DOCSLIB.ORG

Special Functions: Legendre Functions, Spherical Harmonics, and Bessel Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Special Functions: Legendre Functions, Spherical Harmonics, and Bessel Functions Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Physics 212 2010, Electricity and Magnetism Michael Dine Department of Physics University of California, Santa Cruz October 2010 Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions There are several special functions that recur in many branches of physics. You are all familiar, at some level, with spherical harmonics, from angular momentum in quantum mechanics. The spherical harmonics, more generally, are important in problems with spherical symmetry. They occur in electricity and magnetism. They are important also in astrophysics and cosmology, where they play the role of sines and cosines in fourier expanding functions on the sky. Legendre polynomials and legendre functions more generally solve the θ equations. Bessel functions arise in problems with spherical symmetry, but actually occur also more broadly. In quantum mechanics, particular instances solve the free particle radial equation in spherical coordinates, and again in cosmology, they appear as solutions to a number of problems. Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Start with Laplaces’s eqn. in spherical coordinates: 1 @2 1 @ @Φ 1 @2Φ (rΦ) + (sin θ ) + = 0: (1) r @r 2 r 2 sin θ @θ @θ r 2 sin2 θ @φ2 Separate variables: u(r) Φ = P(θ)Q(φ): (2) r Leads to Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions 1 d 2u 1 d dP 1 d 2Q r 2 sin2 θ + (sin θ ) + = 0: (3) u dr 2 r 2 sin θP dθ dθ Q sφ2 The last term must be a constant: d 2Q = −m2Q (4) dφ2 or Q = e±imφ: (5) Singlevaluedness ) m integer. Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Now want equations for P, u. Divide by sin2 θ; 1 d 2u 1 d dP m2 r 2 + (sin θ ) − = 0; (6) u dr 2 sin θP dθ dθ sin2 θ Second and third terms are independent of r, so can again introduce separation constant: d 2u u − `(` + 1) = 0 (7) dr 2 r 2 and 1 d dP m2 (sin θ ) + `(` + 1) − P = 0: (8) sin θ dθ dθ sin2 θ Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions u eqn. has solutions r a, with a(a − 1) = `(` + 1). So u = ar `+1 + Br −`: (9) If solving equation in all of space, can reject solution singular at the origin. Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions For the equation for P, we first write d 1 d x = cos θ; = − (10) dθ sin θ d cos θ so the equation can be rewritten as: d dP m2 (1 − x2) + `(` + 1) − P = 0: (11) dx dx 1 − x2 The case m = 0 is known as the ordinary Legendre differential equation; the case of non-zero m is known as Legendre’s equation. The solutions of the first are known as Legendre polynomials; of the second as associated Legendre functions. Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Solution by series method Starting with the ordinary Legendre equation: d dP (1 − x2) + `(` + 1)P = 0: (12) dx dx try a solution: 1 α X j P(x) = x aj x (13) j=0 Substituting and equating powers of xj , gives the relation: (α + j)(α + j + 1) − `(` + 1) a = a : (14) j+2 (α + j + 2)(α + j + 1) j Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions In addition, either a0 = 0, or α(α − 1) = 0, or a1 = 0 or (α + 1)α = 0. As a result, the series consists either of even terms, starting with x0, or odd terms starting with x1. For j large, a j+2 ! 1; (15) aj 1 this is the same as the series for 1−x2 , and so diverges as x ! 1. In order that the series converge over the whole angular range, we require that the series terminates, which occurs if ` is a positive integer. Exercise: Verify the recursion relation above. With the convention that P(1) = 1, determine the first three Legendre polynomials. Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Properties of the Legendre Polynomials ` P`(x) = (−1) P`(x): (16) Like eik·x in Fourier series, a complete, orthogonal set. Orthogonality: Multiply legendre’s equation by P`0 and integrate over x: Z 0 d 2 dP` dx P 0 (1 − x ) + `(` + 1)P = 0: (17) ` dx dx ` Integrate by parts; then subtract the same equation with ` $ `0. The surface term vanishes since 1 − x2 = 0. Then, for ` 6= `0, Z 1 dxP`(x)P`0 (x) = 0: (18) −1 We will do the case ` = `0 (normalization) shortly. Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Generating Function for the Legendre Polynomials Just what is says: a way of generating the Legendre functions explicitly, without directly solving the differential equation. Start with the fact that 1 G(~x; ~x 0) = (19) j~x − ~x 0j solves the laplace equation, the angular part of which (ignoring φ) is Legendre’s equation. For r 0 < r, expand: 1 1 1 1 = = (20) j~x − ~x 0j (r 2 + r 02 − 2rr 0 cos θ)1=2 r r 02 r 0 1=2 (1 + r 2 − 2 r cos θ) 1 1 X r 0 = ( )`A (cos θ): r r ` `=0 Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Note that the first term is the function u(r). Claim is that A` = P`. Note for starters, that if A` satisfies the the condition A`(1) = 1: The differential equation, in fact, follows by substituting the expansion. For any fixed `, because u` satisfies our previous equation, A` must satisfy the P` equation. The expression, indeed, must hold for all θ and r, so it must hold term by term. Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions We can summarize: 1 1 X ` g(t; x) = = P`(x)t : (21) (1 − 2xt + t2)1=2 `=0 We can extract explicit form for the P`’s by extracting coefficient of t`: E.g. P0 = 1; P1 = x (22) We can extract explicit properties of P`’s by manipulating g(t; x). Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Normalization integral: Z 1 Z 1 1 1 1 X X ` `0 dx = ( )( )P (x)P 0 (x)t t : (23) 1 − 2tx + t2 ` ` −1 −1 `=0 `0=0 Right hand side is Z 1 X 2 2` P`(x) t : (24) −1 Left hand side is elementary: 1 1 + t ln( ): (25) t 1 − t Expand in powers of t, using 1 X (−1)n+1 log(1 + a) = an: (26) n n=1 Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions So we have 1 1 + t 2 t2n+1 log = : (27) t 1 − t t 2n + 1 So Z 1 2 2 dxPn dx = : (28) −1 2n + 1 Exercise: Differentiate g with respect to t and derive the recursion relation: (2n + 1)xPn − (n + 1)Pn+1 − nPn−1 = 0: (29) Similarly, differentiate with respect to x: dP dP `+1 − `−1 − (2` + 1)P = 0: (30) dx dx ` Other recursion formulas, integral formulas, can be derived similarly (see, e.g., Arfken and Weber). Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions With these ingredients, we can expand any function of cos(θ) (i.e. functions which are single-valued and otherwise well behaved as functions of θ) in terms of the Legendre polynomials. X f (cos θ) = a`P`(cos θ) (31) with r 2` + 1 Z 1 a` = f (x)P`(x): (32) 2 −1 Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Associated Legendre Functions Recall the more general differential equation: m2 (1 − x2)v 00 − 2xv 0 + `(` + 1) − v = 0: (33) 1 − x2 This equation is solved by the associated Legendre functions: d m Pm(x) = (−1)m(1 − x2)m=2 P (x): (34) ` dxm ` The constant in front is conventional. That this solves the equation is shown by repeatedly differentiating Legendre’s equation. Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Exercise: Prove that the functions in eqn. 34 solve the full equation for positive m by repeated differentiation. For negative m, argue that the same is true by arguing that −m(x) m P` / P` (x): (35) Not exercise: The normalization integral for the associated Legendre functions is: Z 1 m 2 (` + m)! P`0 (x)P` (x)dx = δ``0 : (36) −1 2` + 1 (` − m)! Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Spherical Harmonics Solve the full θ, φ equations. Convention is to normalize to unity when integrated over the sphere. s 2` + 1 (` − m)! Y (θ; φ) = Pm(cos θ)eimφ: (37) `m 4π (` + m)! ` Very useful is the addition theorem: 1 ` ! 1 X X 1 r ` = 4π < Y ∗ (θ0; φ0)Y (θ; φ): j~x − ~x0j 2` + 1 `+1 `m `m `=0 m=−` r> (38) Physics 212 2010, Electricity and Magnetism Special Functions: Legendre functions, Spherical Harmonics, and Bessel Functions Bessel Functions These arise frequently in problems with cylindrical symmetry.
