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INTRODUCTION TO THE SPECIAL FUNCTIONS OF …

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 Preliminary test 19 Comparison tests 19 The Ratio Test 2

1.15Series solutions of differential equations 28 A simple first order linear differential equation 29 A simple second order linear differential equation 30 1.16Generalized power series 33 Fuchs's conditions 34 2. Analytic continuation 37 2.1The Fundamental Theorem of algebra 37 Conjugate pairs or roots. 38

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