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Preface | |

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A Note on Method | |

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First Order Linear Differential Equations | |

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The Equation a(x)y[prime] + b(x)y = h(x) | |

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First Order Linear Differential Expressions; the Kernel | |

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Finding a Particular Solution by Variation of Parameters | |

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Power Series Review | |

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The Initial Value Problem for a First Order Linear Differential Equation | |

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Second Order Linear Differential Equations | |

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Basic Concepts of Linear Algebra for Function Spaces | |

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The Initial Value Problem for a Second Order Linear Homogeneous Differential Equation | |

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Dimension of the Kernel; General Solution; Abel's Formula | |

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Kernel of Constant-Coefficient Expressions | |

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The Classical Linear Oscillator | |

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Guessing a Particular Solution to a Constant-Coefficient Equation | |

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Particular Solution by Variation of Parameters | |

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The Kernel of Legendre's Differential Expression | |

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The Kernels of Other Classical Expressions | |

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Dirac's Delta Function and Green's Functions | |

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Second Order Linear Differential Equations in the Complex Domain | |

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Hilbert Space | |

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The Vibrating Wire | |

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Fourier Series | |

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Fourier Sine and Cosine Series | |

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Fourier Series over Other Intervals | |

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The Vibrating Wire, Revisited | |

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The Inner Product | |

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Schwarz's Inequality | |

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The Mean-Square Metric; Orthogonal Bases | |

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L[superscript 2] Spaces | |

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Hilbert Space | |

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Linear Second Order Differential Operators in L[superscript 2] Spaces and Their Eigenvalues and Eigenfunctions | |

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Compatibility | |

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Eigenvalues and Eigenfunctions | |

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Hermitian Operators | |

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Some General Operator Theory | |

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The One-Dimensional Laplacian | |

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Legendre's Operator and Its Eigenfunctions, the Legendre Polynomials | |

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Solving Operator Equations with Legendre's Operator | |

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More on Legendre Polynomials: Rodrigues' Formula, the Recursion Relation, and the Generating Function | |

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Hermite's Operator and Its Eigenfunctions, the Hermite Polynomials | |

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Solving Operator Equations with Hermite's Operator | |

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More on Hermite Polynomials: Rodrigues' Formula, the Recursion Relation, and the Generating Function | |

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Mathematical Aspects of Differential Operators in L[superscript 2] Spaces | |

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Schrodinger's Equations in One Dimension | |

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The Wave Equation by the Hilbert Space Method | |

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The Heat Equation by the Hilbert Space Method | |

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Quanta as Eigenvalues-the Time-Independent Schrodinger Equation in One Dimension | |

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Interpretation of the [Psi] Function. The Time-Dependent Schrodinger Equation in One Dimension | |

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The Quantum Linear Oscillator | |

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Solution of the Time-Dependent Schrodinger Equation | |

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A Brief History of Matrix Mechanics | |

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A General Formulation of Quantum Mechanics: States | |

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A General Formulation of Quantum Mechanics: Observables | |

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Bessel's Operator and Bessel Functions | |

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The Wave Equation and Other Equations in Higher Dimensions; Polar Coordinates | |

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Bessel's Equation and Bessel's Operator of Order Zero | |

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J[subscript 0](x): The Bessel Function of the First Kind of Order Zero | |

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J[subscript 0](x): Calculating Its Values and Finding Its Zeros | |

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The Eigenvalues and Eigenfunctions of Bessel's Operator of Order Zero | |

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The Vibrating Drumhead, the Heated Disk, and the Quantum Particle Confined to a Circular Region (the [theta]-Independent Case) | |

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[theta] Dependence: Bessel's Equation and Bessel's Operator of Integral Order p | |

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J[subscript p](x): The Bessel Functions of the First Kind of Integral Order p | |

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The Eigenvalues and Eigenfunctions of Bessel's Operator of Integral Order p | |

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Project on Bessel Functions of Nonintegral Order | |

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Mathematical Theory of Bessel's Operator | |

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Eigenvalues of the Laplacian, with Applications | |

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The Laplacian as a Hilbert Space Operator | |

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Differential Forms, the Stokes Theorem, and Integration by Parts in Two Variables | |

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The Laplacian Is Hermitian | |

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General Facts About the Eigenvalues of the Laplacian | |

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Eigenvalues of the Rectangle | |

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Eigenvalues of the Disk | |

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The Laplacian on the Sphere | |

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Eigenvalues of the Sphere; Spherical Harmonics | |

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The Hydrogen Atom | |

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Project on Laguerre's Operator and Laguerre Polynomials | |

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Laplace's Equation and Harmonic Polynomials | |

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The Legendre, Laguerre, and Schrodinger Operators | |

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The Fourier Transform | |

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Complex Methods in Fourier Series; the Fourier Transform | |

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Plancherel's Theorem; Examples of Fourier Transforms | |

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Fourier Sine and Cosine Transforms | |

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The Fourier Transform Is a Unitary Operator on L[superscript 2](- [infinity], [infinity]) | |

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The Fourier Transform Converts Differentiation into Multiplication by the Independent Variable | |

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The Eigenvalues and Eigenfunctions of the Fourier Transform | |

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[Characters not reproducible]((sin x)/x)dx = [pi] | |

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Index of Symbols | |

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Index | |