DOCSLIB.ORG

Integrability, Nonintegrability and the Poly-Painlevé Test

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Integrability, Nonintegrability and the Poly-Painlevé Test Integrability, Nonintegrability and the Poly-Painlev´eTest Rodica D. Costin Sept. 2005 Nonlinearity 2003, 1997, preprint, Meth.Appl.An. 1997, Inv.Math. 2001 Talk dedicated to my professor, Martin Kruskal. Happy birthday! Historical perspective: Motivated by the idea that new functions can be defined as solutions of linear differential equations (Airy, Bessel, Mathieu, Lam´e,Legendre...), a natural question arose: Can nonlinear equations define new, \transcendental" functions? 1 Fuchs' intuition: A given equation does define functions if movable singularities in C of general solution are no worse than poles (no branch points, no essential singularities). Such an equation has solutions meromorphic on a (common) Riemann surface. 1 u0 = gen sol u = (x − C)1=2 movable branch point 2u 1 u0 = gen sol u = x1=2 + C fixed branch point 2x1=2 The property of a diff.eq. to have general solution without movable singularities (except, perhaps, poles) is now called \the Painlev´eProperty"(PP). 2 Directions: discovery of Painlev´eequations (1893-1910), development of Kowalevski integrability test (1888). • Q: Fuchs 1884: find the nonlinear equations with the Painlev´eproperty. Among first and second order equations class: F (w0; w; z) = 0, algebraic; w00 = F (z; w; w0) rational in w, w0, analytic in z equations with (PP) were classified. Among them 6 define new functions: the Painlev´etranscendents. 3 00 2 00 3 PI w = 6w + zP II w = 2w + zw + a 1 1 1 d P w00 = (w0)2 − w0 + aw2 + b + cw3 + III w z z w 1 3 b P w00 = (w0)2 + w3 + 4zw2 + 2(z2 − a)w + IV 2w 2 w 1 1 1 (w − 1)2 b w w(w + 1) P w00 = + (w0)2 − w0 + aw + + c + d V 2w w − 1 z z2 w z w − 1 1 1 1 1 1 1 1 P w00 = + + (w0)2 − + + w0 VI 2 w w − 1 w − z z z − 1 w − z w(w − 1)(w − z) bz z − 1 z(z − 1) + a + + c + d z2(z − 1)2 w2 (w − 1)2 (w − z)2 Painlev´etranscendents: in many physical applications (critical behavior in stat mech, in random matrices, reductions of integrable PDEs,...). 4 In another direction: Sophie Kowalevski linked the idea of absence of movable singularities (other than poles) to integrability. She developed a method of identifying integrable cases, and applied it to the equations describing the motion of a rigid body (a top) rotating about a fixed point. Kowalevski determines parameters for which there are no movable branch points (in C). She found 3 cases: 1) Euler, 2) Lagrange, 3) Kowalevski. In the new case Kowalevski expressed solutions using theta-functions of two variables. 1888 5 The method: of finding parameters for absence of movable branch points has been refined (Painlev´e-Kowalevski test), generalized to PDEs, used extensively, and successfully to identify integrable systems. Kruskal, Ablowitz, Clarkson, Deift, Flaschka, Grammaticos, Newell, Ramani, Segur, Tabor, Weiss... 6 Many aspects of the Painlev´e-Kowalevski test remain, however, not completely understood. Kruskal: • Is not clear if all the movable singularities of a given eq. have been determined. • Movable essential sing are usually not looked for. In fact, this is a very difficult question. • Painlev´e property is not invariant under coordinate transformations. 0 1 1=2 2 E.g. u = 2u has gen sol u = (x − C) : Set v = u : • A general connection to existence of first integrals is not rigorously known. 