DOCSLIB.ORG

Computing Transcendental Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Computing Transcendental Functions Computing Transcendental Functions Iris Oved [email protected] 29 April 1999 Exponential, logarithmic, and trigonometric functions are transcendental. They are peculiar in that their values cannot be computed in terms of finite compositions of arithmetic and/or algebraic operations. So how might one go about computing such functions? There are several approaches to this problem, three of which will be outlined below. We will begin with Taylor approxima- tions, then take a look at a method involving Minimax Polynomials, and we will conclude with a brief consideration of CORDIC approximations. While the discussion will specifically cover methods for calculating the sine function, the procedures can be generalized for the calculation of other transcendental functions as well. 1 Taylor Approximations One method for computing the sine function is to find a polynomial that allows us to approximate the sine locally. A polynomial is a good approximation to the sine function near some point, x = a, if the derivatives of the polynomial at the point are equal to the derivatives of the sine curve at that point. The more derivatives the polynomial has in common with the sine function at the point, the better the approximation. Thus the higher the degree of the polynomial, the better the approximation. A polynomial that is found by this method is called a Taylor Polynomial. Suppose we are looking for the Taylor Polynomial of degree 1 approximating the sine function near x = 0. We will call the polynomial p(x) and the sine function f(x). We want to find a linear function p(x) such that p(0) = f(0) and p0(0) = f 0(0). Since we are looking for a polynomial of degree 1, we know that it will be of the form p(x) = Bx + A: Now we set p(0) = f(0): Since we know that f(0) = 0, we have p(0) = 0: 1 Substituting x = 0 into p(x) gives p(0) = 0B + A: Thus, in our polynomial, we know the coefficient A = 0. To find the value of the coefficient B, we take the derivative of our linear function and we have: p0(x) = B: Now we set p0(0) = f 0(0): Since we know that the derivative of the sine function is the cosine function, we know that f 0(0) = cos(0) = 1. So we have p0(0) = 1: Substituting x = 0 into p0(x), we get p0(0) = B: Thus we have the coefficient B = 1. Now that we have both coefficients, we arrive at the Taylor Polynomial of degree 1 that approximates the sine curve near x = 0: p(x) = x: Notice that this is the line tangent to the sine function at x = 0.1 1Indeed, the degree 1 Taylor Polynomial of any function is necessarily the line tangent to the function at the point about which the above procedures are performed. 2 4 3 2 -6 -4 -2 2 4 6 -2 -4 Figure 1.1. Sine curve and degree 1 Taylor with −2π ≤ x ≤ 2π We can see in Figure 1.1 that the degree 1 Taylor Polynomial is a good approximation to the sine curve near x = 0, but clearly it gets worse as we move farther away from that point. Higher degree polynomials have a higher number of meaningful derivatives (derivatives not equal to zero). Higher degree Taylor Polynomials thus have more derivatives in common with the function that is being approximated. This means that the higher the degree of the Taylor Polynomial, the more accurately it approximates the sine function. To demonstrate this, let us find the Taylor Polynomial of degree 3, for the sine curve, centered at x = 0. Since we are looking for a polynomial of degree 3, we know that it will be of the form p(x) = Dx3 + Cx2 + Bx + A: As before, we set p(0) = f(0); and conclude that in our polynomial we have the coefficient A = 0. Next, to find the coefficient B, we take the first derivative of our polynomial and get p0(x) = 3Dx2 + 2Cx + B: Now we set p0(0) = f 0(0); and we have the coefficient B = 1. Now, to find the coefficient C, we take the second derivative of our polyno- mial and get p00(x) = 6Dx + 2C: Next we set p00(0) = f 00(0): Since we know that the derivative of the cosine function is the negative of the sine function, we know that f 00(0) = − sin(0) = 0. So we have p00(0) = 0: Substituting x = 0 into p00(x), we get p00(0) = 2C: Thus we have the coefficient C = 0. Finally, to find the coefficient D, we take the third derivative of our polyno- mial and get p000(x) = 6D: Now we set p000(0) = f 000(0): 4 Since we know that the derivative of the negative of the sine function is the negative of the cosine function, we know that f 000(0) = − cos(0) = −1. So we have p000(0) = −1: Substituting x = 0 into p000(x), we get p000(0) = 6D: Thus we have the coefficient D = −1=6. By this method, we have discovered that the degree 3 Taylor Polynomial that approximates the sine function near x = 0 is 1 3 2 p(x) = − 6 x + 0x + x + 0: 5 4 6 2 -6 -4 -2 2 4 6 -2 -4 Figure 1.2. Sine curve and degree 3 Taylor with −2π ≤ x ≤ 2π Although the degree 3 Taylor Polynomial is a better approximation than the degree 1 Taylor Polynomial, it still becomes very inaccurate as we move farther away from x = 0. The general formula for the Taylor Polynomial of degree n approximating some function f(x) for x near 0 is f 00(0) f 000(0) f (n)(0) f(0) + f 0(0)x + x2 + x3 + ··· + xn: 2! 3! n! So suppose we want to find the Taylor polynomial of degree 9 that approxi- mates the sine function near x = 0.2 Substituting f(x) = sin(x) and n = 9 into the general formula gives x3 x5 x7 x9 x − + − + : 3! 5! 7! 9! 2Since sin(−x) = − sin(x), there are no interesting even-degree Taylor Polynomials at x = 0. 7 4 8 2 -6 -4 -2 2 4 6 -2 -4 Figure 1.3. Sine curve and degree 9 Taylor Polynomial with −2π ≤ x ≤ 2π Figure 1.3 shows that the Taylor Polynomial of degree 9 is a better approxi- mation to the sine curve than the linear and cubic polynomials. Still, even this high degree polynomial provides approximations that become worse as we move farther away from x = 0. We can arrive at better and better approximations to the sine curve if we find Taylor Polynomials of higher and higher degree. However, no matter how large the degree of the polynomial, since the sine curve is periodic, while polynomials are not, there will always be a point at which the graph of the polynomial will deviate significantly from the sine curve in some direction. This is to be expected because the sine function is transcendental. No finite composition of arithmetic and/or algebraic operations will capture the behavior of a transcendental function. One solution to this problem is to limit the sine curve to a domain of x- values from which we can infer the sine of any real number. Since the sine function has a period of 2π, it is clear that we can infer the sine of any real number if we can calculate the sine of any real x in the domain 0 ≤ x ≤ 2π. In this domain, the sine function has a finite number of turns, so a high degree polynomial would be an adequate approximation to the curve for all real x in the domain. However, we would benefit by further reducing the domain of inputs so that each of the inputs will be even closer to x = 0. This is important because Taylor Polynomials are good approximations near a point, but not so good farther away from that point. 9 1 10 0.5 1 2 3 4 5 6 -0.5 -1 Figure 1.4. Sine curve with 0 ≤ x ≤ 2π If we look at the sine curve in the domain 0 ≤ x ≤ 2π, we can see that the domain can be further limited. Notice that all of the sine-values at inputs between π and 2π are negations of values between 0 and π. As long as we make a note to negate our solution, we can infer the sine of all of the x-values in the domain π ≤ x ≤ 2π, from the sine of the x-values in the domain 0 ≤ x ≤ π. A polynomial of high degree that resembles the sine curve in this domain will provide information from which we can infer a reasonable appproximation to the sine of any real x. Still, there is some symmetry in the sine curve that we can π exploit to shrink the domain further. The sine of the x-values between 2 and π π can be inferred from the sine of the x-values between 0 and 2 by reflecting π across the line x = 2 . Now the Taylor Polynomial only needs to be a good π approximation to the sine curve over the limited domain 0 ≤ x ≤ 2 in order for us to compute the sine function for any real input. 11 1 12 0.8 0.6 0.4 0.2 0.25 0.5 0.75 1 1.25 1.5 π Figure 1.5.