Load more

Recommended publications
  • 32 FA15 Abstracts
    32 FA15 Abstracts IP1 metric simple exclusion process and the KPZ equation. In Vector-Valued Nonsymmetric and Symmetric Jack addition, the experiments of Takeuchi and Sano will be and Macdonald Polynomials briefly discussed. For each partition τ of N there are irreducible representa- Craig A. Tracy tions of the symmetric group SN and the associated Hecke University of California, Davis algebra HN (q) on a real vector space Vτ whose basis is [email protected] indexed by the set of reverse standard Young tableaux of shape τ. The talk concerns orthogonal bases of Vτ -valued polynomials of x ∈ RN . The bases consist of polyno- IP6 mials which are simultaneous eigenfunctions of commuta- Limits of Orthogonal Polynomials and Contrac- tive algebras of differential-difference operators, which are tions of Lie Algebras parametrized by κ and (q, t) respectively. These polynomi- als reduce to the ordinary Jack and Macdonald polynomials In this talk, I will discuss the connection between superin- when the partition has just one part (N). The polynomi- tegrable systems and classical systems of orthogonal poly- als are constructed by means of the Yang-Baxter graph. nomials in particular in the expansion coefficients between There is a natural bilinear form, which is positive-definite separable coordinate systems, related to representations of for certain ranges of parameter values depending on τ,and the (quadratic) symmetry algebras. This connection al- there are integral kernels related to the bilinear form for lows us to extend the Askey scheme of classical orthogonal the group case, of Gaussian and of torus type. The mate- polynomials and the limiting processes within the scheme.
    [Show full text]
  • Orthogonal Functions: the Legendre, Laguerre, and Hermite Polynomials
    ORTHOGONAL FUNCTIONS: THE LEGENDRE, LAGUERRE, AND HERMITE POLYNOMIALS THOMAS COVERSON, SAVARNIK DIXIT, ALYSHA HARBOUR, AND TYLER OTTO Abstract. The Legendre, Laguerre, and Hermite equations are all homogeneous second order Sturm-Liouville equations. Using the Sturm-Liouville Theory we will be able to show that polynomial solutions to these equations are orthogonal. In a more general context, finding that these solutions are orthogonal allows us to write a function as a Fourier series with respect to these solutions. 1. Introduction The Legendre, Laguerre, and Hermite equations have many real world practical uses which we will not discuss here. We will only focus on the methods of solution and use in a mathematical sense. In solving these equations explicit solutions cannot be found. That is solutions in in terms of elementary functions cannot be found. In many cases it is easier to find a numerical or series solution. There is a generalized Fourier series theory which allows one to write a function f(x) as a linear combination of an orthogonal system of functions φ1(x),φ2(x),...,φn(x),... on [a; b]. The series produced is called the Fourier series with respect to the orthogonal system. While the R b a f(x)φn(x)dx coefficients ,which can be determined by the formula cn = R b 2 , a φn(x)dx are called the Fourier coefficients with respect to the orthogonal system. We are concerned only with showing that the Legendre, Laguerre, and Hermite polynomial solutions are orthogonal and can thus be used to form a Fourier series. In order to proceed we must define an inner product and define what it means for a linear operator to be self- adjoint.