7 Classical problem: For a given DE establish or disprove existence of first integrals (smooth functions constant on the integral curves). Important: exact number of first integrals determines the dimension of the set filled by an integral curve. Extensive research (theoretical and applications). Poincar´e series: identify (or introduce) a small parameter in equation, assume first integral analytic, and expand in power series. Ziglin, extended and generalized by Ramis, Morales: expand eq. around special solutions. Study first approximation using differential Galois theory: if it does not have a meromorphic integral, then original DE does not have a meromorphic integral. However: this is a restrictive assumption, generic integrals are not meromorphic. 8 Kruskal- The Poly-Painlev´eTest Q: When does a diff. eq. have single-valued first integrals? Will illustrate the concepts on two equations which can be explicitly solved. 0 a Ex 1: u (x) = x−b has gen sol u(x) = a ln(x − b) + C A single-valued first integral F (x; u) = Φ(C) satisfies Φ(C) = Φ(u−a ln(x−b)) = Φ(u−a ln(x−b)−2nπia) = Φ(C−2nπia) for all n 2 Z We found the monodromy: how constants change upon analytic continuation along closed loops. exp(u=a) Hence we may take Φ(C) = exp(C=a) , so F (x; u) = x−b 9 Ex 2: u0(x) = a1 + a2 + a3 gen. sol. x−b1 x−b2 x−b3 u = a1 ln(x − b1) + a2 ln(x − b2) + a3 ln(x − b3) + C A single-valued first integral F (x; u) = Φ(C) should satisfy Φ(C) = Φ(C − 2n1πia1 − 2n2πia2 − 2n3πia3) ; n1;2;3 2 Z so Φ(C) = Φ(C +s) for all s 2 S = f2n1πia1 + 2n2πia2 + 2n3πia3 ; n1;2;3 2 Zg For generic (a1; a2; a3) the set S is dense in C. Hence Φ ≡const: no single-valued first integrals. Obstructions to integrability are encoded in the monodromy | branching properties of solutions . Dense branching no single-valued first integrals. 10 How can one find branching properties? Study equations in singular regions. Results: non-integrability analysis, and criteria for • systems with one regular singular point, • with several reg sing points, also: obstructions to linearization, classification; • for polynomial systems; new monodromy groups. •Irregular singular point: general structure of movable singularities solutions develop. 11 Region with one regular singular point xu0 = Mu + h(x; u) n 0 00 u; h 2 C , x 2 C, M 2 Mn;n(C), h analytic on D = f(x; u); juj < r ; r < jxj < r000g, having a zero of order 2 at u = 0. Theorem (RDC, Nonlin.97) For generic matrices M a complete description of the number of single-valued independent integrals on D is given. Generically integrals exist on D, and are not meromorphic. 12 Illustrate approach on 1-d: xu0 − µu = h(x; u). Set u = U. Eq. becomes xU 0 − µU = U 2h~(x; U) 2 Solutions U(x) = U0(x) + U1(x) + U2(x) + ::: 0 µ Then xU0 − µU0 = 0 so gen. sol. U0(x) = Cx It is relatively easy to see if this first approximation is integrable. (i) µ 2 C n R is the generic case. Then F0(x; U0) = P(ln U0 − µ ln x) is single-valued if P is doubly periodic, with periods 2πi, 2πiµ. Note: First integral is not meromorphic (singularities accumulate at U0 = 0). There is no meromorphic integral. 13 q −p p (ii) If µ 2 Q then F0(x; U0) = U0 x for µ = q . (iii) No single-valued integrals if µ 2 R n Q: µ µ 2nπiµ 2nπiµ Φ(C) = F0(x; Cx ) = F0(x; Cx e ) = Φ Ce 8n 2 Z. The set fe2nπiµ ; n 2 Zg is dense in S1; hence F (x; u) ≡const. No single-valued integrals. If µ 2 R n Q no single-valued first integrals. Conjecture: It is natural to expect that the original, nonlinear equation xu0 − µu = h(x; u) is nonintegrable as well. Q: How to prove? 14 A1: If one addresses the question of existence of regular integrals, they can be easily ruled out. Indeed, if there was one, then 2 F (x; u) = F (x; U) = F (x; U0 + U1 + :::) = F0(x) + F1(x)U0 + ::: j Then F0=const and the first j with Fj 6= 0 would give a first integral Fj(x)U0 for the reduced equation. A similar argument works for F meromorphic. However: in our case we can not expect meromorphic first integrals (we saw that first approximation had, generically, nonmeromorphic integrals). 