Lecture 19: Ordinary Differential Equations: Special Functions
|
|
- Christopher Blankenship
- 5 years ago
- Views:
Transcription
1 Lecture 19: Ordinary Differential Equations: Special Functions Key points Hermite differential equation: Legendre's differential equation: Bessel's differential equation: Modified Bessel differential equation: Spherical Bessel differential equation: Airy differential equation: Laguerre differential equation: Maple HermiteH(n,x) LegendreP(n,x), LegendreQ(n,x) BesselJ(n,x), BesselY(n,x) HankelH1(n,x), HankelH2(n,x) BesselI(n,x), BesselK(n,x) AiryAi(x), AiryBi(x) LaguerreL(n,x) KummerM(n,x), KummerU(n,x) Hermite equation Hermite differential equation General solution by Maple where and are the Kummer functions. The Kummer functions are also called confluent hypergeometric functions. In Maple, they are predifined functions, and. (2.1) The two independent solutions for the Hermite differential equation is and
2 For integer, Hermite polynomial. is a solution to the Hermite differential equation.. For, (2.3) In Maple, Hermite polynomials are predefined as HermiteH(n,x) The first few Hermite polynomials are: 1
3 The Hermite polynomials Hermite equation. determined by the following recursive relation are solution to the Orthogonality The second solution to the Hermite equation is the second kind Hermite function which exponentially diverges as. Since it is rarely used in physics, we don't discuss it here. Legendre equation Legendre's differential equation of degree n (0th order)
4 General Solution (3.1) For, (3.2) Two linearly independent solutions to this ODE is known as the first kind of Legendre polynomials and the second kind of Legendre function. is not popular in Physics because it is defined for and. (It is possible to extend to but it is not our interest.) First kind Legendre polynomials In Maple, Legendre polynomials are predefined as. = 1 = x Note that diverges logarithmically at and.
5 Orthogonality forms an orthonormal basis set for. Recursive equation General Legendre equation Legendre's differential equation
6 where and are integers and. (mathematically speaking non-integer values are allowed but not popular in physics.) Associate Legendre functions, are solution to the general Legendre differential equation. 1 Solution by Maple
7 (4.1) (4.2) Bessel equation Bessel's differential equation General solution by Maple (5.1) Two linearly independent solutions are known as Bessel function, and Weber function. The second solution is also called Neumann function and denoted as. Hankel functions independent solutions. Even for integer, there is no simplex expression: For, are also a pair of linearly (5.2) In Maple, these functions are predefined as BesselJ(n,x), BesselY(n,x), BesselH1(n,x), and BesselH2 (n,x). These functions can be expressed only with infinite series (Maple cannot express them in simple forms but you can evaluate numerical values with Maple.)
8 Bessel functions are not orthogonal! Bessel function Weber function Modified Bessel equation Modified Bessel differential equation General solution by Maple Two linearly independent solutions are the first kind and second kind of modified Bessel functions, and, respectively. (6.1) In Maple, the modified Bessel functions are predefined as BesselI(n,x) and BesselK(n,x). There is no
9 simple expression of the modified Bessel functions even for integer. (6.2) The modified Bessel functions are related to the regular Bessel functions as follows: In Maple, the modified Bessel functions are predefined as BesselI(n,x) and BesselK(n,x). There is no simplex expression for these functions. Spherical Bessel equation Spherical Bessel differential equation General Solution
10 (7.1) Two linearly independent solutions are spherical Bessel functions: Although there is no simple expression of Bessel functions in general, the spherical Bessel functions can be written in simple closed form: For, (7.2) For general integer, Spherical Bessel functions can be expressed in simple form. For example, symbolic x symbolic x 2
11 symbolic symbolic Note that the spherical Neumann functions diverge at. Spherical Bessel Spherical Neumann function Airy equation Airy differential equation General Solution
12 Two linearly independent solutions are the first and second kind of Airy functions, Ai(x) and Bi(x), respectively. They are related to modified Bessel functions as follows: (8.1) Since, the second term is usually eliminated by physical boundary condition. In Maple, the Airy functions are predefined as AiryAi(x) and AiryBi(x). 1st kind of Airy function, Ai(x) 2nd kind of Airy function, Bi(x) Laguerre equation Laguerre differential equation General Solution The Kummer functions are the two independent solutions for the Laguerre equation. (9.1)
13 For, (9.2) Similar to the Hermite differential equation, the general solution to the Laguerre equation is linear combination of Kummer functions. This particular Kummer function, has a special name, Laguerre function which can be expressed in simple form when is integer. In Maple, Laguerre polynomials are predefined as LaguerreL(n,x) The first few Laguerre polynomials are: 1
14 Orthogonality forms an orthonormal basis set for :
Lecture 16: Special Functions. In Maple, Hermite polynomials are predefined as HermiteH(n,x) The first few Hermite polynomials are: simplify.
