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Methods of Mathematical Physics by Hilbert and Courant

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  1. Jan 28, 2013 #1

    micromass

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    Table of Contents for Volume I:
    Code (Text):

    [LIST]
    [*] The Algebra of Linear Transformations and Quadratic Forms
    [LIST]
    [*] Linear equations and linear transformations
    [LIST]
    [*] Vectors
    [*] Orthogonal systems of vectors. Completeness
    [*] Linear transformations. Matrices
    [*] Bilinear, quadratic, and Hermitian forms
    [*] Orthogonal and unitary transformations
    [/LIST]
    [*] Linear transformations with a linear parameter
    [*] Transformation to principal axes of quadratic and Hermitian forms
    [LIST]
    [*] Transformation to principal axes on the basis of a maximum principle
    [*] Eigenvalues
    [*] Generalization to Hermitian forms
    [*] Inertial theorem for quadratic forms
    [*] Representation of the resolvent of a form
    [*] Solution of systems of linear equations associated with forms
    [/LIST]
    [*] Minimum-maximum property of eigenvalues
    [LIST]
    [*] Characterization of eigenvalues by a minimum-maximum problem
    [*] Applications. Constraints
    [/LIST]
    [*] Supplement and problems
    [LIST]
    [*] Linear independence and the Gram determinant
    [*] Hadamard's inequality for determinants
    [*] Generalized treatment of canonical transformations
    [*] Bilinear and quadratic forms of infinitely many variables
    [*] Infinitesimal linear transformations
    [*] Perturbations
    [*] Constraints
    [*] Elementary divisors of a matrix or a bilinear form
    [*] Spectrum of a unitary matrix
    [/LIST]
    [*] References
    [/LIST]
    [*] Series Expansions of Arbitrary Functions
    [LIST]
    [*] Orthogonal systems of functions
    [LIST]
    [*] Definitions
    [*] Orthogonalization of functions
    [*] Bessel's inequality. Completeness relation. Approximation in the mean
    [*] Spectral decomposition by Fourier series and integrals
    [*] Dense systems of functions
    [*] A Theorem of H. Muntz on the completeness of powers
    [*] Fejer's summation theorem
    [*] The Mellin inversion formulas
    [*] The Gibbs phenomenon
    [*] A theorem on Gram's determinant
    [*] Application of the Lebesgue integral
    [/LIST]
    [*] References
    [/LIST]
    [*] Linear Integral Equations
    [LIST]
    [*] Introduction
    [LIST]
    [*] Notation and basic concepts
    [*] Functions in integral representation
    [*] Degenerate kernels
    [/LIST]
    [*] Fredholm's theorems for degenerate kernels
    [*] Fredholm's theorems for arbitrary kernels
    [*] Symmetric kernels and their eigenvalues
    [LIST]
    [*] Existence of an eigenvalue of a symmetric kernel
    [*] The totality of eigenfunctions and eigenvalues
    [*] Maximum-minimum property of eigenvalues
    [/LIST]
    [*] The expansion theorem and its applications
    [LIST]
    [*] Expansion theorem
    [*] Solution of the inhomogeneous linear integral equation
    [*] Bilinear formula for iterated kernels
    [*] Mercer's theorem
    [/LIST]
    [*] Neumann series and the reciprocal kernel
    [*] The Fredholm formulas
    [*] Another derivation of the theory
    [LIST]
    [*] A lemma
    [*] Eigenfunctions of a symmetric kernel
    [*] Unsymmetric kernels
    [*] Continuous dependence of eigenvalues and eigenfunctions on the kernel
    [/LIST]
    [*] Extensions of the theory
    [*] Supplement and problems for Chapter III
    [LIST]
    [*] Problems
    [*] Singular integral equations
    [*] E. Schmidt's derivation of the Fredholm theorems