15 A2 : Show that the original, nonlinear equation is analytically equivalent to its linear part. Normal Form of diff.eq. in an annulus around a regular singular point (Nonlinearity, 1997) Theorem Let xu0 = Mu + h(x; u) n 0 00 000 u; h 2 C , x 2 C, M 2 Mn;n(C), h holomorphic for juj < r , r < jxj < r , having a zero of order 2 at u = 0. Assume eigenvalues µ1; :::; µn of M satisfy the Diophantine condition: n n X X −ν n j kjµj +l−µsj > C( kj +jlj) for all s 2 f1; :::; ng; l 2 Z; k 2 N ; jkj ≥ 2 j=1 j=1 Then the system is biholomorphically equivalent to its linear part in D0 ⊂ D. 16 The proof has some similarity with the case of periodic coefficients. Steps: 1) Find equivalence map as a formal power series. 2) Showing convergence. There is a small denominator problem, overcome by a generalization of Newton's method. Note: The set of all Diophantine M has full measure. Consequences: For such matrices, study of nonlinear case reduces to the study of the linear equation (done in the paper). Generically: equations do have first integrals in D, and they are not meromorphic. Q: What about wider regions, containing several regular singular points? 17 Several regular singular points 2 0 1 1 u f~(x;u) u = Au + f(x; u) A = xµ1 + x−1µ2, f = x(x−1) ; x 2 D ⊃ f0; 1g ; juj < r Valid in more dimensions (at least for special classes of matrices µ1;2). Theorem Obstructions to analytic linearization: (RDC, preprint) Assume <µ1;2 > 0.
Load more

Recommended publications
  • Tsinghua Lectures on Hypergeometric Functions (Unfinished and Comments Are Welcome)
    Tsinghua Lectures on Hypergeometric Functions (Unfinished and comments are welcome) Gert Heckman Radboud University Nijmegen [email protected] December 8, 2015 1 Contents Preface 2 1 Linear differential equations 6 1.1 Thelocalexistenceproblem . 6 1.2 Thefundamentalgroup . 11 1.3 The monodromy representation . 15 1.4 Regular singular points . 17 1.5 ThetheoremofFuchs .. .. .. .. .. .. 23 1.6 TheRiemann–Hilbertproblem . 26 1.7 Exercises ............................. 33 2 The Euler–Gauss hypergeometric function 39 2.1 The hypergeometric function of Euler–Gauss . 39 2.2 The monodromy according to Schwarz–Klein . 43 2.3 The Euler integral revisited . 49 2.4 Exercises ............................. 52 3 The Clausen–Thomae hypergeometric function 55 3.1 The hypergeometric function of Clausen–Thomae . 55 3.2 The monodromy according to Levelt . 62 3.3 The criterion of Beukers–Heckman . 66 3.4 Intermezzo on Coxeter groups . 71 3.5 Lorentzian Hypergeometric Groups . 75 3.6 Prime Number Theorem after Tchebycheff . 87 3.7 Exercises ............................. 93 2 Preface The Euler–Gauss hypergeometric function ∞ α(α + 1) (α + k 1)β(β + 1) (β + k 1) F (α, β, γ; z) = · · · − · · · − zk γ(γ + 1) (γ + k 1)k! Xk=0 · · · − was introduced by Euler in the 18th century, and was well studied in the 19th century among others by Gauss, Riemann, Schwarz and Klein. The numbers α, β, γ are called the parameters, and z is called the variable. On the one hand, for particular values of the parameters this function appears in various problems. For example α (1 z)− = F (α, 1, 1; z) − arcsin z = 2zF (1/2, 1, 3/2; z2 ) π K(z) = F (1/2, 1/2, 1; z2 ) 2 α(α + 1) (α + n) 1 z P (α,β)(z) = · · · F ( n, α + β + n + 1; α + 1 − ) n n! − | 2 with K(z) the Jacobi elliptic integral of the first kind given by 1 dx K(z)= , Z0 (1 x2)(1 z2x2) − − p (α,β) and Pn (z) the Jacobi polynomial of degree n, normalized by α + n P (α,β)(1) = .