Load more

Recommended publications
  • Some Transcendental Functions with an Empty Exceptional Set 3
    KYUNGPOOK Math. J. 00(0000), 000-000 Some transcendental functions with an empty exceptional set F. M. S. Lima Institute of Physics, University of Brasilia, Brasilia, DF, Brazil e-mail : [email protected] Diego Marques∗ Department of Mathematics, University de Brasilia, Brasilia, DF, Brazil e-mail : [email protected] Abstract. A transcendental function usually returns transcendental values at algebraic points. The (algebraic) exceptions form the so-called exceptional set, as for instance the unitary set {0} for the function f(z)= ez , according to the Hermite-Lindemann theorem. In this note, we give some explicit examples of transcendental entire functions whose exceptional set are empty. 1 Introduction An algebraic function is a function f(x) which satisfies P (x, f(x)) = 0 for some nonzero polynomial P (x, y) with complex coefficients. Functions that can be con- structed using only a finite number of elementary operations are examples of alge- braic functions. A function which is not algebraic is, by definition, a transcendental function — e.g., basic trigonometric functions, exponential function, their inverses, etc. If f is an entire function, namely a function which is analytic in C, to say that f is a transcendental function amounts to say that it is not a polynomial. By evaluating a transcendental function at an algebraic point of its domain, one usually finds a transcendental number, but exceptions can take place. For a given transcendental function, the set of all exceptions (i.e., all algebraic numbers of the arXiv:1004.1668v2 [math.NT] 25 Aug 2012 function domain whose image is an algebraic value) form the so-called exceptional set (denoted by Sf ).
    [Show full text]
  • A Computational Approach to Solve a System of Transcendental Equations with Multi-Functions and Multi-Variables
    mathematics Article A Computational Approach to Solve a System of Transcendental Equations with Multi-Functions and Multi-Variables Chukwuma Ogbonnaya 1,2,* , Chamil Abeykoon 3 , Adel Nasser 1 and Ali Turan 4 1 Department of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Manchester M13 9PL, UK; [email protected] 2 Faculty of Engineering and Technology, Alex Ekwueme Federal University, Ndufu Alike Ikwo, Abakaliki PMB 1010, Nigeria 3 Aerospace Research Institute and Northwest Composites Centre, School of Materials, The University of Manchester, Manchester M13 9PL, UK; [email protected] 4 Independent Researcher, Manchester M22 4ES, Lancashire, UK; [email protected] * Correspondence: [email protected]; Tel.: +44-(0)74-3850-3799 Abstract: A system of transcendental equations (SoTE) is a set of simultaneous equations containing at least a transcendental function. Solutions involving transcendental equations are often problematic, particularly in the form of a system of equations. This challenge has limited the number of equations, with inter-related multi-functions and multi-variables, often included in the mathematical modelling of physical systems during problem formulation. Here, we presented detailed steps for using a code- based modelling approach for solving SoTEs that may be encountered in science and engineering problems. A SoTE comprising six functions, including Sine-Gordon wave functions, was used to illustrate the steps. Parametric studies were performed to visualize how a change in the variables Citation: Ogbonnaya, C.; Abeykoon, affected the superposition of the waves as the independent variable varies from x1 = 1:0.0005:100 to C.; Nasser, A.; Turan, A.
