulli-numbers by matrixinversion of (P-I) ....................................................................................... 6 5. J contains the eigenvalues, and G m resp. G p contain the eigenvectors of P z resp. P s ......................................................... 7 6. The Binomial-Matrix and the Matrixexponential.............................................................................................................. 8 7. Bernoulli-vectors and the Matrixexponential.................................................................................................................... 8 8. The structure of the remaining coefficients in the matrices G m - and G p........................................................................... 9 9. The original problem of Jacob Bernoulli: "Powersums" - from G -
Mathematics (MATH)
Course Descriptions MATH 1080 QL MATH 1210 QL Mathematics (MATH) Precalculus Calculus I 5 5 MATH 100R * Prerequisite(s): Within the past two years, * Prerequisite(s): One of the following within Math Leap one of the following: MAT 1000 or MAT 1010 the past two years: (MATH 1050 or MATH 1 with a grade of B or better or an appropriate 1055) and MATH 1060, each with a grade of Is part of UVU’s math placement process; for math placement score. C or higher; OR MATH 1080 with a grade of C students who desire to review math topics Is an accelerated version of MATH 1050 or higher; OR appropriate placement by math in order to improve placement level before and MATH 1060. Includes functions and placement test. beginning a math course. Addresses unique their graphs including polynomial, rational, Covers limits, continuity, differentiation, strengths and weaknesses of students, by exponential, logarithmic, trigonometric, and applications of differentiation, integration, providing group problem solving activities along inverse trigonometric functions. Covers and applications of integration, including with an individual assessment and study inequalities, systems of linear and derivatives and integrals of polynomial plan for mastering target material. Requires nonlinear equations, matrices, determinants, functions, rational functions, exponential mandatory class attendance and a minimum arithmetic and geometric sequences, the functions, logarithmic functions, trigonometric number of hours per week logged into a Binomial Theorem, the unit circle, right functions, inverse trigonometric functions, and preparation module, with progress monitored triangle trigonometry, trigonometric equations, hyperbolic functions. Is a prerequisite for by a mentor. May be repeated for a maximum trigonometric identities, the Law of Sines, the calculus-based sciences. -
The Discovery of the Series Formula for Π by Leibniz, Gregory and Nilakantha Author(S): Ranjan Roy Source: Mathematics Magazine, Vol
The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha Author(s): Ranjan Roy Source: Mathematics Magazine, Vol. 63, No. 5 (Dec., 1990), pp. 291-306 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2690896 Accessed: 27-02-2017 22:02 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to Mathematics Magazine This content downloaded from 195.251.161.31 on Mon, 27 Feb 2017 22:02:42 UTC All use subject to http://about.jstor.org/terms ARTICLES The Discovery of the Series Formula for 7r by Leibniz, Gregory and Nilakantha RANJAN ROY Beloit College Beloit, WI 53511 1. Introduction The formula for -r mentioned in the title of this article is 4 3 57 . (1) One simple and well-known moderm proof goes as follows: x I arctan x = | 1 +2 dt x3 +5 - +2n + 1 x t2n+2 + -w3 - +(-I)rl2n+1 +(-I)l?lf dt. The last integral tends to zero if Ix < 1, for 'o t+2dt < jt dt - iX2n+3 20 as n oo. -
1. More Examples Example 1. Find the Slope of the Tangent Line at (3,9)
1. More examples Example 1. Find the slope of the tangent line at (3; 9) to the curve y = x2. P = (3; 9). So Q = (3 + ∆x; 9 + ∆y). 9 + ∆y = (3 + ∆x)2 = (9 + 6∆x + (∆x)2. Thus, we have ∆y = 6 + ∆x. ∆x If we let ∆ go to zero, we get ∆y lim = 6 + 0 = 6. ∆x→0 ∆x Thus, the slope of the tangent line at (3; 9) is 6. Let's consider a more abstract example. Example 2. Find the slope of the tangent line at an arbitrary point P = (x; y) on the curve y = x2. Again, P = (x; y), and Q = (x + ∆x; y + ∆y), where y + ∆y = (x + ∆x)2 = x2 + 2x∆x + (∆x)2. So we get, ∆y = 2x + ∆x. ∆x Letting ∆x go to zero, we get ∆y lim = 2x + 0 = 2x. ∆x→0 ∆x Thus, the slope of the tangent line at an arbitrary point (x; y) is 2x. Example 3. Find the slope of the tangent line at an arbitrary point P = (x; y) on the curve y = ax3, where a is a real number. Q = (x + ∆x; y + ∆y). We get, 1 2 y + ∆y = a(x + ∆x)3 = a(x3 + 3x2∆x + 3x(∆x)2 + (∆x)3). After simplifying we have, ∆y = 3ax2 + 3xa∆x + a(∆x)2. ∆x Letting ∆x go to 0 we conclude, ∆y 2 2 mP = lim = 3ax + 0 + 0 = 3ax . x→0 ∆x 2 Thus, the slope of the tangent line at a point (x; y) is mP = 3ax . Example 4. -