    [Show full text]
  • Children's Drawings and the Riemann-Hilbert Problem
    CHILDREN'S DRAWINGS AND THE RIEMANN-HILBERT PROBLEM DRIMIK ROY CHOWDHURY Abstract. Dessin d'enfants (French for children's drawings) serve as a unique standpoint of studying classical complex analysis under the lens of combinatorial constructs. A thor- ough development of the background of this theory is developed with an emphasis on the relationship of monodromy to Dessins, which serve as a pathway to the Riemann Hilbert problem. This paper investigates representations of Dessins by permutations, the connec- tion of Dessins to a particular class of Riemann surfaces established by Belyi's theorem and how these combinatorial objects provide another perspective of solving the discrete Riemann-Hilbert problem. Contents 1. History and Motivation 2 2. Background 3 2.1. Group Theory 3 2.2. Introduction to Topology 4 2.3. Covering Theory 5 2.4. Complex Analysis 7 2.5. Combinatorics: Hypermaps and Maps 8 3. Dessin d'enfants: Construction 9 3.1. Encoding hypermaps by triple of permutations 9 3.2. Critical points and critical values for complex polynomials 11 3.3. Dessins and Belyi Pairs 12 3.4. Riemann's Existence Theorem into Belyi's Theorem 13 3.5. Other applications of Dessins: Galois group of rationals 14 4. Dessins and Fuchsian Differential Equations 14 4.1. Differential Equations and Monodromy 15 4.2. How Dessins play into the Riemann-Hilbert problem 16 4.3. Riemann Hilbert problem for Dessins as Trees 17 Acknowledgements 18 References 18 Date: October 25, 2019. 1 2 DRIMIK ROY CHOWDHURY 1. History and Motivation The enigmatic world of Dessins actually started in another form under the explorations of Felix Klein (with some of his work relating to the material investigated here [2]).
    [Show full text]
  • Communications in Commun
    Communications in Commun. math. Phys.76, 65-116 (1980) Mathematical Physics © by Springer-Verlag 1980 Monodromy- and Spectrum-Preserving Deformations I Hermann Flaschka* and Alan C. Newell Department of Mathematics and Computer Science, Clarkson College of Technology, Potsdam N. Y. 13676 USA Abstract. A method for solving certain nonlinear ordinary and partial differential equations is developed. The central idea is to study monodromy preserving deformations of linear ordinary differential equations with regular and irregular singular points. The connections with isospectral deformations and with classical and recent work on monodromy preserving deformations are discussed. Specific new results include the reduction of the general initial value problem for the Painleve equations of the second type and a special case of the third type to a system of linear singular integral equations. Several classes of solutions are discussed, and in particular the general expression for rational solutions for the second Painleve equation family is shown to be — — ln(A+/A__\ where Δ+ and Δ_ are determinants. We also demonstrate that each of these equations is an exactly integrable Hamiltonian system.* The basic ideas presented here are applicable to a broad class of ordinary and partial differential equations additional results will be presented in a sequence of future papers. 1. Introduction and Outline This paper is the first in what is planned to be a series of studies on deformations of linear ordinary differential equations with coefficients rational on a Riemann surface. The deformations in question preserve the monodromy at singular points of the linear equation, and this requirement forces the coefficients of the linear equation to satisfy certain nonlinear ordinary or partial differential equations of considerable interest.