Lecture 16: Special Functions 1. Key points Hermite differential equation: Legendre's differential equation: Bessel's differential equation: Modified Bessel differential equation: Spherical Bessel differential
More informationLecture 4: Series expansions with Special functions
Lecture 4: Series expansions with Special functions 1. Key points 1. Legedre polynomials (Legendre functions) 2. Hermite polynomials(hermite functions) 3. Laguerre polynomials (Laguerre functions) Other
More informationINTEGRAL TRANSFORMS and THEIR APPLICATIONS
INTEGRAL TRANSFORMS and THEIR APPLICATIONS Lokenath Debnath Professor and Chair of Mathematics and Professor of Mechanical and Aerospace Engineering University of Central Florida Orlando, Florida CRC Press
More informationSPECIAL FUNCTIONS OF MATHEMATICS FOR ENGINEERS
SPECIAL FUNCTIONS OF MATHEMATICS FOR ENGINEERS Second Edition LARRY C. ANDREWS OXFORD UNIVERSITY PRESS OXFORD TOKYO MELBOURNE SPIE OPTICAL ENGINEERING PRESS A Publication of SPIE The International Society
More informationwhich implies that we can take solutions which are simultaneous eigen functions of
Module 1 : Quantum Mechanics Chapter 6 : Quantum mechanics in 3-D Quantum mechanics in 3-D For most physical systems, the dynamics is in 3-D. The solutions to the general 3-d problem are quite complicated,
More information1 Solutions in cylindrical coordinates: Bessel functions
1 Solutions in cylindrical coordinates: Bessel functions 1.1 Bessel functions Bessel functions arise as solutions of potential problems in cylindrical coordinates. Laplace s equation in cylindrical coordinates
More informationSpecial Functions of Mathematical Physics
Arnold F. Nikiforov Vasilii B. Uvarov Special Functions of Mathematical Physics A Unified Introduction with Applications Translated from the Russian by Ralph P. Boas 1988 Birkhäuser Basel Boston Table
More informationLecture 4.6: Some special orthogonal functions
Lecture 4.6: Some special orthogonal functions Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4340, Advanced Engineering Mathematics
More informationarxiv:math/ v1 [math.ca] 19 Apr 1994
arxiv:math/9404220v1 [math.ca] 19 Apr 1994 Algorithmic Work with Orthogonal Polynomials and Special Functions Wolfram Koepf Konrad-Zuse-Zentrum für Informationstechnik Berlin, Heilbronner Str. 10, D-10711
More informationBessel s and legendre s equations
Chapter 12 Bessel s and legendre s equations 12.1 Introduction Many linear differential equations having variable coefficients cannot be solved by usual methods and we need to employ series solution method
More informationModified Bessel functions : Iα, Kα
Modified Bessel functions : Iα, Kα The Bessel functions are valid even for complex arguments x, and an important special case is that of a purely imaginary argument. In this case, the solutions to the
More informationMATH 543: FUCHSIAN DIFFERENTIAL EQUATIONS HYPERGEOMETRIC FUNCTION
SET 7 MATH 543: FUCHSIAN DIFFERENTIAL EQUATIONS HYERGEOMETRIC FUNCTION References: DK and Sadri Hassan. Historical Notes: lease read the book Linear Differential Equations and the Group Theory by Jeremy
More informationPARTIAL DIFFERENTIAL EQUATIONS and BOUNDARY VALUE PROBLEMS
PARTIAL DIFFERENTIAL EQUATIONS and BOUNDARY VALUE PROBLEMS NAKHLE H. ASMAR University of Missouri PRENTICE HALL, Upper Saddle River, New Jersey 07458 Contents Preface vii A Preview of Applications and
More informationIndex. for Ɣ(a, z), 39. convergent asymptotic representation, 46 converging factor, 40 exponentially improved, 39
Index Abramowitz function computed by Clenshaw s method, 74 absolute error, 356 Airy function contour integral for, 166 Airy functions algorithm, 359 asymptotic estimate of, 18 asymptotic expansions, 81,
More informationInner Product Spaces An inner product on a complex linear space X is a function x y from X X C such that. (1) (2) (3) x x > 0 for x 0.