    [*] Enskog's method for solving symmetric integral equations
    [*] Kellogg's method for the determination of eigenfunctions
    [*] Symbolic functions of a kernel and their eigenvalues
    [*] Example of an unsymmetric kernel without null solutions
    [*] Volterra integral equation
    [*] Abel's integral equation
    [*] Adjoint orthogonal systems belonging to an unsymmetric kernel
    [*] Integral equations of the first kind
    [*] Method of infinitely many variables
    [*] Minimum properties of eigenfunctions
    [*] Polar integral equations
    [*] Symmetrizable kernels
    [*] Determination of the resolvent kernel by functional equations
    [*] Continuity of definite kernels
    [*] Hammerstein's theorem
    [/LIST]
    [*] References
    [/LIST]
    [*] The Calculus of Variations
    [LIST]
    [*] Problems of the calculus of variations
    [LIST]
    [*] Maxima and minima of functions
    [*] Functionals
    [*] Typical problems of the calculus of variations
    [*] Characteristic difficulties of the calculus of variations
    [/LIST]
    [*] Direct solutions
    [LIST]
    [*] The isoperimetric problem
    [*] The Rayleigh-Ritz method. Minimizing sequences
    [*] Other direct methods. Method of finite differences. Infinitely many variables
    [*] General remarks on direct methods of the calculus of variations
    [/LIST]
    [*] The Euler equations
    [LIST]
    [*] "Simplest problem" of the variational calculus
    [*] Several unknown functions
    [*] Higher derivatives
    [*] Several independent variables
    [*] Identical vanishing of the Euler differential expression
    [*] Euler equations in homogeneous form
    [*] Relaxing of conditions. Theorems of du Bois-Reymond and Haar
    [*] Variational problems and functional equations
    [/LIST]
    [*] Integration of the Euler differential equation
    [*] Boundary conditions
    [LIST]
    [*] Natural boundary conditions for free boundaries
    [*] Geometrical problems. Transversality
    [/LIST]
    [*] The second variation and the Legendre condition
    [*] Variational problems with subsidiary conditions
    [LIST]
    [*] Isoperimetric problems
    [*] Finite subsidiary conditions
    [*] Differential equations as subsidiary conditions
    [/LIST]
    [*] Invariant character of the Euler equations
    [LIST]
    [*] The Euler expression as a gradient in function space. Invariance of the Euler expression
    [*] Transformation of \Delta u. Spherical coordinates
    [*] Ellipsoidal coordinates
    [/LIST]
    [*] Transformation of variational problems to canonical and involutory form
    [LIST]
    [*] Transformation of an ordinary minimum problem with subsidiary conditions
    [*] Involutory transformation of the simplest variational problems
    [*] Transformation of variational problems to canonical form
    [*] Generalizations
    [/LIST]
    [*] Variational calculus and the differential equations of mathematical physics
    [LIST]
    [*] General remarks
    [*] The vibrating string and the vibrating rod
    [*] Membrane and plate
    [/LIST]
    [*] Reciprocal quadratic variational problems
    [*] Supplementary remarks and exercises
    [LIST]
    [*] Variational problem for a given differential equation
    [*] Reciprocity for isoperimetric problems
    [*] Circular light rays
    [*] The problem of Dido
    [*] Examples of problems in space
    [*] The indicatrix and applications
    [*] Variable domains
    [*] E. Noether's theorem on invariant variational problems. Integrals in particle mechanics
    [*] Transversality for multiple integrals
    [*] Euler's differential expressions on surfaces
    [*] Thomson's principle in electrostatics
