Table of Contents Page List of Figures viii Acknowledgements ix Chapter 1. Introduction 1 1.1 The Vibrating String 1 1.2 Network Theory 2 1.3 Random Walk With Discrete Time Process 4 Chapter 2. Regular Sturm-Liouville Problem 7 2.1 Set Up 7 2.2 Preliminary Results 9 2.3 Orthogonality, Eigenfunction Expansion, Spectral Function, and Green's Function 17 Chapter 3. Singular Sturm-Liouville Problem 28 3.1 Definition 28 3.2 CV Circles 28 3.3 Ca Circles 35 3.4 Existence of Boundary Conditions 38 3.5 Singular Boundary Value Problems 40 3.6 Green's Function 41 3.7 Self-Adjointness 43 3.8 A-Independence of Boundary Conditions 45 3.9 Green's Formulas 50 3.10 Spectral Resolution 53 3.11 Limit-Point and Limit-Circle Tests 70 Chapter 4. Polynomial Solutions 74 V
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