    [Show full text]
  • Some Results on the Arithmetic Behavior of Transcendental Functions
    Algebra: celebrating Paulo Ribenboim’s ninetieth birthday SOME RESULTS ON THE ARITHMETIC BEHAVIOR OF TRANSCENDENTAL FUNCTIONS Diego Marques University of Brasilia October 25, 2018 After I found the famous Ribenboim’s book (Chapter 10: What kind of p p 2 number is 2 ?): Algebra: celebrating Paulo Ribenboim’s ninetieth birthday My first transcendental steps In 2005, in an undergraduate course of Abstract Algebra, the professor (G.Gurgel) defined transcendental numbers and asked about the algebraic independence of e and π. Algebra: celebrating Paulo Ribenboim’s ninetieth birthday My first transcendental steps In 2005, in an undergraduate course of Abstract Algebra, the professor (G.Gurgel) defined transcendental numbers and asked about the algebraic independence of e and π. After I found the famous Ribenboim’s book (Chapter 10: What kind of p p 2 number is 2 ?): In 1976, Mahler wrote a book entitled "Lectures on Transcendental Numbers". In its chapter 3, he left three problems, called A,B, and C. The goal of this lecture is to talk about these problems... Algebra: celebrating Paulo Ribenboim’s ninetieth birthday Kurt Mahler Kurt Mahler (Germany, 1903-Australia, 1988) Mahler’s works focus in transcendental number theory, Diophantine approximation, Diophantine equations, etc In its chapter 3, he left three problems, called A,B, and C. The goal of this lecture is to talk about these problems... Algebra: celebrating Paulo Ribenboim’s ninetieth birthday Kurt Mahler Kurt Mahler (Germany, 1903-Australia, 1988) Mahler’s works focus in transcendental number theory, Diophantine approximation, Diophantine equations, etc In 1976, Mahler wrote a book entitled "Lectures on Transcendental Numbers".
    [Show full text]
  • Original Research Article Hypergeometric Functions On
    Original Research Article Hypergeometric Functions on Cumulative Distribution Function ABSTRACT Exponential functions have been extended to Hypergeometric functions. There are many functions which can be expressed in hypergeometric function by using its analytic properties. In this paper, we will apply a unified approach to the probability density function and corresponding cumulative distribution function of the noncentral chi square variate to extract and derive hypergeometric functions. Key words: Generalized hypergeometric functions; Cumulative distribution theory; chi-square Distribution on Non-centrality Parameter. I) INTRODUCTION Higher-order transcendental functions are generalized from hypergeometric functions. Hypergeometric functions are special function which represents a series whose coefficients satisfy many recursion properties. These functions are applied in different subjects and ubiquitous in mathematical physics and also in computers as Maple and Mathematica. They can also give explicit solutions to problems in economics having dynamic aspects. The purpose of this paper is to understand the importance of hypergeometric function in different fields and initiating economists to the large class of hypergeometric functions which are generalized from transcendental function. The paper is organized in following way. In Section II, the generalized hypergeometric series is defined with some of its analytical properties and special cases. In Sections 3 and 4, hypergeometric function and Kummer’s confluent hypergeometric function are discussed in detail which will be directly used in our results. In Section 5, the main result is proved where we derive the exact cumulative distribution function of the noncentral chi sqaure variate. An appendix is attached which summarizes notational abbreviations and function names. The paper is introductory in nature presenting results reduced from general formulae.
    [Show full text]
  • Pre-Calculus-Honors-Accelerated
    PUBLIC SCHOOLS OF EDISON TOWNSHIP OFFICE OF CURRICULUM AND INSTRUCTION Pre-Calculus Honors/Accelerated/Academic Length of Course: Term Elective/Required: Required Schools: High School Eligibility: Grade 10-12 Credit Value: 5 Credits Date Approved: September 23, 2019 Pre-Calculus 2 TABLE OF CONTENTS Statement of Purpose 3 Course Objectives 4 Suggested Timeline 5 Unit 1: Functions from a Pre-Calculus Perspective 11 Unit 2: Power, Polynomial, and Rational Functions 16 Unit 3: Exponential and Logarithmic Functions 20 Unit 4: Trigonometric Function 23 Unit 5: Trigonometric Identities and Equations 28 Unit 6: Systems of Equations and Matrices 32 Unit 7: Conic Sections and Parametric Equations 34 Unit 8: Vectors 38 Unit 9: Polar Coordinates and Complex Numbers 41 Unit 10: Sequences and Series 44 Unit 11: Inferential Statistics 47 Unit 12: Limits and Derivatives 51 Pre-Calculus 3 Statement of Purpose Pre-Calculus courses combine the study of Trigonometry, Elementary Functions, Analytic Geometry, and Math Analysis topics as preparation for calculus. Topics typically include the study of complex numbers; polynomial, logarithmic, exponential, rational, right trigonometric, and circular functions, and their relations, inverses and graphs; trigonometric identities and equations; solutions of right and oblique triangles; vectors; the polar coordinate system; conic sections; Boolean algebra and symbolic logic; mathematical induction; matrix algebra; sequences and series; and limits and continuity. This course includes a review of essential skills from algebra, introduces polynomial, rational, exponential and logarithmic functions, and gives the student an in-depth study of trigonometric functions and their applications. Modern technology provides tools for supplementing the traditional focus on algebraic procedures, such as solving equations, with a more visual perspective, with graphs of equations displayed on a screen.