    [Show full text]
  • Generalizations and Specializations of Generating Functions for Jacobi, Gegenbauer, Chebyshev and Legendre Polynomials with Definite Integrals
    Journal of Classical Analysis Volume 3, Number 1 (2013), 17–33 doi:10.7153/jca-03-02 GENERALIZATIONS AND SPECIALIZATIONS OF GENERATING FUNCTIONS FOR JACOBI, GEGENBAUER, CHEBYSHEV AND LEGENDRE POLYNOMIALS WITH DEFINITE INTEGRALS HOWARD S. COHL AND CONNOR MACKENZIE Abstract. In this paper we generalize and specialize generating functions for classical orthogo- nal polynomials, namely Jacobi, Gegenbauer, Chebyshev and Legendre polynomials. We derive a generalization of the generating function for Gegenbauer polynomials through extension a two element sequence of generating functions for Jacobi polynomials. Specializations of generat- ing functions are accomplished through the re-expression of Gauss hypergeometric functions in terms of less general functions. Definite integrals which correspond to the presented orthogonal polynomial series expansions are also given. 1. Introduction This paper concerns itself with analysis of generating functions for Jacobi, Gegen- bauer, Chebyshev and Legendre polynomials involving generalization and specializa- tion by re-expression of Gauss hypergeometric generating functions for these orthog- onal polynomials. The generalizations that we present here are for two of the most important generating functions for Jacobi polynomials, namely [4, (4.3.1–2)].1 In fact, these are the first two generating functions which appear in Section 4.3 of [4]. As we will show, these two generating functions, traditionally expressed in terms of Gauss hy- pergeometric functions, can be re-expressed in terms of associated Legendre functions (and also in terms of Ferrers functions, associated Legendre functions on the real seg- ment ( 1,1)). Our Jacobi polynomial generating function generalizations, Theorem 1, Corollary− 1 and Corollary 2, generalize the generating function for Gegenbauer polyno- mials.
    [Show full text]
  • Arxiv:2008.08079V2 [Math.FA] 29 Dec 2020 Hypergroups Is Not Required)
    HARMONIC ANALYSIS OF LITTLE q-LEGENDRE POLYNOMIALS STEFAN KAHLER Abstract. Many classes of orthogonal polynomials satisfy a specific linearization prop- erty giving rise to a polynomial hypergroup structure, which offers an elegant and fruitful link to harmonic and functional analysis. From the opposite point of view, this allows regarding certain Banach algebras as L1-algebras, associated with underlying orthogonal polynomials or with the corresponding orthogonalization measures. The individual be- havior strongly depends on these underlying polynomials. We study the little q-Legendre polynomials, which are orthogonal with respect to a discrete measure. Their L1-algebras have been known to be not amenable but to satisfy some weaker properties like right character amenability. We will show that the L1-algebras associated with the little q- Legendre polynomials share the property that every element can be approximated by linear combinations of idempotents. This particularly implies that these L1-algebras are weakly amenable (i. e., every bounded derivation into the dual module is an inner deriva- tion), which is known to be shared by any L1-algebra of a locally compact group. As a crucial tool, we establish certain uniform boundedness properties of the characters. Our strategy relies on continued fractions, character estimations and asymptotic behavior. 1. Introduction 1.1. Motivation. One of the most famous results of mathematics, the ‘Banach–Tarski paradox’, states that any ball in d ≥ 3 dimensions can be split into a finite number of pieces in such a way that these pieces can be reassembled into two balls of the original size. It is also well-known that there is no analogue for d 2 f1; 2g, and the Banach–Tarski paradox heavily relies on the axiom of choice [37].
    [Show full text]
  • Shifted Jacobi Polynomials and Delannoy Numbers
    SHIFTED JACOBI POLYNOMIALS AND DELANNOY NUMBERS GABOR´ HETYEI A` la m´emoire de Pierre Leroux Abstract. We express a weigthed generalization of the Delannoy numbers in terms of shifted Jacobi polynomials. A specialization of our formulas extends a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago [8], [13], [14], to all Delannoy numbers and certain Jacobi polynomials. Another specializa- tion provides a weighted lattice path enumeration model for shifted Jacobi polynomials and allows the presentation of a combinatorial, non-inductive proof of the orthogonality of Jacobi polynomials with natural number parameters. The proof relates the orthogo- nality of these polynomials to the orthogonality of (generalized) Laguerre polynomials, as they arise in the theory of rook polynomials. We also find finite orthogonal polynomial sequences of Jacobi polynomials with negative integer parameters and expressions for a weighted generalization of the Schr¨odernumbers in terms of the Jacobi polynomials. Introduction It has been noted more than fifty years ago [8], [13], [14] that the diagonal entries of the Delannoy array (dm,n), introduced by Henri Delannoy [5], and the Legendre polynomials Pn(x) satisfy the equality (1) dn,n = Pn(3), but this relation was mostly considered a “coincidence”. An important observation of our present work is that (1) can be extended to (α,0) (2) dn+α,n = Pn (3) for all α ∈ Z such that α ≥ −n, (α,0) where Pn (x) is the Jacobi polynomial with parameters (α, 0). This observation in itself is a strong indication that the interaction between Jacobi polynomials (generalizing Le- gendre polynomials) and the Delannoy numbers is more than a mere coincidence.