Inner Product Spaces An inner product on a complex linear space X is a function x y from X X C such that (1) () () (4) x 1 + x y = x 1 y + x y y x = x y x αy = α x y x x > 0 for x 0 Consequently, (5) (6)
More information5.4 Bessel s Equation. Bessel Functions
SEC 54 Bessel s Equation Bessel Functions J (x) 87 # with y dy>dt, etc, constant A, B, C, D, K, and t 5 HYPERGEOMETRIC ODE At B (t t )(t t ), t t, can be reduced to the hypergeometric equation with independent
More informationPHYS 404 Lecture 1: Legendre Functions
PHYS 404 Lecture 1: Legendre Functions Dr. Vasileios Lempesis PHYS 404 - LECTURE 1 DR. V. LEMPESIS 1 Legendre Functions physical justification Legendre functions or Legendre polynomials are the solutions
More informationChapter 5.3: Series solution near an ordinary point
Chapter 5.3: Series solution near an ordinary point We continue to study ODE s with polynomial coefficients of the form: P (x)y + Q(x)y + R(x)y = 0. Recall that x 0 is an ordinary point if P (x 0 ) 0.
More informationMA22S3 Summary Sheet: Ordinary Differential Equations
MA22S3 Summary Sheet: Ordinary Differential Equations December 14, 2017 Kreyszig s textbook is a suitable guide for this part of the module. Contents 1 Terminology 1 2 First order separable 2 2.1 Separable
More informationContents. I Basic Methods 13
Preface xiii 1 Introduction 1 I Basic Methods 13 2 Convergent and Divergent Series 15 2.1 Introduction... 15 2.1.1 Power series: First steps... 15 2.1.2 Further practical aspects... 17 2.2 Differential
More information4 Power Series Solutions: Frobenius Method
4 Power Series Solutions: Frobenius Method Now the ODE adventure takes us to series solutions for ODEs, a technique A & W, that is often viable, valuable and informative. These can be readily applied Sec.
More informationGENERAL CONSIDERATIONS 1 What functions are. Organization of the Atlas. Notational conventions. Rules of the calculus.
Every chapter has sections devoted to: notation, behavior, definitions, special cases, intrarelationships, expansions, particular values, numerical values, limits and approximations, operations of the
More informationThis ODE arises in many physical systems that we shall investigate. + ( + 1)u = 0. (λ + s)x λ + s + ( + 1) a λ. (s + 1)(s + 2) a 0
Legendre equation This ODE arises in many physical systems that we shall investigate We choose We then have Substitution gives ( x 2 ) d 2 u du 2x 2 dx dx + ( + )u u x s a λ x λ a du dx λ a λ (λ + s)x
More informationPhysics 6303 Lecture 11 September 24, LAST TIME: Cylindrical coordinates, spherical coordinates, and Legendre s equation
Physics 6303 Lecture September 24, 208 LAST TIME: Cylindrical coordinates, spherical coordinates, and Legendre s equation, l l l l l l. Consider problems that are no axisymmetric; i.e., the potential depends
More information1954] BOOK REVIEWS 185
1954] BOOK REVIEWS 185 = i*(7r w +i(i n )) where i:k n ~*K n+l is the identity map. When, in addition, ir n +i(k)=0 and n>3, T n + 2 (K) is computed to be H n {K)/2H n {K). This is equivalent to the statement
More informationElectromagnetism HW 1 math review
Electromagnetism HW math review Problems -5 due Mon 7th Sep, 6- due Mon 4th Sep Exercise. The Levi-Civita symbol, ɛ ijk, also known as the completely antisymmetric rank-3 tensor, has the following properties:
More informationMathematics for Chemistry: Exam Set 1
Mathematics for Chemistry: Exam Set 1 June 18, 017 1 mark Questions 1. The minimum value of the rank of any 5 3 matrix is 0 1 3. The trace of an identity n n matrix is equal to 1-1 0 n 3. A square matrix
More information1 angle.mcd Inner product and angle between two vectors. Note that a function is a special case of a vector. Instructor: Nam Sun Wang. x.