    [Show full text]
  • Algebraic Values of Analytic Functions Michel Waldschmidt
    Algebraic values of analytic functions Michel Waldschmidt To cite this version: Michel Waldschmidt. Algebraic values of analytic functions. International Conference on Special Functions and their Applications (ICSF02), Sep 2002, Chennai, India. pp.323-333. hal-00411427 HAL Id: hal-00411427 https://hal.archives-ouvertes.fr/hal-00411427 Submitted on 27 Aug 2009 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Algebraic values of analytic functions Michel Waldschmidt Institut de Math´ematiques de Jussieu, Universit´eP. et M. Curie (Paris VI), Th´eorie des Nombres Case 247, 175 rue du Chevaleret 75013 PARIS [email protected] http://www.institut.math.jussieu.fr/∼miw/ Abstract Given an analytic function of one complex variable f, we investigate the arithmetic nature of the values of f at algebraic points. A typical question is whether f(α) is a transcendental number for each algebraic number α. Since there exist transcendental (t) entire functions f such that f (α) ∈ Q[α] for any t ≥ 0 and any algebraic number α, one needs to restrict the situation by adding hypotheses, either on the functions, or on the points, or else on the set of values.
    [Show full text]
  • Transcendence of Values at Algebraic Points for Certain Higher Order Hypergeometric Functions (Updated)
    TRANSCENDENCE OF VALUES AT ALGEBRAIC POINTS FOR CERTAIN HIGHER ORDER HYPERGEOMETRIC FUNCTIONS (UPDATED) PIERRE-ANTOINE DESROUSSEAUX, MARVIN D. TRETKOFF, AND PAULA TRETKOFF Abstract. In this paper, we describe the exceptional set of a function studied by Picard. This is the set of algebraic points at which the function takes algebraic values. In particular, we deduce necessary and su±cient conditions for the ¯niteness of the exceptional set. Some of our results depend on recent conjectures of Pink [21],[22]. 1. Introduction The purpose of this paper is to analyze the exceptional set of a function studied by Picard in the course of his investigation of Appell's hypergeomet- ric function. In particular, we describe the set of algebraic points at which this function assumes algebraic values. Necessary and su±cient conditions for this set to be ¯nite are also derived. Some background for this type of problem is provided in the present section. One of the recurrent themes in the theory of transcendental numbers is the problem of determining the set of algebraic numbers at which a given transcendental function assumes algebraic values. This set has come to be known as the exceptional set of the function. The classical work of Hermite (1873), Lindemann (1882) and Weierstrass (1885) established that the exceptional set associated to the exponential function exp(x), x 2 C, is trivial, that is, consists only of x = 0. These results stand among the highlights of 19th century mathematics because they imply that ¼ is a transcendental number, thus proving the impossibility of squaring the circle, a problem dating to the ancient Greeks.