    [*] Equilibrium problems for elastic bodies. Castigliano's principle
    [*] The variational problem of buckling
    [/LIST]
    [*] References
    [/LIST]
    [*] Vibration and Eigenvalue Problems
    [LIST]
    [*] Preliminary remarks about linear differential equations
    [LIST]
    [*] Principle of superposition
    [*] Homogeneous and nonhomogeneous problems. Boundary conditions
    [*] Formal relations. Adjoint differential expressions. Green's formulas
    [*] Linear functional equations as limiting cases and analogues of systems of linear equations
    [/LIST]
    [*] Systems of a finite number of degrees of freedom
    [LIST]
    [*] Normal modes of vibration. Normal coordinates. General theory of motion
    [*] General properties of vibrating systems
    [/LIST]
    [*] The vibrating string
    [LIST]
    [*] Free motion of the homogeneous string
    [*] Forced motion
    [*] The general nonhomogeneous string and the Sturm-Liouville eigenvalue problem
    [/LIST]
    [*] The vibrating rod
    [*] The vibrating membrane
    [LIST]
    [*] General eigenvalue problem for the homogeneous membrane
    [*] Forced motion
    [*] Nodal lines
    [*] Rectangular membrane
    [*] Circular membrane. Bessel functions
    [*] Nonhomogeneous membrane
    [/LIST]
    [*] The vibrating plate
    [LIST]
    [*] General remarks
    [*] Circular boundary
    [/LIST]
    [*] General remarks on the eigenfunction method
    [LIST]
    [*] Vibration and equilibrium problems
    [*] Heat conduction and eigenvalue problems
    [/LIST]
    [*] Vibration of three-dimensional continua. Separation of variables
    [*] Eigenfunctions and the boundary value problem of potential theory
    [LIST]
    [*] Circle, sphere, and spherical shell
    [*] Cylindrical domain
    [*] The Lame problem
    [/LIST]
    [*] Problems of the Sturm-Liouville type. Singular boundary points
    [LIST]
    [*] Bessel functions
    [*] Legendre functions of arbitrary order
    [*] Jacobi and Tchebycheff polynomials
    [*] Hermite and Laguerre polynomials
    [/LIST]
    [*] The asymptotic behavior of the solutions of Sturm-Liouville equations
    [LIST]
    [*] Boundedness of the solution as the independent variable tends to infinity
    [*] A sharper result. (Bessel functions)
    [*] Boundedness as the parameter increases
    [*] Asymptotic representation of the solutions
    [*] Asymptotic representation of Sturm-Liouville eigenfunctions
    [/LIST]
    [*] Eigenvalue problems with a continuous spectrum
    [LIST]
    [*] Trigonometric functions
    [*] Bessel functions
    [*] Eigenvalue problem of the membrane equation for the infinite plane
    [*] The Schrodinger eigenvalue problem
    [/LIST]
    [*] Perturbation theory
    [LIST]
    [*] Simple eigenvalues
    [*] Multiple eigenvalues
    [*] An example
    [/LIST]
    [*] Green's function (influence function) and reduction of differential equations to integral equations
    [LIST]
    [*] Green's function and boundary value problem for ordinary differential equations
    [*] Construction of Green's function; Green's function in the generalized sense
    [*] Equivalence of integral and differential equations
    [*] Ordinary differential equations of higher order
    [*] Partial differential equations
    [/LIST]
    [*] Examples of Green's function
    [LIST]
    [*] Ordinary differential equations
    [*] Green's function for \Delta u: circle and sphere
    [*] Green's function and conformal mapping
    [*] Green's function for the potential equation on the surface of a sphere
    [*] Green's function for \Delta u = 0 in a rectangular parallelepiped
    [*] Green's function for \Delta u in the interior o
    [/LIST]
    [*] Supplement to Chapter V
    [LIST]