angle.mcd Inner product and angle between two vectors. Note that a function is a special case of a vector. Instructor: Nam Sun Wang Define angle between two vectors & y:. y. y. cos( ) (, y). y. y Projection
More informationList of mathematical functions
List of mathematical functions From Wikipedia, the free encyclopedia In mathematics, a function or groups of functions are important enough to deserve their own names. This is a listing of articles which
More informationSection 5.2 Series Solution Near Ordinary Point
DE Section 5.2 Series Solution Near Ordinary Point Page 1 of 5 Section 5.2 Series Solution Near Ordinary Point We are interested in second order homogeneous linear differential equations with variable
More information221A Lecture Notes Steepest Descent Method
Gamma Function A Lecture Notes Steepest Descent Method The best way to introduce the steepest descent method is to see an example. The Stirling s formula for the behavior of the factorial n! for large
More informationBessel function - Wikipedia, the free encyclopedia
Bessel function - Wikipedia, the free encyclopedia Bessel function Page 1 of 9 From Wikipedia, the free encyclopedia In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli
More informationDie Grundlehren der mathematischen Wissenschaften
Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Beriicksichtigung der Anwendungsgebiete Band 52 H erau.fgegeben von J. L. Doob. E. Heinz F. Hirzebruch. E. Hopf H.
More informationCourse Outline. Date Lecture Topic Reading
Course Outline Date Lecture Topic Reading Graduate Mathematical Physics Tue 24 Aug Linear Algebra: Theory 744 756 Vectors, bases and components Linear maps and dual vectors Inner products and adjoint operators
More informationThe Perrin Conjugate and the Laguerre Orthogonal Polynomial
The Perrin Conjugate and the Laguerre Orthogonal Polynomial In a previous chapter I defined the conjugate of a cubic polynomial G(x) = x 3 - Bx Cx - D as G(x)c = x 3 + Bx Cx + D. By multiplying the polynomial
More informationMathematics for Chemistry: Exam Set 1
Mathematics for Chemistry: Exam Set 1 March 19, 017 1 mark Questions 1. The maximum value of the rank of any 5 3 matrix is (a (b3 4 5. The determinant of an identity n n matrix is equal to (a 1 (b -1 0
More informationENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS. Special Functions GEORGE E. ANDREWS RICHARD ASKEY RANJAN ROY CAMBRIDGE UNIVERSITY PRESS
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS Special Functions GEORGE E. ANDREWS RICHARD ASKEY RANJAN ROY CAMBRIDGE UNIVERSITY PRESS Preface page xiii 1 The Gamma and Beta Functions 1 1.1 The Gamma
More informationAND NONLINEAR SCIENCE SERIES. Partial Differential. Equations with MATLAB. Matthew P. Coleman. CRC Press J Taylor & Francis Croup
CHAPMAN & HALL/CRC APPLIED MATHEMATICS AND NONLINEAR SCIENCE SERIES An Introduction to Partial Differential Equations with MATLAB Second Edition Matthew P Coleman Fairfield University Connecticut, USA»C)
More information221B Lecture Notes Steepest Descent Method
Gamma Function B Lecture Notes Steepest Descent Method The best way to introduce the steepest descent method is to see an example. The Stirling s formula for the behavior of the factorial n! for large
More informationFOURIER SERIES, TRANSFORMS, AND BOUNDARY VALUE PROBLEMS
fc FOURIER SERIES, TRANSFORMS, AND BOUNDARY VALUE PROBLEMS Second Edition J. RAY HANNA Professor Emeritus University of Wyoming Laramie, Wyoming JOHN H. ROWLAND Department of Mathematics and Department
More informationSolutions to Laplace s Equations- II
Solutions to Laplace s Equations- II Lecture 15: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay Laplace s Equation in Spherical Coordinates : In spherical coordinates
More informationwhich arises when we compute the orthogonal projection of a vector y in a subspace with an orthogonal basis. Hence assume that P y = A ij = x j, x i
MODULE 6 Topics: Gram-Schmidt orthogonalization process We begin by observing that if the vectors {x j } N are mutually orthogonal in an inner product space V then they are necessarily linearly independent.
