DOCSLIB.ORG

University of Kota, Kota

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
University of Kota, Kota A STUDY OF GENERATING FUNCTION INVOLVING GENERALIZED HYPERGEOMETRIC FUNCTIONS OF ONE AND MORE VARIABLES A THESIS Submitted for the Award of Ph.D. degree of University of Kota, Kota in (Mathematics-Faculty of Science) by DEEPAK KUMAR KABRA Under the supervision of Dr. YASMEEN (Lecturer Selection Scale) Department of Mathematics Government College Kota, Kota 324001(India) UNIVERSITY OF KOTA, KOTA August 2015 Dr. Yasmeen Lecturer (Selection Scale) (M.Phil. Ph.D.) Dept. of Mathematics Govt. College Kota-Kota, 324 001 CERTIFICATE I feel great pleasure in certifying that the thesis entitled \A STUDY OF GENERATING FUNCTION INVOLVING GENERALIZED HYPERGEOMETRIC FUNCTIONS OF ONE AND MORE VARI- ABLES", embodies a record of the results of investigations carried out by Mr. Deepak Kumar Kabra under my supervision. I am sat- isfied with the findings of present work. He has completed the residential requirement as per rules. I recommend the submission of thesis. (Dr. Yasmeen) Date: August 2015 Research Supervisor ii DECLARATION I hereby declare that the (i) The thesis entitled \A STUDY OF GENERATING FUNC- TION INVOLVING GENERALIZED HYPERGEOMETRIC FUNC- TIONS OF ONE AND MORE VARIABLES" submitted by me is an original piece of research work, carried out under the supervision of Dr. Yasmeen. (ii) The above thesis has not been submitted to this univer- sity or any other university for any degree. Date: August 2015 (Deepak Kumar Kabra) iii DECLARATION I hereby declare that the (i) The thesis entitled \A STUDY OF GENERATING FUNCTION IN- VOLVING GENERALIZED HYPERGEOMETRIC FUNCTIONS OF ONE AND MORE VARIABLES" submitted by me is an original piece of research work, car- ried out under the supervision of Dr. Yasmeen. (ii) The above thesis has not been submitted to this university or any other university for any degree. Date: August 2015 (Deepak Kumar Kabra) iv Acknowledgement All praise be to Almighty GOD, the most beneficent, merciful, who bestowed upon me the strength, courage and patience to embark upon this work and carry it to its completion. I express my thanks to GOD and bow my head before Him in gratitude. Words and lexicons cannot do full justice in expressing a sense of gratitude which flows from my heart towards my esteemed supervisor Dr. Yasmeen, Department of Mathematics, Government College Kota, Kota. I learned and ventured to write under her guidance. Her help and advice have gone into the preparation of this thesis. I shall remain indebted to for all that I have gained from her. I wish to put on record the sense of gratitude to Dr. M. I. Qureshi, Profes- sor, Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia (A Central University), New Delhi, In- dia for his inspiring enthusiasm and helpful discussion without which the study could have been abandoned. His expert advice, guidance, suggestions, motiva- tion, discussions, and feedback have enabled me to complete the present work successfully. My sincere thanks and gratitude are due to Prof. M. A. Pathan, Department of Mathematics, Aligarh Muslim University, Aligarh, U.P. for fruitful discussions and help on the subject. My sincere thanks and gratitude are due to Dr. D. N. Vyas, of M. L. V. Textile & Engineering College, Bhilwara (an autonomous institute of Govt. of Rajasthan) for his learned comments, valuable discussions and help on the subject. v I deem it my pride to spell out the words of gratitude for my teachers Dr. V. K. Vaidya, Dr. J. P. N. Ojha, whom I consider my prime mentors. I wish to express my gratitude to Prof. T. C. Loya, Principal, Mrs. Hemlata Lahoti, Vice Principal, Government College, Kota for their constant encourage- ment and providing all the necessary facilities for conducting the present research work. I am grateful