    [*] Examples for the vibrating string
    [*] Vibrations of a freely suspended rope; Bessel functions
    [*] Examples for the explicit solution of the vibration equation. Mathieu functions
    [*] Boundary conditions with parameters
    [*] Green's tensors for systems of differential equations
    [*] Analytic continuation of the solutions of the equation \Delta u + \lambda u =0
    [*] A theorem on the nodal curves of the solutions of \Delta u +\lambda u = 0
    [*] An example of eigenvalues of infinite multiplicity
    [*] Limits for the validity of the expansion theorems
    [/LIST]
    [*] References
    [/LIST]
    [*] Application of the Calculus of Variations to Eigenvalue Problems
    [LIST]
    [*] Extremum properties of eigenvalues
    [LIST]
    [*] Classical extremum properties
    [*] Generalizations
    [*] Eigenvalue problems for regions with separate components
    [*] The maximum-minimum property of eigenvalues
    [/LIST]
    [*] General consequences of the extremum properties of the eigenvalues
    [LIST]
    [*] General theorems
    [*] Infinite growth of the eigenvalues
    [*] Asymptotic behavior of the eigenvalues in the Sturm-Liouville problem
    [*] Singular differential equations
    [*] Further remarks concerning the growth of eigenvalues. Occurrence of negative eigenvalues
    [*] Continuity of eigenvalues
    [/LIST]
    [*] Completeness and expansion theorems
    [LIST]
    [*] Completeness of the eigenfunctions
    [*] The expansion theorem
    [*] Generalization of the expansion theorem
    [/LIST]
    [*] Asymptotic distribution of eigenvalues
    [LIST]
    [*] The equation \Delta u + \lambda u = 0 for a rectangl
    [*] The equation \Delta u + \lambda u = 0 for domains consisting of a finite number of squares or cubes
    [*] Extension to the general differential equation L[u] + \lambda \rho u = 0
    [*] Asymptotic distribution of eigenvalues for an arbitrary domain
    [*] Sharper form of the laws of asymptotic distribution of eigenvalues for the differential equation \Delta u + \lambda u = 0
    [/LIST]
    [*] Eigenvalue problems of the Schrodinger type
    [*] Nodes of eigenfunctions
    [*] Supplementary remarks and problems
    [LIST]
    [*] Minimizing properties of eigenvalues. Derivation from completeness
    [*] Characterization of the first eigenfunction by absence of nodes
    [*] Further minimizing properties of eigenvalues
    [*] Asymptotic distribution of eigenvalues
    [*] Parameter eigenvalue problems
    [*] Boundary conditions containing parameters
    [*] Eigenvalue problems for closed surfaces
    [*] Estimates of eigenvalues when singular points occur
    [*] Minimum theorems for the membrane and plate
    [*] Minimum problems for variable mass distribution
    [*] Nodal points for the Sturm-Liouville problem. Maximum-minimum principle
    [/LIST]
    [*] References
    [/LIST]
    [*] Special Functions Defined by Eigenvalue Problems
    [LIST]
    [*] Preliminary discussion of linear second order differential equations
    [*] Bessel functions
    [LIST]
    [*] Application of the integral transformation
    [*] Hankel functions
    [*] Bessel and Neumann functions
    [*] Integral representations of Bessel functions
    [*] Another integral representation of the Hankel and Bessel functions
    [*] Power series expansion of Bessel functions
    [*] Relations between Bessel functions
    [*] Zeros of Bessel functions
    [*] Neumann functions
    [/LIST]
    [*] Legendre functions
    [LIST]
    [*] Schlafli's integral
    [*] Integral representations of Laplace
    [*] Legendre functions of the second kind
    [*] Associated Legendre functions. (Legendre functions of higher order.)
