INTRODUCTION TO THE SPECIAL FUNCTIONS OF ... - Physics

1 INTRODUCTION TO THE SPECIAL FUNCTIONS OF MATHEMATICAL Physics with applications to the physical and applied sciences John Michael Finn April 13, 2005 CONTENTS Contents iii Preface xi Dedication xvii 1. Infinite Series 1 1 cautionary tale 2 series 6 Proof by mathematical induction 6 of an infinite series 7 Convergence of the chessboard problem 8 Distance traveled by A bouncing ball 9 remainder of a series 11 about series 12 Formal definition of convergence 13 series 13 Alternating Harmonic Series 14 Convergence 16 Distributive Law for scalar multiplication 18 Scalar multiplication 18 Addition of series 18 for convergence 19 iv Contents

2 Preliminary test 19 Comparison tests 19 The Ratio Test 20 The Integral Test 20 of convergence 21 Evaluation techniques 23 of FUNCTIONS in power series 23 The binomial expansion 24 Repeated Products 25 properties of power series 26 techniques 27 solutions of differential equations 28 A simple first order linear differential equation 29 A simple second order linear differential equation 30 power series 33 Fuchs's conditions 34 2. Analytic continuation 37 Fundamental Theorem of algebra 37 Conjugate pairs or roots. 38 Transcendental FUNCTIONS 38 Quadratic Formula 38 Definition of the square root 39 Definition of the square root of -1 40 The geometric interpretation of multiplication 41 complex plane 42 coordinates 44 Contents v of complex numbers 45 roots of 1/nz 47 infinite series 49 of complex FUNCTIONS 50 exponential function 53 natural logarithm 54 power function 55 under-damped harmonic oscillator 55

3 And hyperbolic FUNCTIONS 58 hyperbolic FUNCTIONS 59 trigonometric FUNCTIONS 60 trigonometric and hyperbolic FUNCTIONS 61 Cauchy Riemann conditions 63 to Laplace equation in two dimensions 64 3. Gamma and Beta FUNCTIONS 67 Gamma function 67 Extension of the Factorial function 68 Gamma FUNCTIONS for negative values of p 70 Evaluation of definite integrals 72 Beta Function 74 Error Function 76 Series 78 Sterling s formula 81 4. Elliptic Integrals 83 integral of the second kind 84 Integral of the first kind 88 vi Contents Elliptic FUNCTIONS 92 integral of the third kind 96 5.

4 Fourier Series 99 a string 99 solution to a simple eigenvalue equation 100 Orthogonality 101 of Fourier series 103 Completeness of the series 104 Sine and cosine series 104 complex form of Fourier series 105 intervals 106 106 The Full wave Rectifier 106 The Square wave 110 Gibbs Phenomena 112 Non-symmetric intervals and period doubling 114 and differentiation 119 Differentiation 119 Integration 120 s Theorem 123 Generalized Parseval s Theorem 125 to infinite series 125 6.

5 Orthogonal function spaces 127 of variables 127 s equation in polar coordinates 127 s equation 130 Contents vii theory 133 Linear self-adjoint differential operators 135 Orthogonality 137 Completeness of the function basis 139 Comparison to Fourier Series 139 Convergence of a Sturm-Liouville series 141 Vector space representation 142 7. Spherical Harmonics 145 polynomials 146 Series expansion 148 Orthogonality and Normalization 151 A second solution 154 s formula 156 Leibniz s rule for differentiating products 156 function 159 relations 162 Legendre Polynomials 164 Normalization of Associated Legendre polynomials 168 Parity of the Associated Legendre polynomials 168 Recursion relations 169 Harmonics 169 equation in spherical coordinates 172 8.

6 Bessel FUNCTIONS 175 solution of Bessel s equation 175 Neumann or Weber FUNCTIONS 178 Bessel FUNCTIONS 180 viii Contents Hankel FUNCTIONS 181 Zeroes of the Bessel FUNCTIONS 182 Orthogonality of Bessel FUNCTIONS 183 Orthogonal series of Bessel FUNCTIONS 183 Generating function 186 Recursion relations 186 Bessel FUNCTIONS 188 Modified Bessel FUNCTIONS of the second kind 190 Recursion formulas for modified Bessel FUNCTIONS 191 to other differential equations 192 Bessel FUNCTIONS 193 Definitions 194 Recursion relations 198 Orthogonal series of spherical Bessel FUNCTIONS 199 9.

7 Laplace equation 205 of Laplace equation 205 equation in Cartesian coordinates 207 Solving for the coefficients 210 equation in polar coordinates 214 to steady state temperature distribution 215 spherical capacitor, revisited 217 Charge distribution on a conducting surface 219 equation with cylindrical boundary conditions 221 Solution for a clyindrical capacitor 225 10. Time dependent differential equations 227 of partial differential equations 227 Contents ix equation 232 equation 236 Pressure waves: standing waves in a pipe 239 The struck string 240 The normal modes of a vibrating drum head 242 dinger equation 245 with spherical boundary conditions 246 Quantum mechanics in a spherical bag 246 Heat flow in a sphere 247 with cylindrical boundary conditions 250 Normal modes in a cylindrical cavity 250 Temperature distribution in a cylinder 250 11.

8 Green s FUNCTIONS and propagators 252 driven oscillator 253 domain analysis 257 s function solution to Possion s equation 259 expansion of a charge distribution 260 of images 262 Solution for a infinite grounded plane 263 Induced charge distribution on a grounded plane 265 Green s function for a conducting sphere 266 s function solution to the Yakawa interaction 268 PREFACE This text is based on a one semester advanced undergraduate course that I have taught at the College of William and Mary. In the spring semester of 2005, I decided to collect my notes and to present them in a more formal manner. The course covers se-lected topics on mathematical methods in the physical sciences and is cross listed at the senior level in the Physics and applied sciences departments.

9 The intended audience is junior and se-nior science majors intending to continue their studies in the pure and applied sciences at the graduate level. The course, as taught at the College, is hugely successful. The most frequent comment has been that students wished they had been intro-duced to this material earlier in their studies. Any course on mathematical methods necessarily involves a choice from a venue of topics that could be covered. The empha-sis on this course is to introduce students the SPECIAL FUNCTIONS of mathematical Physics with emphasis on those techniques that would be most useful in preparing a student to enter a program of graduate studies in the sciences or the engineering discip-lines.

10 The students that I have taught at the College are the gen-erally the best in their respective programs and have a solid foundation in basic methods. Their mathematical preparation xii Preface includes, at a minimum, courses in ordinary differential equa-tions, linear algebra, and multivariable calculus. The least expe-rienced junior level students have taken at least two semesters of Lagrangian mechanics, a semester of quantum mechanics, and are enrolled in a course in electrodynamics, concurrently.