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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 | |