to Mrs. Usha Pancholi, Head, Department of Mathematics, Gov- ernment College, Kota and all the learned faculty members of the Department for their consistent support provided to me during the entire period of the present study. I express my sincere thanks to my friends Dr. Anuj Nuwal, Dr. Dheeraj Jain, Dr. Amit Chouhan, Dr. Rajeev Mehta, Dr. Mukesh Paliwal, Dr. Preeti Mehta, Dr. C.D. Kumawat, Dr. R.K. Somani, Ms. Rakhi Goyal, Mrs. Seema Bhandari, Mrs. Poonam Maheshwari, Mrs. Nidhi Bhatnagar, Mr. Vaibhav Pancholi, Mr. Rakesh Bhandari, Mr. Vinesh Agarwal for their kind support. I shall be failing in my duty if I do not express my sense of gratitude to Mr. Abdul Salim, husband of Dr. Yasmeen, for his generosity and his prayer to almighty GOD for my welfare and good health. I am indebted to my brothers Mr. Abhijeet and Mr. Koushal and to my sisters Mrs. Sunita Bhargav Inani and Minakshi Sourabh Nuwal for their love, affection and inspiration at every stage of my research work. Without their love and blessing, this research work would probably have never been thought of. I would like to express my deep sense of indebtedness to my beloved mother Smt. Prem Devi, reverend father Sh. Govind Lal Kabra, mother in law Smt. Kamla Devi, uncle and aunti for their inevitable role in my academic endeavour. Their countless blessing, deep love and affection have always worked as a hidden power behind everything I did. I am thankful to Mr. K. L. Soni for their efficient editing and meticulous pro- cessing of the manuscript. vi Besides these, I am thankful to all those who helped me directly or indirectly in the preparation of this thesis. Last, but not least, I thank to my wife Nisha Kabra for her patience, sacrifice and moral support. The quite and innocent smile of loving daughter Bhavya Kabra, who has always been instrumental in bringing me out a stressful situations on many occasions. (DEEPAK KUMAR KABRA) vii Contents 1 A Brief Survey of the Literature 1 1.1 Introduction .............................. 2 1.2 Gamma Function, Pochhammer Symbols and Associated Results . 5 1.3 Ordinary Generalized Hypergeometric Function of One Variable . 9 1.3.1 Power Series Form and Convergence Conditions . 10 1.3.2 Mellin-Barnes type Contour Integral Representation for pFq 11 1.4 Meijer's G-Function of One Variable . 12 1.4.1 Mellin-Barnes type Contour Integral Representation for G- Function ............................ 13 1.4.2 Properties of G-Function ................... 15 1.5 Fox-Wright Generalized Hypergeometric Function of One Variable 16 1.6 Truncated Ordinary Hypergeometric Series . 17 1.7 Bilateral Hypergeometric Function . 18 1.8 Appell's Double Hypergeometric Functions . 19 1.9 Humbert's Double Hypergeometric Functions . 20 1.10 Kamp´e de F´eriet's Double Hypergeometric Functions . 21 1.11 Exton's Double Hypergeometric Functions . 21 1.12 Triple Hypergeometric Functions of Jain, Exton's and Srivastava . 23 1.13 Srivastava's General Triple Hypergeometric Functions F (3) . 23 (4) 1.14 Megumi Saigo's General Quadruple Hypergeometric Function FM 25 1.15 Lauricella Hypergeometric Function of r Variables . 26 1.16 Karlsson's Multivariable Hypergeometric Functions . 27 1.17 Multi-Variable Extension of Kamp´e de F´eriet's Functions . 28 1.18 Srivastava-Daoust Multi-Variable Hypergeometric Functions . 29 viii 1.19 Hypergeometric Forms of Some Elementary and Composite Func- tions .................................. 30 1.20 Hypergeometric Forms of Some Special Functions . 32 1.21 Transformation and Reduction Formulas . 37 1.22 Multiple Series Identities ....................... 41 1.23 Some Hypergeometric Polynomials, their Relationships and Gen- erating Relations ........................... 43 2 Some Linear, Bilinear, Bilateral Generating Relations and Sum- mation Formulae 61 2.1 Introduction .............................. 