Leonhard Euler: His Life, the Man, and His Works∗
SIAM REVIEW c 2008 Walter Gautschi Vol. 50, No. 1, pp. 3–33 Leonhard Euler: His Life, the Man, and His Works∗ Walter Gautschi† Abstract. On the occasion of the 300th anniversary (on April 15, 2007) of Euler’s birth, an attempt is made to bring Euler’s genius to the attention of a broad segment of the educated public. The three stations of his life—Basel, St. Petersburg, andBerlin—are sketchedandthe principal works identified in more or less chronological order. To convey a flavor of his work andits impact on modernscience, a few of Euler’s memorable contributions are selected anddiscussedinmore detail. Remarks on Euler’s personality, intellect, andcraftsmanship roundout the presentation. Key words. LeonhardEuler, sketch of Euler’s life, works, andpersonality AMS subject classification. 01A50 DOI. 10.1137/070702710 Seh ich die Werke der Meister an, So sehe ich, was sie getan; Betracht ich meine Siebensachen, Seh ich, was ich h¨att sollen machen. –Goethe, Weimar 1814/1815 1. Introduction. It is a virtually impossible task to do justice, in a short span of time and space, to the great genius of Leonhard Euler. All we can do, in this lecture, is to bring across some glimpses of Euler’s incredibly voluminous and diverse work, which today fills 74 massive volumes of the Opera omnia (with two more to come). Nine additional volumes of correspondence are planned and have already appeared in part, and about seven volumes of notebooks and diaries still await editing! We begin in section 2 with a brief outline of Euler’s life, going through the three stations of his life: Basel, St. -
Sums of Powers and the Bernoulli Numbers Laura Elizabeth S
Eastern Illinois University The Keep Masters Theses Student Theses & Publications 1996 Sums of Powers and the Bernoulli Numbers Laura Elizabeth S. Coen Eastern Illinois University This research is a product of the graduate program in Mathematics and Computer Science at Eastern Illinois University. Find out more about the program. Recommended Citation Coen, Laura Elizabeth S., "Sums of Powers and the Bernoulli Numbers" (1996). Masters Theses. 1896. https://thekeep.eiu.edu/theses/1896 This is brought to you for free and open access by the Student Theses & Publications at The Keep. It has been accepted for inclusion in Masters Theses by an authorized administrator of The Keep. For more information, please contact [email protected]. THESIS REPRODUCTION CERTIFICATE TO: Graduate Degree Candidates (who have written formal theses) SUBJECT: Permission to Reproduce Theses The University Library is rece1v1ng a number of requests from other institutions asking permission to reproduce dissertations for inclusion in their library holdings. Although no copyright laws are involved, we feel that professional courtesy demands that permission be obtained from the author before we allow theses to be copied. PLEASE SIGN ONE OF THE FOLLOWING STATEMENTS: Booth Library of Eastern Illinois University has my permission to lend my thesis to a reputable college or university for the purpose of copying it for inclusion in that institution's library or research holdings. u Author uate I respectfully request Booth Library of Eastern Illinois University not allow my thesis -
The Structure of Bernoulli Numbers
The structure of Bernoulli numbers Bernd C. Kellner Abstract We conjecture that the structure of Bernoulli numbers can be explicitly given in the closed form n 2 −1 −1 n−l −1 −1 Bn = (−1) |n|p |p (χ(p,l) − p−1 )|p p p,l ∈ irr p−1∤n ( ) Ψ1 p−1|n Y n≡l (modY p−1) Y irr where the χ(p,l) are zeros of certain p-adic zeta functions and Ψ1 is the set of irregular pairs. The more complicated but improbable case where the conjecture does not hold is also handled; we obtain an unconditional structural formula for Bernoulli numbers. Finally, applications are given which are related to classical results. Keywords: Bernoulli number, Kummer congruences, irregular prime, irregular pair of higher order, Riemann zeta function, p-adic zeta function Mathematics Subject Classification 2000: 11B68 1 Introduction The classical Bernoulli numbers Bn are defined by the power series ∞ z zn = B , |z| < 2π , ez − 1 n n! n=0 X where all numbers Bn are zero with odd index n > 1. The even-indexed rational 1 1 numbers Bn alternate in sign. First values are given by B0 = 1, B1 = − 2 , B2 = 6 , 1 arXiv:math/0411498v1 [math.NT] 22 Nov 2004 B4 = − 30 . Although the first numbers are small with |Bn| < 1 for n = 2, 4,..., 12, these numbers grow very rapidly with |Bn|→∞ for even n →∞. For now, let n be an even positive integer. An elementary property of Bernoulli numbers is the following discovered independently by T. Clausen [Cla40] and K. -
Tight Bounds on the Mutual Coherence of Sensing Matrices for Wigner D-Functions on Regular Grids