More informationMATHEMATICS (MATH) Calendar
MATHEMATICS (MATH) This is a list of the Mathematics (MATH) courses available at KPU. For information about transfer of credit amongst institutions in B.C. and to see how individual courses transfer, go
More informationTyn Myint-U Lokenath Debnath. Linear Partial Differential Equations for Scientists and Engineers. Fourth Edition. Birkhauser Boston Basel Berlin
Tyn Myint-U Lokenath Debnath Linear Partial Differential Equations for Scientists and Engineers Fourth Edition Birkhauser Boston Basel Berlin Preface to the Fourth Edition Preface to the Third Edition
More informationMATHEMATICAL HANDBOOK. Formulas and Tables
SCHAUM'S OUTLINE SERIES MATHEMATICAL HANDBOOK of Formulas and Tables Second Edition MURRAY R. SPIEGEL, Ph.D. Former Professor and Chairman Mathematics Department Rensselaer Polytechnic Institute Hartford
More informationSpherical Harmonics on S 2
7 August 00 1 Spherical Harmonics on 1 The Laplace-Beltrami Operator In what follows, we describe points on using the parametrization x = cos ϕ sin θ, y = sin ϕ sin θ, z = cos θ, where θ is the colatitude
More informationA RECURSION FORMULA FOR THE COEFFICIENTS OF ENTIRE FUNCTIONS SATISFYING AN ODE WITH POLYNOMIAL COEFFICIENTS
Georgian Mathematical Journal Volume 11 (2004), Number 3, 409 414 A RECURSION FORMULA FOR THE COEFFICIENTS OF ENTIRE FUNCTIONS SATISFYING AN ODE WITH POLYNOMIAL COEFFICIENTS C. BELINGERI Abstract. A recursion
More informationLecture 20: ODE V - Examples in Physics
Lecture 20: ODE V - Examples in Physics Helmholtz oscillator The system. A particle of mass is moving in a potential field. Set up the equation of motion. (1.1) (1.2) (1.4) (1.5) Fixed points Linear stability
More informationProblem Set 8 Mar 5, 2004 Due Mar 10, 2004 ACM 95b/100b 3pm at Firestone 303 E. Sterl Phinney (2 pts) Include grading section number
Problem Set 8 Mar 5, 24 Due Mar 1, 24 ACM 95b/1b 3pm at Firestone 33 E. Sterl Phinney (2 pts) Include grading section number Useful Readings: For Green s functions, see class notes and refs on PS7 (esp
More informationRelevant sections from AMATH 351 Course Notes (Wainwright): Relevant sections from AMATH 351 Course Notes (Poulin and Ingalls):
Lecture 5 Series solutions to DEs Relevant sections from AMATH 35 Course Notes (Wainwright):.4. Relevant sections from AMATH 35 Course Notes (Poulin and Ingalls): 2.-2.3 As mentioned earlier in this course,
More informationSolutions of linear ordinary differential equations in terms of special functions
Solutions of linear ordinary differential equations in terms of special functions Manuel Bronstein ManuelBronstein@sophiainriafr INRIA Projet Café 004, Route des Lucioles, BP 93 F-0690 Sophia Antipolis
More informationIndex. Cambridge University Press Essential Mathematical Methods for the Physical Sciences K. F. Riley and M. P. Hobson.
absolute convergence of series, 547 acceleration vector, 88 addition rule for probabilities, 618, 623 addition theorem for spherical harmonics Yl m (θ,φ), 340 adjoint, see Hermitian conjugate adjoint operators,
More informationPower Series Solutions And Special Functions: Review of Power Series
Power Series Solutions And Special Functions: Review of Power Series Pradeep Boggarapu Department of Mathematics BITS PILANI K K Birla Goa Campus, Goa September, 205 Pradeep Boggarapu (Dept. of Maths)
More information14 EE 2402 Engineering Mathematics III Solutions to Tutorial 3 1. For n =0; 1; 2; 3; 4; 5 verify that P n (x) is a solution of Legendre's equation wit
EE 0 Engineering Mathematics III Solutions to Tutorial. For n =0; ; ; ; ; verify that P n (x) is a solution of Legendre's equation with ff = n. Solution: Recall the Legendre's equation from your text or
More informationPower series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) a n z n. n=0
Lecture 22 Power series solutions for 2nd order linear ODE s (not necessarily with constant coefficients) Recall a few facts about power series: a n z n This series in z is centered at z 0. Here z can
More informationPOCKETBOOKOF MATHEMATICAL FUNCTIONS
POCKETBOOKOF MATHEMATICAL FUNCTIONS Abridged edition of Handbook of Mathematical Functions Milton Abramowitz and Irene A. Stegun (eds.) Material selected by Michael Danos and Johann Rafelski 1984 Verlag
More informationMATHEMATICAL FORMULAS AND INTEGRALS
HANDBOOK OF MATHEMATICAL FORMULAS AND INTEGRALS Second Edition ALAN JEFFREY Department of Engineering Mathematics University of Newcastle upon Tyne Newcastle upon Tyne United Kingdom ACADEMIC PRESS A Harcourt
More informationChapter 1. Preliminaries. The purpose of this chapter is to provide some basic background information. Linear Space. Hilbert Space.
