Mathematical Methods in Science and Engineering - Selcuk S Bayin

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Résumé :

Preface xix

1 Legendre Equation and Polynomials 1

1.1 Second-Order Differential Equations of Physics 1

1.2 Legendre Equation 2

1.2.1 Method of Separation of Variables 4

1.2.2 Series Solution of the Legendre Equation 4

1.2.3 Frobenius Method - Review 7

1.3 Legendre Polynomials 8

1.3.1 Rodriguez Formula 10

1.3.2 Generating Function 10

1.3.3 Recursion Relations 12

1.3.4 Special Values 12

1.3.5 Special Integrals 13

1.3.6 Orthogonality and Completeness 14

1.3.7 Asymptotic Forms 17

1.4 Associated Legendre Equation and Polynomials 18

1.4.1 Associated Legendre Polynomials Pm l (x) 20

1.4.2 Orthogonality 21

1.4.3 Recursion Relations 22

1.4.4 Integral Representations 24

1.4.5 Associated Legendre Polynomials for m < 0 26

1.5 Spherical Harmonics 27

1.5.1 Addition Theorem of Spherical Harmonics 30

1.5.2 Real Spherical Harmonics 33

Bibliography 33

Problems 34

2 Laguerre Polynomials 39

2.1 Central Force Problems in Quantum Mechanics 39

2.2 Laguerre Equation and Polynomials 41

2.2.1 Generating Function 42

2.2.2 Rodriguez Formula 43

2.2.3 Orthogonality 44

2.2.4 Recursion Relations 45

2.2.5 Special Values 46

2.3 Associated Laguerre Equation and Polynomials 46

2.3.1 Generating Function 48

2.3.2 Rodriguez Formula and Orthogonality 49

2.3.3 Recursion Relations 49

Bibliography 49

Problems 50

3 Hermite Polynomials 53

3.1 Harmonic Oscillator in Quantum Mechanics 53

3.2 Hermite Equation and Polynomials 54

3.2.1 Generating Function 56

3.2.2 Rodriguez Formula 56

3.2.3 Recursion Relations and Orthogonality 57

Bibliography 61

Problems 62

4 Gegenbauer and Chebyshev Polynomials 65

4.1 Wave Equation on a Hypersphere 65

4.2 Gegenbauer Equation and Polynomials 68

4.2.1 Orthogonality and the Generating Function 68

4.2.2 Another Representation of the Solution 69

4.2.3 The Second Solution 70

4.2.4 Connection with the Gegenbauer Polynomials 71

4.2.5 Evaluation of the Normalization Constant 72

4.3 Chebyshev Equation and Polynomials 72

4.3.1 Chebyshev Polynomials of the First Kind 72

4.3.2 Chebyshev and Gegenbauer Polynomials 73

4.3.3 Chebyshev Polynomials of the Second Kind 73

4.3.4 Orthogonality and Generating Function 74

4.3.5 Another Definition 75

Bibliography 76

Problems 76

5 Bessel Functions 81

5.1 Bessel's Equation 83

5.2 Bessel Functions 83

5.2.1 Asymptotic Forms 84

5.3 Modified Bessel Functions 86

5.4 Spherical Bessel Functions 87

5.5 Properties of Bessel Functions 88

5.5.1 Generating Function 88

5.5.2 Integral Definitions 89

5.5.3 Recursion Relations of the Bessel Functions 89

5.5.4 Orthogonality and Roots of Bessel Functions 90

5.5.5 Boundary Conditions for the Bessel Functions 91

5.5.6 Wronskian of Pairs of Solutions 94

5.6 Transformations of Bessel Functions 95

5.6.1 Critical Length of a Rod 96

Bibliography 98

Problems 99

6 Hypergeometric Functions 103

6.1 Hypergeometric Series 103

6.2 Hypergeometric Representations of Special Functions 107

6.3 Confluent Hypergeometric Equation 108

6.4 Pochhammer Symbol and Hypergeometric Functions 109

6.5 Reduction of Parameters 113

Bibliography 115

Problems 115

7 Sturm-Liouville Theory 119

7.1 Self-Adjoint Differential Operators 119

7.2 Sturm-Lio...

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