    [Show full text]
  • Algorithms for Classical Orthogonal Polynomials
    Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustr. 7, D-14195 Berlin - Dahlem Wolfram Ko epf Dieter Schmersau Algorithms for Classical Orthogonal Polynomials at Berlin Fachb ereich Mathematik und Informatik der Freien Universit Preprint SC Septemb er Algorithms for Classical Orthogonal Polynomials Wolfram Ko epf Dieter Schmersau koepfzibde Abstract In this article explicit formulas for the recurrence equation p x A x B p x C p x n+1 n n n n n1 and the derivative rules 0 x p x p x p x p x n n+1 n n n n1 n and 0 p x p x x p x x n n n n1 n n resp ectively which are valid for the orthogonal p olynomial solutions p x of the dierential n equation 00 0 x y x x y x y x n of hyp ergeometric typ e are develop ed that dep end only on the co ecients x and x which themselves are p olynomials wrt x of degrees not larger than and resp ectively Partial solutions of this problem had b een previously published by Tricomi and recently by Yanez Dehesa and Nikiforov Our formulas yield an algorithm with which it can b e decided whether a given holonomic recur rence equation ie one with p olynomial co ecients generates a family of classical orthogonal p olynomials and returns the corresp onding data density function interval including the stan dardization data in the armative case In a similar way explicit formulas for the co ecients of the recurrence equation and the dierence rule x rp x p x p x p x n n n+1 n n n n1 of the classical orthogonal p olynomials of a discrete variable are given that dep end only
    [Show full text]
  • A Uniqueness Theorem for the Legendre and Hermite
    A UNIQUENESSTHEOREM FOR THELEGENDRE ANDHERMITE POLYNOMIALS* BY K. P. WILLIAMS 1. If we replace y in the expansion of (l-\-y)~v by 2xz-\-z2, the coefficient of zn will, when x is replaced by —x, be the generalized polynomial L'ñ(x) of Legendre. It is also easy to show that the Hermitian polynomial Hn(x), usually defined by is the coefficient of 2"/«! in the series obtained on replacing y in the expansion of e~y by the same expression 2xz-\-z2. Furthermore, there is a simple recursion formula between three successive Legrendre polynomials and between three successive Hermitian polynomials. These facts suggest the following problem. Let a> ftg <p(y) = «o + «i 2/ +y,-2/2+3y ¡y3+ ••• and put (p(2xz 4z2) = P„ + Px (x) t + Ps(x) z2+---. To what extent is the generating function q>(y) determined if it is known that a simple recursion relation exists between three of the successive polynomials P0, Pi(x), P%(x), •••? We shall find that the generalized Legendre polynomials and those of Hermite possess a certain uniqueness in this regard. 2. We have PÁX) = «! 7i«íp(2!rí + £,).._0- When we make use of the formula for the «th derivative of a function of a function given by Faà de Bruno,t we find without difficulty P»(x) = Zjtt^xy, 'Presented to the Society, October 25, 1924. "¡■Quarterly Journal of Mathematics, vol. 1, p. 359. 441 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 442 K- p- WILLIAMS [October where the summation extends to all values of i and j subject to the relation i + 2j = n.