Chapter 1 Preliminaries The purpose of this chapter is to provide some basic background information. Linear Space Hilbert Space Basic Principles 1 2 Preliminaries Linear Space The notion of linear space
More informationClassroom Tips and Techniques: Eigenvalue Problems for ODEs - Part 1
Classroom Tips and Techniques: Eigenvalue Problems for DEs - Part 1 Initializations restart with LinearAlgebra : with Student Calculus1 : with RootFinding : Introduction Robert J. Lopez Emeritus Professor
More informationTwo special equations: Bessel s and Legendre s equations. p Fourier-Bessel and Fourier-Legendre series. p
LECTURE 1 Table of Contents Two special equations: Bessel s and Legendre s equations. p. 259-268. Fourier-Bessel and Fourier-Legendre series. p. 453-460. Boundary value problems in other coordinate system.
More informationCYK\2010\PH402\Mathematical Physics\Tutorial Find two linearly independent power series solutions of the equation.
CYK\010\PH40\Mathematical Physics\Tutorial 1. Find two linearly independent power series solutions of the equation For which values of x do the series converge?. Find a series solution for y xy + y = 0.
More informationNormalization integrals of orthogonal Heun functions
Normalization integrals of orthogonal Heun functions Peter A. Becker a) Center for Earth Observing and Space Research, Institute for Computational Sciences and Informatics, and Department of Physics and
More informationFredholm determinant with the confluent hypergeometric kernel
Fredholm determinant with the confluent hypergeometric kernel J. Vasylevska joint work with I. Krasovsky Brunel University Dec 19, 2008 Problem Statement Let K s be the integral operator acting on L 2
More informationSolutions: Problem Set 3 Math 201B, Winter 2007
Solutions: Problem Set 3 Math 201B, Winter 2007 Problem 1. Prove that an infinite-dimensional Hilbert space is a separable metric space if and only if it has a countable orthonormal basis. Solution. If
More informationPolynomial Solutions of the Laguerre Equation and Other Differential Equations Near a Singular
Polynomial Solutions of the Laguerre Equation and Other Differential Equations Near a Singular Point Abstract Lawrence E. Levine Ray Maleh Department of Mathematical Sciences Stevens Institute of Technology
More informationGAUSS-LAGUERRE AND GAUSS-HERMITE QUADRATURE ON 64, 96 AND 128 NODES
GAUSS-LAGUERRE AND GAUSS-HERMITE QUADRATURE ON 64, 96 AND 128 NODES RICHARD J. MATHAR Abstract. The manuscript provides tables of abscissae and weights for Gauss- Laguerre integration on 64, 96 and 128
More informationSeries Solutions of ODEs. Special Functions
C05.tex 6/4/0 3: 5 Page 65 Chap. 5 Series Solutions of ODEs. Special Functions We continue our studies of ODEs with Legendre s, Bessel s, and the hypergeometric equations. These ODEs have variable coefficients
More informationSecond-Order Linear ODEs (Textbook, Chap 2)
Second-Order Linear ODEs (Textbook, Chap ) Motivation Recall from notes, pp. 58-59, the second example of a DE that we introduced there. d φ 1 1 φ = φ 0 dx λ λ Q w ' (a1) This equation represents conservation
More informationSecond-order ordinary differential equations Special functions, Sturm-Liouville theory and transforms R.S. Johnson
Second-order ordinary differential equations Special functions, Sturm-Liouville theory and transforms R.S. Johnson R.S. Johnson Second-order ordinary differential equations Special functions, Sturm-Liouville
More informationThe Quantum Harmonic Oscillator
The Classical Analysis Recall the mass-spring system where we first introduced unforced harmonic motion. The DE that describes the system is: where: Note that throughout this discussion the variables =
More informationhomogeneous 71 hyperplane 10 hyperplane 34 hyperplane 69 identity map 171 identity map 186 identity map 206 identity matrix 110 identity matrix 45
address 12 adjoint matrix 118 alternating 112 alternating 203 angle 159 angle 33 angle 60 area 120 associative 180 augmented matrix 11 axes 5 Axiom of Choice 153 basis 178 basis 210 basis 74 basis test
More informationEXACT SOLUTIONS for ORDINARY DIFFERENTIAL EQUATIONS SECOND EDITION
HANDBOOK OF EXACT SOLUTIONS for ORDINARY DIFFERENTIAL EQUATIONS SECOND EDITION Andrei D. Polyanin Valentin F. Zaitsev CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C.