    [Show full text]
  • Tsinghua Lectures on Hypergeometric Functions (Unfinished and Comments Are Welcome)
    Tsinghua Lectures on Hypergeometric Functions (Unfinished and comments are welcome) Gert Heckman Radboud University Nijmegen [email protected] September 9, 2015 1 Contents Preface 2 1 Linear differential equations 6 1.1 Thelocalexistenceproblem . 6 1.2 Thefundamentalgroup . 11 1.3 The monodromy representation . 15 1.4 Regular singular points . 17 1.5 ThetheoremofFuchs .. .. .. .. .. .. 23 1.6 TheRiemann–Hilbertproblem . 26 1.7 Exercises ............................. 28 2 The Euler–Gauss hypergeometric function 34 2.1 The hypergeometric function of Euler–Gauss . 34 2.2 The monodromy according to Schwarz–Klein . 38 2.3 The Euler integral revisited . 43 2.4 Exercises ............................. 46 3 The Clausen–Thomae hypergeometric function 47 3.1 The hypergeometric function of Clausen–Thomae . 47 3.2 The monodromy according to Levelt . 50 3.3 The criterion of Beukers–Heckman . 54 3.4 Intermezzo on Coxeter groups . 57 3.5 Prime Number Theorem after Tchebycheff . 60 3.6 Exercises ............................. 66 2 Preface The Euler–Gauss hypergeometric function ∞ α(α + 1) (α + k 1)β(β + 1) (β + k 1) F (α, β, γ; z) = · · · − · · · − zk γ(γ + 1) (γ + k 1)k! Xk=0 · · · − was introduced by Euler in the 18th century, and was well studied in the 19th century among others by Gauss, Riemann, Schwarz and Klein. The numbers α, β, γ are called the parameters, and z is called the variable. On the one hand, for particular values of the parameters this function appears in various problems. For example α (1 z)− = F (α, 1, 1; z) − arcsin z = 2zF (1/2, 1, 3/2; z2 ) π K(z) = F (1/2, 1/2, 1; z2 ) 2 α(α + 1) (α + n) 1 z P (α,β)(z) = · · · F ( n, α + β + n + 1; α + 1 − ) n n! − | 2 with K(z) the Jacobi elliptic integral of the first kind given by 1 dx K(z)= , Z0 (1 x2)(1 z2x2) − − p (α,β) and Pn (z) the Jacobi polynomial of degree n, normalized by α + n P (α,β)(1) = .
    [Show full text]
  • A Second-Order Linear Ordinary Differential Equation (ODE)
    Mohammed Mousa Mohsin, International Journal of Computer Science and Mobile Computing, Vol.3 Issue.3, March- 2014, pg. 748-761 Available Online at www.ijcsmc.com International Journal of Computer Science and Mobile Computing A Monthly Journal of Computer Science and Information Technology ISSN 2320–088X IJCSMC, Vol. 3, Issue. 3, March 2014, pg.748 – 761 RESEARCH ARTICLE A Second-Order Linear Ordinary Differential Equation (ODE) Mohammed Mousa Mohsin Department of Statistics & university of Basra, Iraq Email: [email protected] Abstract The term "hypergeometric function" sometimes refers to the generalized hypergeometric function. In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as special or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ Keywords….. Hypergeometric function Introduction The term "hypergeometric series" was first used by John Wallis in his 1655 book Arithmetica Infinitorum. Hypergeometric series were studied by Euler, but the first full systematic treatment was given by Gauss (1813), Studies in the nineteenth century included those of Ernst Kummer (1836), and the fundamental characterisation by Bernhard Riemann of the hypergeometric function by means of the differential equation it satisfies. Riemann showed that the second-order differential equation (in z) for the 2F1, examined in the complex plane, could be characterised (on the Riemann sphere) by its three regular singularities. The cases where the solutions are algebraic functions were found by H.