    [/LIST]
    [*] Application of the method of integral transformation to Legendre, Tchebycheff, Hermite, and Laguerre equations
    [LIST]
    [*] Legendre functions
    [*] Tchebycheff functions
    [*] Hermite functions
    [*] Laguerre functions
    [/LIST]
    [*] Laplace spherical harmonics
    [LIST]
    [*] Determination of 2n + 1 spherical harmonics of n-th order
    [*] Completeness of the system of functions
    [*] Expansion theorem
    [*] The Poisson integral
    [*] The Maxwell-Sylvester representation of spherical harmonics
    [/LIST]
    [*] Asymptotic expansions
    [LIST]
    [*] Stirling's formula
    [*] Asymptotic calculation of Hankel and Bessel functions for large values of the arguments
    [*] The saddle point method
    [*] Application of the saddle point method to the calculation of Hankel and Bessel functions for large parameter and large argument
    [*] General remarks on the saddle point method
    [*] The Darboux method
    [*] Application of the Darboux method to the asymptotic expansion of Legendre polynomials
    [/LIST]
    [*] Appendix to Chapter VII. Transformation of Spherical Harmonics
    [LIST]
    [*] Introduction and notation
    [*] Orthogonal transformations
    [*] A generating function for spherical harmonics
    [*] Transformation formula
    [*] Expressions in terms of angular coordinates
    [/LIST]
    [/LIST]
    [*] Additional Bibliography
    [*] Index
    [/LIST]
     
     
    Last edited by a moderator: May 6, 2017
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  3. Feb 3, 2013 #2

    micromass

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    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Table of Contents for Volume II:
    Code (Text):

    [LIST]
    [*] Introductory Remarks
    [LIST]
    [*] General Information about the Variety of Solutions
    [LIST]
    [*] Examples
    [*] Differential Equations for Given Families of Functions
    [/LIST]
    [*] Systems of Differential Equations
    [LIST]
    [*] The Question of Equivalence of a System of Differential Equations and a Single Differential Equation
    [*] Elimination from a Linear System with Constant Coefficients
    [*] Determined, Overdetermined, Underdetermined Systems
    [/LIST]
    [*] Methods of Integration for Special Differential Equations
    [LIST]
    [*] Separation of Variables
    [*] Construction of Further Solutions by Superposition. Fundamental Solution of the Heat Equation. Poisson's Integral
    [/LIST]
    [*] Geometric Interpretation of a First Order Partial Differential Equation in Two Independent Variables. The Complete Integral
    [LIST]
    [*] Geometric Interpretation of a First Order Partial Differential Equation
    [*] The Complete Integral
    [*] Singular Integrals
    [*] Examples
    [/LIST]
    [*] Theory of Linear and Quasi-Linear Differential Equations of First Order
    [LIST]
    [*] Linear Differential Equations
    [*] Quasi-Linear Differential Equations
    [/LIST]
    [*] The Legendre Transformation
    [LIST]
    [*] The Legendre Transformation for Functions of Two Variables
    [*] The Legendre Transformation for Functions of n Variables
    [*] Application of the Legendre Transformation to Partial Differential Equations
    [/LIST]
    [*] The Existence Theorem of Cauchy and Kowalewsky
    [LIST]
    [*] Introduction and Examples
    [*] Reduction to a System of Quasi-Linear Differential Equations
    [*] Determination of Derivatives Along the Initial Manifold
    [*] Existence Proof for Solutions of Analytic Differential Equations
    [LIST]
    [*] Observation About Linear Differential Equations
    [*] Remark About Konanalytic Differential Equations
    [/LIST]
    [*] Remarks on Critical Initial Data. Characteristics
    [/LIST]
    [*] Appendix: Laplace's Differential Equation for the Support Function of a Minimal Surface
    [*] Appendix: Systems of Differential Equations of First Order and Differential Equations of Higher Order
    [LIST]
    [*] Plausibility Considerations
    [*] Conditions of Equivalence for Systems of Two First Order Partial Differential Equations and a Differential Equation of Second Order
    [/LIST]
    [/LIST]
    [*] General Theory of Partial Differential Equations of First Order
    [LIST]
    [*] Geometric Theory of Quasi-Linear Differential Equations in Two Independent Variables
    [LIST]
    [*] Characteristic Curves
    [*] Initial Value Problem
    [*] Examples
    [/LIST]
    [*] Quasi-Linear Differential Equations in n Independent Variables
    [*] General Differential Equations in Two Independent Variables
    [LIST]
    [*] Characteristic Curves and Focal Curves. The Monge Cone
    [*] Solution of the Initial Value Problem
    [*] Characteristics as Branch Elements. Supplementary Remarks. Integral Conoid. Caustics
    [*] The Complete Integral
    [/LIST]
    [*] Focal Curves and the Monge Equation
    [*] Examples
    [LIST]
    [*] The Differential Equation of Straight Light Rays, (grad u)^2 = 1
    [*] The Equation F (u_x , u_y) = 0
    [*] Clairaut's Differential Equation
    [*] Differential Equation of Tubular Surfaces
    [*] Homogeneity Relation
    [/LIST]
    [*] General Differential Equation in n Independent Variables
    [*] Complete Integral and Hamilton-Jacobi Theory
    [LIST]
    [*] Construction of Envelopes and Characteristic Curve
    [*] Canonical Form of the Characteristic Differential Equations
    [*] Hamilton-Jacobi Theory
    [*] Example. The Two-Body Problem
    [*] Example. Geodesics on an Ellipsoid
    [/LIST]
    [*] Hamilton-Jacobi Theory and the Calculus of Variations
    [LIST]
    [*] Euler's Differential Equations in Canonical Form
    [*] Geodetic Distance or Eiconal and Its Derivatives. Hamilton-Jacobi Partial Differential Equation
    [*] Homogeneous Integrands
    [*] Fields of Extremals. Hamilton-Jacohi Differential Equation.