    [Show full text]
  • ME201/MTH281/ME400/CHE400 Legendre Polynomials
    ME201/MTH281/ME400/CHE400 Legendre Polynomials 1. Introduction This notebook has three objectives: (1) to summarize some useful information about Legendre polynomials, (2) to show how to use Mathematica in calculations with Legendre polynomials, and (3) to present some examples of the use of Legendre polynomials in the solution of Laplace's equation in spherical coordinates. In our course, the Legendre polynomials arose from separation of variables for the Laplace equation in spherical coordinates, so we begin there. The basic spherical coordinate system is shown below. The location of a point P is specified by the distance r of the point from the origin, the angle f between the position vector and the z-axis, and the angle q from the x-axis to the projection of the position vector onto the xy plane. The Laplace equation for a function F(r, f, q) is given by 1 ¶ ¶F 1 ¶ ¶F 1 ¶2 F “2 F = r2 + sinf + = 0 . (1) r2 ¶r ¶r r2 sinf ¶f ¶f r2 sin2 f ¶q2 In this notebook, we will consider only axisymmetric solutions of (1) -- that is, solutions which depend on r and f but not on q. Then equation (1) reduces to 1 ¶ ¶F 1 ¶ ¶F r2 + sinf = 0 . (2) r2 ¶r ¶r r2 sinf ¶f ¶f As we showed in class by a rather lengthy analysis, equation (2) has separated solutions of the form 2 legendre.nb n -Hn+1L r Pn HcosfL and r Pn HcosfL , (3) where n is a non-negative integer and Pn is the nth Legendre polynomial.
    [Show full text]
  • Orthogonal Polynomials and Classical Orthogonal Polynomials
    International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue 10, October 2018, pp. 1613–1630, Article ID: IJMET_09_10_164 Available online at http://iaeme.com/Home/issue/IJMET?Volume=9&Issue=10 ISSN Print: 0976-6340 and ISSN Online: 0976-6359 © IAEME Publication Scopus Indexed ORTHOGONAL POLYNOMIALS AND CLASSICAL ORTHOGONAL POLYNOMIALS DUNIA ALAWAI JARWAN Education for Girls College, Al-Anbar University, Ministry of Higher Education and Scientific Research, Iraq ABSTRACT The focus of this project is to clarify the concept of orthogonal polynomials in the case of continuous internal and discrete points on R and the Gram – Schmidt orthogonalization process of conversion to many orthogonal limits and the characteristics of this method. We have highlighted the classical orthogonal polynomials as an example of orthogonal polynomials because of they are great importance in physical practical applications. In this project, we present 3 types (Hermite – Laguerre – Jacobi) of classical orthogonal polynomials by clarifying the different formulas of each type and how to reach some formulas, especially the form of the orthogonality relation of each. Keywords: Polynomials, Classical Orthogonal, Monic Polynomial, Gram – Schmidt Cite this Article Dunia Alawai Jarwan, Orthogonal Polynomials and Classical Orthogonal Polynomials, International Journal of Mechanical Engineering and Technology, 9(10), 2018, pp. 1613–1630. http://iaeme.com/Home/issue/IJMET?Volume=9&Issue=10 1. INTRODUCTION The mathematics is the branch where the lots of concepts are included. An orthogonality is the one of the concept among them. Here we focuse on the orthogonal polynomial sequence. The orthogonal polynomial are divided in two classes i.e. classical orthogonal polynomials, Discrete orthogonal polynomials and Sieved orthogonal polynomials .There are different types of classical orthogonal polynomials such that Jacobi polynomials, Associated Laguerre polynomials and Hermite polynomials.
    [Show full text]
  • Orthogonal Polynomials: an Illustrated Guide
    Orthogonal Polynomials: An Illustrated Guide Avaneesh Narla December 10, 2018 Contents 1 Definitions 1 2 Example 1 2 3 Three-term Recurrence Relation 3 4 Christoffel-Darboux Formula 5 5 Zeros 6 6 Gauss Quadrature 8 6.1 Lagrange Interpolation . .8 6.2 Gauss quadrature formula . .8 7 Classical Orthogonal Polynomials 11 7.1 Hermite Polynomials . 11 7.2 Laguerre Polynomials . 12 7.3 Legendre Polynomials . 14 7.4 Jacobi Polynomials . 16 7.5 Chebyshev Polynomials of the First Kind . 17 7.6 Chebyshev Polynomials of the Second Kind . 19 7.7 Gegenbauer polynomials . 20 1 Definitions Orthogonal polynomials are orthogonal with respect to a certain function, known as the weight function w(x), and a defined interval. The weight function must be continuous and positive such that its moments (µn) exist. Z b n µn := w(x)x dx; n = 0; 1; 2; ::: a The interval may be infinite. We now define the inner product of two polynomials as follows Z 1 hf; giw(x) := w(x)f(x)g(x) dx −∞ 1 We will drop the subscript indicating the weight function in future cases. Thus, as always, a 1 sequence of polynomials fpn(x)gn=0 with deg(pn(x)) = n are called orthogonal polynomials for a weight function w if hpm; pni = hnδmn Above, the delta function is the Kronecker Delta Function There are two possible normalisations: If hn = 1 8n 2 f0; 1; 2:::g, the sequence is orthonormal. If the coefficient of highest degree term is 1 for all elements in the sequence, the sequence is monic.
