Applied Analysis by the Hilbert Space Method An Introduction with Applications to the Wave, Heat, and Schrodinger Equations

ISBN-10: 0486458016

ISBN-13: 9780486458014

Edition: 2007

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Description:

Numerous examples and exercises highlight this unified treatment of the Hermitian operator theory in its Hilbert space setting. Its simple explanations of difficult subjects make it intuitively appealing to students in applied mathematics, physics, and engineering. It is also a fine reference for professionals. 1990 edition.
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Book details

List price: $29.95
Copyright year: 2007
Publisher: Dover Publications, Incorporated
Publication date: 6/5/2007
Binding: Paperback
Pages: 576
Size: 5.50" wide x 8.50" long x 1.25" tall
Weight: 1.232
Language: English

Preface
A Note on Method
First Order Linear Differential Equations
The Equation a(x)y[prime] + b(x)y = h(x)
First Order Linear Differential Expressions; the Kernel
Finding a Particular Solution by Variation of Parameters
Power Series Review
The Initial Value Problem for a First Order Linear Differential Equation
Second Order Linear Differential Equations
Basic Concepts of Linear Algebra for Function Spaces
The Initial Value Problem for a Second Order Linear Homogeneous Differential Equation
Dimension of the Kernel; General Solution; Abel's Formula
Kernel of Constant-Coefficient Expressions
The Classical Linear Oscillator
Guessing a Particular Solution to a Constant-Coefficient Equation
Particular Solution by Variation of Parameters
The Kernel of Legendre's Differential Expression
The Kernels of Other Classical Expressions
Dirac's Delta Function and Green's Functions
Second Order Linear Differential Equations in the Complex Domain
Hilbert Space
The Vibrating Wire
Fourier Series
Fourier Sine and Cosine Series
Fourier Series over Other Intervals
The Vibrating Wire, Revisited
The Inner Product
Schwarz's Inequality
The Mean-Square Metric; Orthogonal Bases
L[superscript 2] Spaces
Hilbert Space
Linear Second Order Differential Operators in L[superscript 2] Spaces and Their Eigenvalues and Eigenfunctions
Compatibility
Eigenvalues and Eigenfunctions
Hermitian Operators
Some General Operator Theory
The One-Dimensional Laplacian
Legendre's Operator and Its Eigenfunctions, the Legendre Polynomials
Solving Operator Equations with Legendre's Operator
More on Legendre Polynomials: Rodrigues' Formula, the Recursion Relation, and the Generating Function
Hermite's Operator and Its Eigenfunctions, the Hermite Polynomials
Solving Operator Equations with Hermite's Operator
More on Hermite Polynomials: Rodrigues' Formula, the Recursion Relation, and the Generating Function
Mathematical Aspects of Differential Operators in L[superscript 2] Spaces
Schrodinger's Equations in One Dimension
The Wave Equation by the Hilbert Space Method
The Heat Equation by the Hilbert Space Method
Quanta as Eigenvalues-the Time-Independent Schrodinger Equation in One Dimension
Interpretation of the [Psi] Function. The Time-Dependent Schrodinger Equation in One Dimension
The Quantum Linear Oscillator
Solution of the Time-Dependent Schrodinger Equation
A Brief History of Matrix Mechanics
A General Formulation of Quantum Mechanics: States
A General Formulation of Quantum Mechanics: Observables
Bessel's Operator and Bessel Functions
The Wave Equation and Other Equations in Higher Dimensions; Polar Coordinates
Bessel's Equation and Bessel's Operator of Order Zero
J[subscript 0](x): The Bessel Function of the First Kind of Order Zero
J[subscript 0](x): Calculating Its Values and Finding Its Zeros
The Eigenvalues and Eigenfunctions of Bessel's Operator of Order Zero
The Vibrating Drumhead, the Heated Disk, and the Quantum Particle Confined to a Circular Region (the [theta]-Independent Case)
[theta] Dependence: Bessel's Equation and Bessel's Operator of Integral Order p
J[subscript p](x): The Bessel Functions of the First Kind of Integral Order p
The Eigenvalues and Eigenfunctions of Bessel's Operator of Integral Order p
Project on Bessel Functions of Nonintegral Order
Mathematical Theory of Bessel's Operator
Eigenvalues of the Laplacian, with Applications
The Laplacian as a Hilbert Space Operator
Differential Forms, the Stokes Theorem, and Integration by Parts in Two Variables
The Laplacian Is Hermitian
General Facts About the Eigenvalues of the Laplacian
Eigenvalues of the Rectangle
Eigenvalues of the Disk
The Laplacian on the Sphere
Eigenvalues of the Sphere; Spherical Harmonics
The Hydrogen Atom
Project on Laguerre's Operator and Laguerre Polynomials
Laplace's Equation and Harmonic Polynomials
The Legendre, Laguerre, and Schrodinger Operators
The Fourier Transform
Complex Methods in Fourier Series; the Fourier Transform
Plancherel's Theorem; Examples of Fourier Transforms
Fourier Sine and Cosine Transforms
The Fourier Transform Is a Unitary Operator on L[superscript 2](- [infinity], [infinity])
The Fourier Transform Converts Differentiation into Multiplication by the Independent Variable
The Eigenvalues and Eigenfunctions of the Fourier Transform
[Characters not reproducible]((sin x)/x)dx = [pi]
Index of Symbols
Index
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