    [*] Cone of Hays. Huyghens' Construction
    [*] Hilbert's Invariant Integral for the Representation of the Eiconal
    [*] Theorem of Hamilton and Jacobi
    [/LIST]
    [*] Canonical Transformations and Applications
    [LIST]
    [*] The Canonical Transformation
    [*] New Proof of the Hamilton-Jacobi Theorem
    [*] Variation of Constants (Canonical Perturbation Theory)
    [/LIST]
    [*] Appendix
    [LIST]
    [*] Further Discussion of Characteristic Manifolds
    [LIST]
    [*] Remarks on Differentiation in n Dimensions
    [*] Initial Value Problem. Characteristic Manifolds
    [/LIST]
    [*] Systems of Quasi-Linear Differential Equations with the Same Principal Part. New Derivation of the Theory
    [*] Haar's Uniqueness Proof
    [/LIST]
    [*] Appendix: Theory of Conservation Laws
    [/LIST]
    [*] Differential Equations of Higher Order
    [LIST]
    [*] Normal Forms for Linear and Quasi-Linear Differential Operators of Second Order in Two Independent Variables
    [LIST]
    [*] Elliptic, Hyperbolic, and Parabolic Normal Forms. Mixed Types
    [*] Examples
    [*] Normal Forms for Quasi-Linear Second Order Differential Equations in Two Variables
    [*] Example. Minimal Surfaces
    [*] Systems of Two Differential Equations of First Order
    [/LIST]
    [*] Classification in General and Characteristics
    [LIST]
    [*] Notations
    [*] Systems of First Order with Two Indepebdebt Variables. Characteristics
    [*] Systems of First Order with n Independent Variables
    [*] Differential Equations of Higher Order. Hyperbolicity
    [*] Supplementary Remarks
    [*] Examples. Maxwell's and Dirac's Equations
    [/LIST]
    [*] Linear Differcntial Equations with Constant Coefficients
    [LIST]
    [*] Normal Form and Classification for Equations of Second Order
    [*] Fundamental Solutions for Equations of Second Order
    [*] Plane Waves
    [*] Plane Waves Continued. Progressing Waves. Dispersion
    [*] Examples. Telegraph Equation. Undistorted Waves in Cables
    [*] Cylindrical and Spherical Waves
    [/LIST]
    [*] Initial Value Problems. Radiation Problems for the Wave Equation
    [LIST]
    [*] Initial Value Problems for Heat Conduction. Transformation of the Theta Function
    [*] Initial Value Problems for the Wave Equation
    [*] Duhamel's Principle. Nonhomogeneous Equations. Retarded
    [LIST]
    [*] Duhamel's Principle for Systems of First Order
    [/LIST]
    [*] Initial Value Problem for the Wave Equation in Two-Dimensional Space. Method of Descent
    [*] The Radiation Problem
    [*] Propagation Phenomena and Huyghens' Principle
    [/LIST]
    [*] Solution of Initial Value Problems by Fourier Integrals
    [LIST]
    [*] Cauchy's Method of the Fourier Integral
    [*] Example
    [*] Justification of Cauchy's Method
    [/LIST]
    [*] Typical Problems in Differential Equations of Mathematical Physics
    [LIST]