More informationd 1 µ 2 Θ = 0. (4.1) consider first the case of m = 0 where there is no azimuthal dependence on the angle φ.
4 Legendre Functions In order to investigate the solutions of Legendre s differential equation d ( µ ) dθ ] ] + l(l + ) m dµ dµ µ Θ = 0. (4.) consider first the case of m = 0 where there is no azimuthal
More informationOn an Eigenvalue Problem Involving Legendre Functions by Nicholas J. Rose North Carolina State University
On an Eigenvalue Problem Involving Legendre Functions by Nicholas J. Rose North Carolina State University njrose@math.ncsu.edu 1. INTRODUCTION. The classical eigenvalue problem for the Legendre Polynomials
More informationHANDBOOK OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS for ENGINEERS and SCIENTISTS
HANDBOOK OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS for ENGINEERS and SCIENTISTS Andrei D. Polyanin Chapman & Hall/CRC Taylor & Francis Group Boca Raton London New York Singapore Foreword Basic Notation
More informationChapter 6. Legendre and Bessel Functions
Chapter 6 Legendre and Bessel Functions Legendre's Equation Legendre's Equation (order n): legendre d K y''k y'c n n C y = : is an important ode in applied mathematics When n is a non-negative integer,
More information14 Fourier analysis. Read: Boas Ch. 7.
14 Fourier analysis Read: Boas Ch. 7. 14.1 Function spaces A function can be thought of as an element of a kind of vector space. After all, a function f(x) is merely a set of numbers, one for each point
More informationLecture 14: Ordinary Differential Equation I. First Order
Lecture 14: Ordinary Differential Equation I. First Order 1. Key points Maple commands dsolve 2. Introduction We consider a function of one variable. An ordinary differential equations (ODE) specifies
More informationFunction Space and Convergence Types
Function Space and Convergence Types PHYS 500 - Southern Illinois University November 1, 2016 PHYS 500 - Southern Illinois University Function Space and Convergence Types November 1, 2016 1 / 7 Recall
More informationAN INTRODUCTION TO THE FRACTIONAL CALCULUS AND FRACTIONAL DIFFERENTIAL EQUATIONS
AN INTRODUCTION TO THE FRACTIONAL CALCULUS AND FRACTIONAL DIFFERENTIAL EQUATIONS KENNETH S. MILLER Mathematical Consultant Formerly Professor of Mathematics New York University BERTRAM ROSS University
More informationBackground and Definitions...2. Legendre s Equation, Functions and Polynomials...4 Legendre s Associated Equation and Functions...
Legendre Polynomials and Functions Reading Problems Outline Background and Definitions...2 Definitions...3 Theory...4 Legendre s Equation, Functions and Polynomials...4 Legendre s Associated Equation and
More informationODE Homework Series Solutions Near an Ordinary Point, Part I 1. Seek power series solution of the equation. n(n 1)a n x n 2 = n=0
ODE Homework 6 5.2. Series Solutions Near an Ordinary Point, Part I 1. Seek power series solution of the equation y + k 2 x 2 y = 0, k a constant about the the point x 0 = 0. Find the recurrence relation;
More informationRecall that any inner product space V has an associated norm defined by
Hilbert Spaces Recall that any inner product space V has an associated norm defined by v = v v. Thus an inner product space can be viewed as a special kind of normed vector space. In particular every inner
More informationClosed-form Second Solution to the Confluent Hypergeometric Difference Equation in the Degenerate Case
International Journal of Difference Equations ISS 973-669, Volume 11, umber 2, pp. 23 214 (216) http://campus.mst.edu/ijde Closed-form Second Solution to the Confluent Hypergeometric Difference Equation
More informationPhysics 342 Lecture 22. The Hydrogen Atom. Lecture 22. Physics 342 Quantum Mechanics I
Physics 342 Lecture 22 The Hydrogen Atom Lecture 22 Physics 342 Quantum Mechanics I Friday, March 28th, 2008 We now begin our discussion of the Hydrogen atom. Operationally, this is just another choice
More informationMath Assignment 11
Math 2280 - Assignment 11 Dylan Zwick Fall 2013 Section 8.1-2, 8, 13, 21, 25 Section 8.2-1, 7, 14, 17, 32 Section 8.3-1, 8, 15, 18, 24 1 Section 8.1 - Introduction and Review of Power Series 8.1.2 - Find