    [Show full text]
  • Isomonodromy, Painlevé Transcendents And
    April 22, 2014 Isomonodromy, Painlev´eTranscendents and Scattering off of Black Holes F´abioNovaes and Bruno Carneiro da Cunha Departamento de F´ısica, Universidade Federal de Pernambuco, 50670-901, Recife, Pernam- buco, Brazil E-mail: [email protected], [email protected] Abstract: We apply the method of isomonodromy to study the scattering of a generic Kerr-NUT-(A)dS black hole. For generic values of the charges, the problem is related to the connection problem of the Painlev´eVI transcendent. We review a few facts about Painlev´eVI, Garnier systems and the Hamiltonian structure of flat connections in the Riemann sphere. We then outline a method for computing the scattering amplitudes based on Hamilton-Jacobi structure of Painlev´e,and discuss the implications of the generic result to black hole complementarity. Keywords: Isomonodromy, Painlev´eTranscendents, Heun Equation, Scattering The- ory, Black Holes. arXiv:1404.5188v1 [hep-th] 21 Apr 2014 Contents 1 Introduction1 2 Killing-Yano and Separability3 2.1 Kerr-NUT-(A)dS case5 2.2 Heun equation from Conformally Coupled Kerr-NUT-(A)dS7 3 Scattering, Isomonodromy and Painlev´eVI 12 4 Flat Connections and Monodromies 17 5 The Classical Mechanics of Monodromies 24 6 The Generic Scattering 28 7 Discussion 31 1 Introduction The scattering of quantum fields around a classical gravitational background has had an important impact in the study of quantum fields in curved spaces [1]. The classical result for Schwarzschild black holes has helped in the development of the Hawking mechanism, as well as fundamental questions of unitarity of evolution and the exact character of the event horizon as a special place in a black hole background.
    [Show full text]
  • Ordinary Differential Equations
    Ordinary Differential Equations Math 3027 (Columbia University Summer 2019) Henry Liu Last updated: August 13, 2019 Contents 0 Introduction 2 1 First-order equations 3 1.1 Slope fields . .3 1.2 Linear equations with constant coefficients . .5 1.3 Separable equations . .6 1.4 Existence and uniqueness . .7 1.5 Autonomous systems . .8 1.6 Exact equations and integrating factors . .9 1.7 Changes of variable . 11 2 Second-order linear equations 13 2.1 Structure of solutions . 13 2.2 Homogeneous equations with constant coefficients . 15 2.3 Non-homogeneous equations . 18 2.4 Higher-order linear equations . 19 3 Singularities and series solutions 20 3.1 Review of power series . 20 3.2 Ordinary points . 22 3.3 Regular singular points . 24 3.4 Singularities at infinity . 29 3.5 Irregular singular points . 31 4 Laplace transform 33 4.1 Solving IVPs . 34 4.2 Piecewise continuous forcing functions . 35 4.3 Impulsive forcing functions . 38 4.4 Convolution and impulse response . 40 5 First-order systems 42 5.1 Review of linear algebra . 43 5.2 Homogeneous systems with constant coefficients . 46 5.3 Matrix exponentials . 50 5.4 Non-homogeneous systems . 52 5.5 Phase plane and stability . 54 6 Boundary value problems 58 6.1 Green's functions . 59 1 0 Introduction The concept of an algebraic equation and its solutions is introduced fairly early in a standard mathematics curriculum. We first learn how to solve linear equations like 3x + 5 = 2, and then quadratic equations like x2 − x − 3 = 0. Solutions are numbers x which satisfy the equation.