    [*] Introductory Remarks
    [*] Basic Principles
    [*] Remarks about "Improperly Posed" Problems
    [*] General Remarks About Linear Problems
    [/LIST]
    [*] Appendix
    [LIST]
    [*] Sobolev's Lemma
    [*] Adjoint Operators
    [LIST]
    [*] Matrix Operators
    [*] Adjoint Differential Operators
    [/LIST]
    [/LIST]
    [*] Appendix: The Uniqueness Theorem of Holmgren
    [/LIST]
    [*] Potential Theory and Elliptic Differential Equations
    [LIST]
    [*]  Basic Notions
    [LIST]
    [*] Equations of Laplace and Poisson, and Related Equations
    [*] Potentials of Mass Distributions
    [*] Green's Formulas and Applications
    [*] Derivatives of Potentials of Mass Distributions
    [/LIST]
    [*] Poisson's Integral and Applications
    [LIST]
    [*] The Boundary Value Problem and Green's Function
    [*] Green's Function for the Circle and Sphere. Poisson's Integral for the Sphere and Half-Space
    [*] Consequences of Poisson's Formula
    [/LIST]
    [*] The Mean Value Theorem and Applications
    [LIST]
    [*] The Homogeneous and Nonhomogeneous Mean Value Equation
    [*] The Converse of the Mean Value Theorems
    [*] Poisson's Equation for Potentials of Spatial Distributions
    [*] Mean Value Theorems for Other Elliptic Differential Equations
    [/LIST]
    [*] The Boundary Value Problem
    [LIST]
    [*] Preliminaries. Continuous Dependence on the Boundary Values and on the Domain
    [*] Solution of the Boundary Value Problem by the Schwarz Alternating Procedure
    [*] The Method of Integral Equations for Plane Regions with Sufficiently Smooth Boundaries
    [*] Remarks on Boundary Values
    [LIST]
    [*] Capacity and Assumption of Boundary Values
    [/LIST]
    [*] Perron's Method of Subharmonic Functions
    [/LIST]
    [*] The Reduced Wave Equation. Scattering
    [LIST]
    [*] Background
    [*] Sommerfeld's Radiation Condition
    [*] Scattering
    [/LIST]
    [*] Boundary Value Problems for More General Elliptic Differential Equations. Uniqueness of the Solutions
    [LIST]
    [*] Linear Differential Equations
    [*] Nonlinear Equations
    [*] Rellich's Theorem on the Monge-Ampere Differential Equation
    [*] The Maximum Principle and Applications
    [/LIST]
    [*] A Priori Estimates of Schauder and Their Applications
    [LIST]
    [*] Schauder's Estimates
    [*] Solution of the Boundary Value Problem
    [*] Strong Barrier Functions and Applications
    [*] Some Properties of Solutions of L[u] = f
    [*] Further Results on Elliptic Equations; Behavior at the Boundary
    [/LIST]
    [*] Solution of the Beltrami Equations
    [*] The Boundary Value Problem for a Special Quasi-Linear Equation. Fixed Point Method of Leray and Schauder
    [*] Solution of Elliptic Differential Equations by Means of Integral Equations
    [LIST]
    [*] Construction of Particular Solutions. Fundamental Solutions. Parametrix.