Tight bounds on the mutual coherence of sensing matrices for Wigner D-functions on regular grids Arya Bangun, Arash Behboodi, and Rudolf Mathar,∗ October 7, 2020 Abstract Many practical sampling patterns for function approximation on the rotation group utilizes regu- lar samples on the parameter axes. In this paper, we relate the mutual coherence analysis for sensing matrices that correspond to a class of regular patterns to angular momentum analysis in quantum mechanics and provide simple lower bounds for it. The products of Wigner d-functions, which ap- pear in coherence analysis, arise in angular momentum analysis in quantum mechanics. We first represent the product as a linear combination of a single Wigner d-function and angular momentum coefficients, otherwise known as the Wigner 3j symbols. Using combinatorial identities, we show that under certain conditions on the bandwidth and number of samples, the inner product of the columns of the sensing matrix at zero orders, which is equal to the inner product of two Legendre polynomials, dominates the mutual coherence term and fixes a lower bound for it. In other words, for a class of regular sampling patterns, we provide a lower bound for the inner product of the columns of the sensing matrix that can be analytically computed. We verify numerically our theoretical results and show that the lower bound for the mutual coherence is larger than Welch bound. Besides, we provide algorithms that can achieve the lower bound for spherical harmonics. 1 Introduction In many applications, the goal is to recover a function defined on a group, say on the sphere S2 and the rotation group SO(3), from only a few samples [1{5]. -
Higher Order Bernoulli and Euler Numbers
David Vella, Skidmore College [email protected] Generating Functions and Exponential Generating Functions • Given a sequence {푎푛} we can associate to it two functions determined by power series: • Its (ordinary) generating function is ∞ 풏 푓 풙 = 풂풏풙 풏=ퟏ • Its exponential generating function is ∞ 풂 품 풙 = 풏 풙풏 풏! 풏=ퟏ Examples • The o.g.f and the e.g.f of {1,1,1,1,...} are: 1 • f(x) = 1 + 푥 + 푥2 + 푥3 + ⋯ = , and 1−푥 푥 푥2 푥3 • g(x) = 1 + + + + ⋯ = 푒푥, respectively. 1! 2! 3! The second one explains the name... Operations on the functions correspond to manipulations on the sequence. For example, adding two sequences corresponds to adding the ogf’s, while to shift the index of a sequence, we multiply the ogf by x, or differentiate the egf. Thus, the functions provide a convenient way of studying the sequences. Here are a few more famous examples: Bernoulli & Euler Numbers • The Bernoulli Numbers Bn are defined by the following egf: x Bn n x x e 1 n1 n! • The Euler Numbers En are defined by the following egf: x 2e En n Sech(x) 2x x e 1 n0 n! Catalan and Bell Numbers • The Catalan Numbers Cn are known to have the ogf: ∞ 푛 1 − 1 − 4푥 2 퐶 푥 = 퐶푛푥 = = 2푥 1 + 1 − 4푥 푛=1 • Let Sn denote the number of different ways of partitioning a set with n elements into nonempty subsets. It is called a Bell number. It is known to have the egf: ∞ 푆 푥 푛 푥푛 = 푒 푒 −1 푛! 푛=1 Higher Order Bernoulli and Euler Numbers th w • The n Bernoulli Number of order w, B n is defined for positive integer w by: w x B w n xn x e 1 n1 n! th w • The n Euler Number of -
Approximation Properties of G-Bernoulli Polynomials
Approximation properties of q-Bernoulli polynomials M. Momenzadeh and I. Y. Kakangi Near East University Lefkosa, TRNC, Mersiin 10, Turkey Email: [email protected] [email protected] July 10, 2018 Abstract We study the q−analogue of Euler-Maclaurin formula and by introducing a new q-operator we drive to this form. Moreover, approximation properties of q-Bernoulli polynomials is discussed. We estimate the suitable functions as a combination of truncated series of q-Bernoulli polynomials and the error is calculated. This paper can be helpful in two different branches, first solving differential equations by estimating functions and second we may apply these techniques for operator theory. 1 Introduction The present study has sough to investigate the approximation of suitable function f(x) as a linear com- bination of q−Bernoulli polynomials. This study by using q−operators through a parallel way has achived to the kind of Euler expansion for f(x), and expanded the function in terms of q−Bernoulli polynomials. This expansion offers a proper tool to solve q-difference equations or normal differential equation as well. There are many approaches to approximate the capable functions. According to properties of q-functions many q-function have been used in order to approximate a suitable function.For example, at [28], some identities and formulae for the q-Bernstein basis function, including the partition of unity property, formulae for representing the monomials were studied. In addition, a kind of approximation of a function in terms of Bernoulli polynomials are used in several approaches for solving differential equations, such as [7, 20, 21].