    [Show full text]
  • Transcendental Functions with a Complex Twist
    TRANSCENDENTAL FUNCTIONS WITH A COMPLEX TWIST Michael Warren Department of Mathematics Tarleton State University Box T-0470 Stephenville, TX 76402 [email protected] Dr. John Gresham Department of Mathematics Tarleton State University Box T-0470 Stephenville, TX 76402 [email protected] Dr. Bryant Wyatt Department of Mathematics Tarleton State University Box T-0470 Stephenville, TX 76402 [email protected] Abstract In our previous paper, Real Polynomials with a Complex Twist [see http://archives.math.utk.edu/ICTCM/VOL28/A040/paper.pdf], we used advancements in computer graphics that allow us to easily illustrate more complete graphs of polynomial functions that are still accessible to students of many different levels. In this paper we examine the 3D graphical representations of selected transcendental functions over subsets of the complex plane for which the functions are real-valued. We visualize and find connections between circular trigonometric functions and hyperbolic functions. Introduction In Warren, Gresham & Wyatt (2016), the authors established that students could benefit from a three-dimensional visualization of polynomial functions with real coefficients whose domain contains all complex inputs that produce real-valued outputs. Connections between the symbolic and graphical representations can stimulate student comprehension of functions (Lipp, 1994; Presmeg, 2014; Rittle-Johnson & Star, 2009). Students need to understand the concepts of domain, range, and the graphical representation of functions in order to be prepared for advanced mathematical study (Martinez-Planell & Gaisman 2012). The process of domain restriction involved in creating three-dimensional graphical representations of commonly studied functions provides a context for students to explore all these ideas. When working with polynomial functions, instructors often introduce the existence of complex roots via the Fundamental Theorem of Algebra followed by symbolic methods for finding those roots.
    [Show full text]
  • Hypergeometric Functions: Application in Distribution Theory
    International Journal of Mathematics Trends and Technology (IJMTT) – Volume 40 Number 2- December2016 Hypergeometric Functions: Application in Distribution Theory Dr. Pooja Singh, Maharaja Surajmal Institute of Technology (Affiliated to GGSIPU), Delhi ABSTRACT: Hypergeometric functions are generalized from exponential functions. There are functions which can also be evaluated analytically and expressed in form of hypergeometric function. In this paper, a unified approach to hypergeometric functions is given to derive the probability density function and corresponding cumulative distribution function of the noncentral F variate. Key words and phrases: Hypergeometric functions; distribution theory; chi-square Distribution, Non- centrality Parameter. I) Introduction extensively in Cox and Hinkley(1974), Craig(1936), Feller(1971), Hardle and The hypergeometric function is a special function Linton(1994), Muellbauer(1983). Further encountered in a variety of application. Higher-order applications in statistics/econometrics and transcendental functions are generalized from economic theory a r e suggested throughout. hypergeometric functions. These functions are Here, Hypergeometric function is applied applied in different subjects like theoretical physics, effectively to Distribution Theory. and also functional in computers as Maple and Mathematica. They are also broadly explored in the II) The generalized Hypergeometric Series area of statistical/econometric distribution theory. Before introducing the hypergeometric function, we The purpose of this paper is to understand define the Pochhammer’s symbol importance of collection of hypergeometric function in different fields especially initiating economists to the wide class of hypergeometric functions. The paper is organized as follows. In Section II, the generalized hypergeometric series is explained with some of its properties. In Sections 3 and 4, some famous special cases are discussed which will be used in our important results.