    [Show full text]
  • On the Interplay Among Hypergeometric Functions, Complete
    On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier–Legendre expansions a, b c John M. Campbell ∗, Jacopo D’Aurizio , Jonathan Sondow aDepartment of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3 bDepartment of Mathematics, University of Pisa, Italy, Largo Bruno Pontecorvo 5 - 56127 c209 West 97th Street, New York, NY 10025 Abstract Motivated by our previous work on hypergeometric functions and the parbelos constant, we perform a deeper investigation on the interplay among general- ized complete elliptic integrals, Fourier–Legendre (FL) series expansions, and pFq series. We produce new hypergeometric transformations and closed-form evaluations for new series involving harmonic numbers, through the use of the integration method outlined as follows: Letting K denote the complete elliptic integral of the first kind, for a suitable function g we evaluate integrals such as 1 K √x g(x) dx Z0 in two different ways: (1) by expanding K as a Maclaurin series, perhaps after a transformation or a change of variable, and then integrating term-by-term; and (2) by expanding g as a shifted FL series, and then integrating term-by-term. Equating the expressions produced by these two approaches often gives us new arXiv:1710.03221v4 [math.NT] 13 Feb 2019 closed-form evaluations, as in the formulas involving Catalan’s constant G 2 4 1 ∞ 2n Hn+ 1 Hn 1 Γ 4G 4 − − 4 = 4 , n 16n 8π2 − π n=0 X 2 2 2m 2n 7ζ(3) 4G m n = − . 16m+n(m + n + 1)(2m + 3) π2 m,n 0 X≥ ∗Corresponding author Email address: [email protected] (John M.
    [Show full text]
  • Legendre Polynomials and Applications
    LEGENDRE POLYNOMIALS AND APPLICATIONS We construct Legendre polynomials and apply them to solve Dirichlet problems in spherical coordinates. 1. Legendre equation: series solutions The Legendre equation is the second order differential equation (1) (1 − x2)y00 − 2xy0 + λy = 0 which can also be written in self-adjoint form as [ ]0 (2) (1 − x2)y0 + λy = 0 : This equation has regular singular points at x = −1 and at x = 1 while x = 0 is an ordinary point. We can find power series solutions centered at x = 0 (the radius of convergence will be at least 1). Now we construct such series solutions. Assume that X1 j y = cjx j=0 be a solution of (1). By substituting such a series in (1), we get X1 X1 X1 2 j−2 j−1 j (1 − x ) j(j − 1)cjx − 2x jcjx + λ cjx = 0 j=2 j=1 j=0 X1 X1 X1 X1 j−2 j j j j(j − 1)cjx − j(j − 1)cjx − 2jcjx + λ cjx = 0 j=2 j=2 j=1 j=0 After re-indexing the first series and grouping the other series, we get X1 X1 j 2 j (j + 2)(j + 1)cj+2x − (j + j − λ)cjx = 0 j=0 j=0 and then X1 [ ] 2 j (j + 2)(j + 1)cj+2 − (j + j − λ)cj x = 0 : j=0 By equating each coefficient to 0, we obtain the recurrence relations (j + 1)j − λ c = c ; j = 0; 1; 2; ··· j+2 (j + 2)(j + 1) j We can obtain two independent solutions as follows.
    [Show full text]