62 2.2 Linear Generating Relation and Summations . 65 2.3 Linear Generating Relation using Series Rearrangement Technique 65 2.4 Some Summations Formulae ..................... 66 2.5 Bilinear and Bilateral Generating Relations using Fractional Deriva- tive ................................... 67 2.6 Linear, Bilinear and Bilateral Generating Relations using the Pro- cess of Confluence ........................... 68 3 Some Generating Relations Involving Multi-Variable Laguerre Polynomials 71 3.1 Introduction .............................. 72 3.2 Partial Differential Equations Associated with Polynomial Sets . 72 3.3 Series Representation of Polynomials . 73 3.4 General Theorem Associated with Generating Relation and Gen- erating Function ............................ 75 3.5 Applications of Theorems 1 and 2 in Generating Relations . 76 4 A General Family of Generating Functions 79 4.1 Introduction .............................. 80 4.2 General Multiple Series Identities . 80 4.3 Applications .............................. 82 5 Some Generating Relations and Hypergeometric Transforma- ix tions using Decomposition Technique 86 5.1 Introduction .............................. 87 5.1.1 Hypergeometric Form of Subuhi Khan Polynomial . 88 5.1.2 Hypergeometric Form of Lagrange's Polynomial in Two Variables ............................ 89 5.2 Linear Generating Relations and Hypergeometric Transformations Associated with Dickinson-Warsi Polynomials . 90 5.3 Linear Generating Relations Involving Subuhi Khan Polynomials . 91 5.4 Linear Generating Relations Associated with Lagrange's Polyno- mials in Two Variables ........................ 92 5.5 Derivation of all Generating Relations and Transformations . 93 6 Multiple Generating Relations Involving Double and Triple Hy- pergeometric Functions 99 6.1 Introduction ..............................100 6.2 Expansion of tn in a Series of Generalized Hypergeometric Poly- nomials ................................101
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]
  • Fresnel Integral Computation Techniques
    FRESNEL INTEGRAL COMPUTATION TECHNIQUES ALEXANDRU IONUT, , JAMES C. HATELEY Abstract. This work is an extension of previous work by Alazah et al. [M. Alazah, S. N. Chandler-Wilde, and S. La Porte, Numerische Mathematik, 128(4):635{661, 2014]. We split the computation of the Fresnel Integrals into 3 cases: a truncated Taylor series, modified trapezoid rule and an asymptotic expansion for small, medium and large arguments respectively. These special functions can be computed accurately and efficiently up to an arbitrary preci- sion. Error estimates are provided and we give a systematic method in choosing the various parameters for a desired precision. We illustrate this method and verify numerically using double precision. 1. Introduction The Fresnel integrals and their simultaneous parametric plot, the clothoid, have numerous applications including; but not limited to, optics and electromagnetic theory [8, 15, 16, 17, 21], robotics [2,7, 12, 13, 14], civil engineering [4,9, 20] and biology [18]. There have been numerous works over the past 70 years computing and numerically approximating Fresnel integrals. Boersma established approximations using the Lanczos tau-method [3] and Cody computed rational Chebyshev approx- imations using the Remes algorithm [5]. Another approach includes a spreadsheet computation by Mielenz [10]; which is based on successive improvements of known relational approximations. Mielenz also gives an improvement of his work [11], where the accuracy is less then 1:e-9. More recently, Alazah, Chandler-Wilde and LaPorte propose a method to compute these integrals via a modified trapezoid rule [1]. Alazah et al. remark after some experimentation that a truncation of the Taylor series is more efficient and accurate than their new method for a small argument [1].