    [Show full text]
  • Some Remarks on the Location of Non-Asymptotic Zeros of Whittaker and Kummer Hypergeometric Functions Islam Boussaada, Guilherme Mazanti, Silviu-Iulian Niculescu
    Some Remarks on the Location of Non-Asymptotic Zeros of Whittaker and Kummer Hypergeometric Functions Islam Boussaada, Guilherme Mazanti, Silviu-Iulian Niculescu To cite this version: Islam Boussaada, Guilherme Mazanti, Silviu-Iulian Niculescu. Some Remarks on the Location of Non-Asymptotic Zeros of Whittaker and Kummer Hypergeometric Functions. 2021. hal-03250426v2 HAL Id: hal-03250426 https://hal.archives-ouvertes.fr/hal-03250426v2 Preprint submitted on 17 Jun 2021 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. Some Remarks on the Location of Non-Asymptotic Zeros of Whittaker and Kummer Hypergeometric Functions Islam Boussaada · Guilherme Mazanti · Silviu-Iulian Niculescu Abstract This paper focuses on the location of the non-asymptotic zeros of Whittaker and Kummer confluent hypergeometric functions. Based on a technique by E. Hille for the analysis of solutions of some second-order ordinary differential equations, we characterize the sign of the real part of zeros of Whittaker and Kummer functions and provide estimates on the regions of the complex plane where those zeros can be located. Our main result is a correction of a previous statement by G. E. Tsvetkov whose propagation has induced mistakes in the literature.
    [Show full text]
  • Transcendence of Special Values of Modular and Hypergeometric Functions
    TRANSCENDENCE OF SPECIAL VALUES OF MODULAR AND HYPERGEOMETRIC FUNCTIONS PAULA TRETKOFF 1. Lecture I: Modular Functions Part 1: Most known transcendence results about functions of one and several complex variables are derived from those for commutative algebraic groups. Varieties and vector spaces are assumed de¯ned over Q, although we also consider their complex points. A commutative algebraic group, also called a group variety, is a variety with a commutative group structure for which the composition and inverse maps are regular. The group Ga has complex ¤ points C, under addition. The group Gm has complex points C = C n f0g, under multiplication. A group variety G has a maximal subgroup L of the r s form Ga £ Gm such that the quotient G=L is a projective group variety, otherwise known as an abelian variety. We will focus almost exclusively on abelian varieties and their moduli spaces. Abelian varieties of complex dimension one are called elliptic curves. The complex points A(C) of an elliptic curve A can be represented as a complex torus C=L where L is a lattice in C. Therefore, L = Z!1 + Z!2 for !1 , !2 complex numbers with z = !2=!1 2 H, where H is the set of complex numbers with positive imaginary part. The Weierstrass elliptic function is de¯ned by 1 X 1 1 }(z) = }(z; L) = + ¡ : z2 (z ¡ !)2 !2 !2L;!6=0 Therefore, }(z) = }(z + !), for all ! 2 L, and }(z) satis¯es the di®erential equation 0 2 3 } (z) = 4}(z) ¡ g2}(z) ¡ g3; for certain algebraic invariants g2 , g3 depending only on L.
    [Show full text]
  • Linear Meromorphic Differential Equations: a Modern Point of View
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 33, Number 1, January 1996 LINEAR MEROMORPHIC DIFFERENTIAL EQUATIONS: A MODERN POINT OF VIEW V. S. VARADARAJAN Abstract. A large part of the modern theory of differential equations in the complex domain is concerned with regular singularities and holonomic systems. However the theory of differential equations with irregular singularities has a long history and has become very active in recent years. Substantial links of this theory to the theory of algebraic groups, commutative algebra, resurgent functions, and Galois differential methods have been discovered. This survey attempts a general introduction to some of these aspects, with emphasis on reduction theory, asymptotic analysis, Stokes phenomena, and certain moduli problems. 1. Introduction I would like to give a brief outline of some of the advances made in recent years in the theory of ordinary linear differential equations in the complex domain with meromorphic coefficients. This is a vast area with connections to many parts of contemporary mathematics such as commutative algebra, algebraic geometry, alge- braic groups, complex analysis (both conventional and resurgent), and so on. To give a comprehensive survey of this subject is thus impossible and I will not at- tempt it here. I shall only restrict myself to a few aspects with which I have some familiarity and which have been discussed in detail in my papers with D. G. Bab- bitt [8(a)–(g)]. For treatments of topics not discussed here, as well as additional perspective and more complete references, I refer the reader to V. Arnold’s ency- clopaedia article [3], my own paper in the Expositiones Mathematicae [4(a)], the Bourbaki talks of D.