More informationLegendre s Equation. PHYS Southern Illinois University. October 18, 2016
Legendre s Equation PHYS 500 - Southern Illinois University October 18, 2016 PHYS 500 - Southern Illinois University Legendre s Equation October 18, 2016 1 / 11 Legendre s Equation Recall We are trying
More informationIntroduction to Mathematical Physics
Introduction to Mathematical Physics Methods and Concepts Second Edition Chun Wa Wong Department of Physics and Astronomy University of California Los Angeles OXFORD UNIVERSITY PRESS Contents 1 Vectors
More informationMETHODS FOR SOLVING MATHEMATICAL PHYSICS PROBLEMS
METHODS FOR SOLVING MATHEMATICAL PHYSICS PROBLEMS V.I. Agoshkov, P.B. Dubovski, V.P. Shutyaev CAMBRIDGE INTERNATIONAL SCIENCE PUBLISHING Contents PREFACE 1. MAIN PROBLEMS OF MATHEMATICAL PHYSICS 1 Main
More informationarxiv:math/ v1 [math.ca] 21 Mar 2006
arxiv:math/0603516v1 [math.ca] 1 Mar 006 THE FOURTH-ORDER TYPE LINEAR ORDINARY DIFFERENTIAL EQUATIONS W.N. EVERITT, D. J. SMITH, AND M. VAN HOEIJ Abstract. This note reports on the recent advancements
More informationTime part of the equation can be separated by substituting independent equation
Lecture 9 Schrödinger Equation in 3D and Angular Momentum Operator In this section we will construct 3D Schrödinger equation and we give some simple examples. In this course we will consider problems where
More informationMATHEMATICAL FORMULAS AND INTEGRALS
MATHEMATICAL FORMULAS AND INTEGRALS ALAN JEFFREY Department of Engineering Mathematics University of Newcastle upon Tyne Newcastle upon Tyne United Kingdom Academic Press San Diego New York Boston London
More informationLimits at Infinity. Horizontal Asymptotes. Definition (Limits at Infinity) Horizontal Asymptotes
Limits at Infinity If a function f has a domain that is unbounded, that is, one of the endpoints of its domain is ±, we can determine the long term behavior of the function using a it at infinity. Definition
More informationJUST THE MATHS UNIT NUMBER DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) A.J.Hobson
JUST THE MATHS UNIT NUMBER.5 DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) by A.J.Hobson.5. Maclaurin s series.5. Standard series.5.3 Taylor s series.5.4 Exercises.5.5 Answers to exercises
More informationLight Scattering Group
Light Scattering Inversion Light Scattering Group A method of inverting the Mie light scattering equation of spherical homogeneous particles of real and complex argument is being investigated The aims
More information96 CHAPTER 4. HILBERT SPACES. Spaces of square integrable functions. Take a Cauchy sequence f n in L 2 so that. f n f m 1 (b a) f n f m 2.
96 CHAPTER 4. HILBERT SPACES 4.2 Hilbert Spaces Hilbert Space. An inner product space is called a Hilbert space if it is complete as a normed space. Examples. Spaces of sequences The space l 2 of square
More informationIntroductions to ExpIntegralEi
Introductions to ExpIntegralEi Introduction to the exponential integrals General The exponential-type integrals have a long history. After the early developments of differential calculus, mathematicians
More informationClass notes: Approximation
Class notes: Approximation Introduction Vector spaces, linear independence, subspace The goal of Numerical Analysis is to compute approximations We want to approximate eg numbers in R or C vectors in R
More informationIntegrals, Series, Eighth Edition. Table of. and Products USA. Daniel Zwillinger, Editor Rensselaer Polytechnic Institute,
Table of Integrals, Series, and Products Eighth Edition I S Gradshteyn and I M Ryzhik Daniel Zwillinger, Editor Rensselaer Polytechnic Institute, USA Victor Moll (Scientific Editor) Tulane University,
More informationSPECIAL FUNCTIONS AN INTRODUCTION TO THE CLASSICAL FUNCTIONS OF MATHEMATICAL PHYSICS
SPECIAL FUNCTIONS AN INTRODUCTION TO THE CLASSICAL FUNCTIONS OF MATHEMATICAL PHYSICS SPECIAL FUNCTIONS AN INTRODUCTION TO THE CLASSICAL FUNCTIONS OF MATHEMATICAL PHYSICS NICO M.TEMME Centrum voor Wiskunde
More information