    [Show full text]
  • Differential Algebraic Equations from Definability
    Differential algebraic equations from definability Thomas Scanlon 24 October 2014 Thomas Scanlon ADEs from definability Is the logarithm a function? × The exponential function exp : C ! C has a many-valued analytic × inverse log : C ! C where log is well-defined only up to the adding an element of 2πiZ. Thomas Scanlon ADEs from definability The logarithmic derivative Treating exp and log as functions on functions does not help: If U × is some connected Riemann surface and f : U ! C is analytic, then we deduce a “function” log(f ): U ! C. However, because log(f ) is well-defined up to an additive constant, d @ log(f ) := dz (log(f )) is a well defined function. That is, for M = M (U) the differential field of meromorphic functions on U we have a well-defined differential-analytic function @ log : M× ! M. f 0 Of course, one computes that @ log(f ) = f is, in fact, differential algebraic. Thomas Scanlon ADEs from definability Explanation? Why is the logarithmic differential algebraic? What a silly question! The logarithm is the very first transcendental function whose derivative is computed in a standard calculus course. The differential algebraicity of d dz (log(f )) is merely a consequence of an elementary calculation. The usual logarithmic derivative is an instance of Kolchin’s general theory of algebraic logarithmic derivatives on algebraic groups in which the differential algebraicity is explained by the triviality of the tangent bundle of an algebraic group. It can also be seen as an instance of the main theorem to be discussed today: certain kinds of differential analytic functions constructed from covering maps are automatically differential algebraic due to two key ideas from logic: elimination of imaginaries and the Peterzil-Starchenko theory of o-minimal complex analysis.
    [Show full text]
  • Transcendental Numbers
    Examples of usage of transcendental numbers 1] Transcendental numbers transcend human experience “… Transcendental Numbers are in a word, profound. For they excel, surpass and transcend human experience . They are synonyms of peerless, incomparable, unequaled, matchless, unrivaled, unparalleled, unique, consummate, paramount, superior, surpassing, supreme, preeminent, sublime, excelling, superb, magnificent, marvelous, and... well, transcendental. Mathematically, they are, by definition, “not capable of being produced by the algebraic operations of addition, multiplication, and involution, or any of the inverse operations.”[1] Philosophically, they are “existing apart from, not subject to the limitations of, the material universe”, not to mention: “ 1 transcendent 2a (in Kantian philosophy) presupposed in and necessary to experience; a priori. b (in Schelling’s philosophy) explaining matter and objective things as products of the subjective mind. c (esp. in Emerson’s philosophy) regarding the divine as the guiding principle in man. 3 a visionary; abstract b vague; obscure.” That should about cover it” http://www.halexandria.org/dward089.htm 2] Transcendental numbers are versatile “… I am in love with the mysterious transcendental numbers. Did you know that there are "more" transcendental numbers than the more familiar algebraic ones? Even so, only a few classes of transcendental numbers are known to humans , and it's very difficult to prove that a particular number is transcendental. In 1844, math genius Joseph Liouville (1809-1882) was the first to prove the existence of transcendental numbers. (More precisely, he was the first to prove that a specific number was transcendental.) Hermite proved that the number e was transcendental in 1873. Lindeman proved that pi was transcendental in 1882.
    [Show full text]
  • Auxiliary Functions in Transcendence Proofs Michel Waldschmidt
    Auxiliary functions in transcendence proofs Michel Waldschmidt To cite this version: Michel Waldschmidt. Auxiliary functions in transcendence proofs. 2009. hal-00405120 HAL Id: hal-00405120 https://hal.archives-ouvertes.fr/hal-00405120 Preprint submitted on 24 Jul 2009 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. SASTRA Ramanujan Lectures The Ramanujan Journal Updated: March 13, 2009 Auxiliary functions in transcendental number theory. Michel Waldschmidt Contents 1 Explicit functions 3 1.1 Liouville . 3 1.2 Hermite . 5 1.3 Pad´eapproximation . 6 1.4 Hypergeometric methods . 7 2 Interpolation methods 7 2.1 Weierstraß question . 8 2.2 Integer–valued entire functions . 9 2.3 Transcendence of eπ ......................... 11 2.4 Lagrange interpolation . 12 3 Auxiliary functions arising from the Dirichlet’s box principle 13 3.1 Thue–Siegel lemma . 13 3.1.1 The origin of the Thue–Siegel Lemma . 14 3.1.2 Siegel, Gel’fond, Schneider . 15 3.1.3 Schneider–Lang Criterion . 18 3.1.4 Higher dimension: several variables . 20 3.1.5 The six exponentials Theorem . 22 3.1.6 Modular functions . 23 3.2 Universal auxiliary functions . 24 3.2.1 A general existence theorem .
    [Show full text]