    [Show full text]
  • Orthogonal Polynomials and Special Functions Comments
    Euler's formula (1) gives Γ(z + 1) z (m + 1)(z + 1) = lim = z Γ(z) z + 1 m!1 m + 1 + z Hence we obtain the most remarkable property for the Gamma function: Γ(z + 1) = zΓ(z) with Γ(1) = 1 = 0! : (3) Orthogonal Polynomials and Special Functions Comments LTCC course, 19 Feb - 20 March, 2018 Ana F. Loureiro ([email protected]) Outline Comments I Part 1. Special Functions I Gamma, Digamma and Beta Function I Hypergeometric Functions and Hypergeometric Series I Confluent Hypergeometric Function I Part 2. Orthogonal Polynomials I Main properties Recurrence relations, zeros, distribution of the zeros and so on and on.... I Classical Orthogonal Polynomials Hermite, Laguerre, Bessel and Jacobi!! I Other notions of "classical orthogonal polynomials" How to identify this on the Askey Scheme? I Semiclassical Orthogonal Polynomials How do these link to Random Matrix Theory, Painlev´eequations and so on? I Part 3. Multiple Orthogonal Polynomials When the orthogonality measure is spread across a vector of measures? Special Functions Comments 1. Gamma function. Introduced by Euler in 1729 in a letter to Golbach, the Gamma function arose to answer the question of finding a function mapping any nonnegative integer n to its factorial n!, that is Γ : N0 −! N0 N0 8 n Y <> n! = n(n − 1) ::: 3 · 2 · 1 = j if n ≥ 1; n 7−! j=1 :> 0! = 1 if n = 0: Gamma Function - Euler's definition Comments +1 z −1! 1 Y 1 z Γ(z) = 1 + 1 + (1) z j j j=1 valid for any z such that z 6= 0; −1; −2;:::.
    [Show full text]
  • Hypergeometric Equation and Monodromy Groups
    HYPERGEOMETRIC EQUATION AND MONODROMY GROUPS EDMUND Y.-M. CHIANG Abstract. We shall introduce the basics of classical Gauss hypergeometric equation and to study its monodromy problem. Historically, Riemann was the first one who worked out the monodromy group of the DE in 1857. 1. Fuchsian-type Differential Equations Theorem 1.1. Let a1; a2; ··· an and 1 be the regular singular points of d2w dw (1.1) + p(z) + q(z)w = 0: dz2 dz Then we have n n X Ak X Bk Ck (1.2) p(z) = ; q(z) = 2 + ; z − ak (z − ak) z − ak k=1 k=1 and the indicial equation at each ak is given by 2 α + (Ak − 1)α + Bk = 0: 0 0 Suppose fαk; αkg and fα1; α1g denote, respectively, the exponents of the corre- sponding indicial equations at a1; a2; ··· an and 1. Then we have n 0 X 0 (α1 + α1) + (αk + αk) = n − 1; k=1 Proof. Since p(z) has at most a first order pole at each of the points a1; a2; ··· an, so the function φ(z) defind in n X Ak p(z) = + '(z) z − ak k=1 where '(z) is an entire function in C, and where the Ak is the residue of p(z) at ak (k = 1; 2; ··· n). On the other hand, the 1 is also a regular singular point of p(z), it follows from Proposition 3.2 in part I and which states 1 is regular singular iff zp and z2q are analytic at 1. That is, p(z) = O(1=z).
    [Show full text]