    [Show full text]
  • Analytical Evaluation and Asymptotic Evaluation of Dawson's Integral And
    Analytical evaluation and asymptotic evaluation of Dawson’s integral and related functions in mathematical physics V. Nijimbere School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada Abstract Dawson’s integral and related functions in mathematical physics that in- clude the complex error function (Faddeeva’s integral), Fried-Conte (plasma dispersion) function, (Jackson) function, Fresnel function and Gordeyev’s in- tegral are analytically evaluated in terms of the confluent hypergeometric function. And hence, the asymptotic expansions of these functions on the complex plane C are derived using the asymptotic expansion of the confluent hypergeometric function. Keywords: Dawson’s integral, Complex error function, Plasma dispersion function, Fresnel functions, Gordeyev’s integral, Confluent hypergeometric function, asymptotic expansion 1. Introduction Let us consider the first-order initial value problem, D′ +2zD =1,D(0) = 0. (1) arXiv:1703.06757v1 [math.CA] 14 Mar 2017 Its solution given by the definite integral z z2 η2 daw z = D(z)= e− e dη (2) Z0 Email address: [email protected] (V. Nijimbere) Preprint submitted to Elsevier March 21, 2017 is known as Dawson’s integral [1, 17, 24, 27]. Dawson’s integral is related to several important functions (in integral form) in mathematical physics that include Faddeeva’s integral (also know as the complex error function or Kramp function) [9, 10, 22, 18, 27] z z2 2i z2 z2 2i η2 w(z)= e− 1+ e daw z = e− 1+ e dη , (3) √π √π Z0 Fried-Conte function (or plasma dispersion function) [4, 11] z z2 2i z2 z2 2i η2 Z(z)= i√πw(z)= i√πe− 1+ e daw z = i√πe− 1+ e dη , √π √π Z0 (4) (Jackson) function [14] G(z)=1+ zZ(z)=1+ i√πzw(z) z2 2i z2 =1+ i√πze− 1+ e daw z √π z z2 2i η2 =1+ i√πze− 1+ e dη , (5) √π Z0 and Fresnel functions C(x) and S(x) [1] defined by the relation z iπz2 e 2 daw √iπz = eiπη dη = C(x)+ iS(x), (6) √iπ Z0 where z z C(x)= cos(πη2)dη and S(x)= sin (πη2)dη.
    [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]
  • 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).
    [Show full text]
  • Oscillatority of Fresnel Integrals and Chirp-Like Functions
    D ifferential E quations & A pplications Volume 5, Number 4 (2013), 527–547 doi:10.7153/dea-05-31 OSCILLATORITY OF FRESNEL INTEGRALS AND CHIRP–LIKE FUNCTIONS MAJA RESMAN,DOMAGOJ VLAH AND VESNA Zˇ UPANOVIC´ Dedicated to Luka Korkut, upon his retirement (Communicated by Yuki Naito) Abstract. In this review article, we present results concerning fractal analysis of Fresnel and gen- eralized Fresnel integrals. The study is related to computation of box dimension and Minkowski content of spirals defined parametrically by Fresnel integrals, as well as computation of box di- mension of the graph of reflected component function which are chirp-like function. Also, we present some results about relationship between oscillatority of the graph of solution of differ- ential equation, and oscillatority of a trajectory of the corresponding system in the phase space. We are concentrated on a class of differential equations with chirp-like solutions, and also spiral behavior in the phase space. 1. Introduction As coauthors of Luka Korkut, we present here an overview of his scientific con- tribution in fractal analysis of Fresnel integrals and chirp-like functions. We cite the main results from joint articles of Korkut, Resman, Vlah, Zubrini´ˇ candZupanovi´ˇ c, [18, 19, 20, 21, 22, 23]. The articles are mostly based on qualitative theory of differen- tial equations. A standard technique of this theory is phase plane analysis. We study trajectories of the corresponding system of differential equations in the phase plane, instead of studying the graph of the solution of the equation directly. Our main interest is fractal analysis of behavior of the graph of oscillatory solution, and of a trajectory of the associated system.
    [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]