    [*] Further Remarks
    [/LIST]
    [*] Appendix: Nonlinear Equations
    [LIST]
    [*] Perturbation Theory
    [*] The Equation \Delta u = f(x, u)
    [/LIST]
    [*] Supplement to Chapter IV. Function Theoretic Aspects of the Theory of Elliptic Partial Differential Equations
    [LIST]
    [*] Definition of Pseudoanalytic Functions
    [*] An Integral Equation
    [*] Similarity Principle
    [*] Applications of the Similarity Principle
    [*] Formal Powers
    [*] Differentiation and Integration of Pseudoanalytic Functions
    [*] Example. Equations of Mixed Type
    [*] General Definition of Pseudoanalytic Functions
    [*] Quasiconformality and a General Representation Theorem
    [*] A Nonlinear Boundary Value Problem
    [*] An Extension of Riemann's Mapping Theorem
    [*] Two Theorems on Minimal Surfaces
    [*] Equations with Analytic Coefficients
    [*] Proof of Privaloff's Theorem
    [*] Proof of the Schauder Fixed Point Theorem
    [/LIST]
    [/LIST]
    [*] Hyperbolic Differential Equations in Two Independent Variables
    [LIST]
    [*] Introduction
    [*] Characteristics for Differential Equations Mainly of Second Order
    [LIST]
    [*] Basic Notions. Quasi-Linear Equations
    [*] Characteristics on Integral Surfaces
    [*] Characteristics as Curves of Discontinuity. Wave Fronts. Propagation of Discontinuities
    [*] General Differential Equations of Second Order
    [*] Differential Equations of Higher Order
    [*] Invariance of Characteristics under Point Transformations
    [*] Reduction to Quasi-Linear Systems of First Order
    [/LIST]
    [*] Characteristic Normal Forms for Hyperbolic Systems of First Order
    [LIST]
    [*] Linear, Semilinear and Quasi-Linear Systems
    [*] The Case k = 2. Linearization by the Hodograph Transformation
    [/LIST]
    [*] Applications to Dynamics of Compressible Fluids
    [LIST]
    [*] One-Dimensional Isentropic Flow
    [*] Spherically Symmetric Flow
    [*] Steady Irrotational Flow
    [*] Systems of Three Equations for Nonisentropic Flow
    [*] Linearized Equations
    [/LIST]
    [*] Uniqueness. Domain of Dependence
    [LIST]
    [*] Domains of Dependence, Influence, and Determinacy
    [*] Uniqueness Proofs for Linear Differential Equations of Second Order
    [*] General Uniqueness Theorem for Linear Systems of First Order
    [*] Uniqueness for Quasi-Linear Systems
    [*] Energy Inequalities
    [/LIST]
    [*] Riemann's Representation of Solutions
    [LIST]
    [*] The Initial Value Problem
    [*] Riemann's Function
    [*] Symmetry of Riemann's Function
    [*] Riemann's Function and Radiation from a Point. Generalization to Higher Order Problems
    [*] Examples
    [/LIST]
    [*] Solution of Hyperbolic Linear and Semilinear Initial Value Problems by Iteration
    [LIST]
    [*] Construction of the Solution for a Second Order Equation
    [*] Notations and Results for Linear and Semilinear Systems of First Order
    [*] Construction of the Solution
    [*] Remarks. Dependence of Solutions on Parameters
    [*] Mixed Initial and Boundary Value Problems
    [/LIST]
    [*] Cauchy's Problem for Quasi-Linear Systems
    [*] Cauchy's Problem for Single Hyperbolic Differential Equations of Higher Order
    [LIST]
    [*] Reduction to a Characteristic System of First Order
    [*] Characteristic Representation of L[u]
    [*] Solution of Cauchy's Problem
    [*] Other Variants for the Solution. A Theorem by P. Ungar
    [*] Remarks
    [/LIST]
    [*] Discontinuities of Solutions. Shocks
    [LIST]
    [*] Generalized Solutions. Weak Solutions
    [*] Discontinuities for Quasi-Linear Systems Expressing Conservation Laws. Shocks
    [/LIST]
    [*] Appendix: Applications of Characteristics as Coordinates
    [LIST]
    [*] Additional Remarks on General Nonlinear Equations of Second Order
    [LIST]
    [*] The Quasi-Linear Differential Equation
    [*] The General Nonlinear Equation
    [/LIST]
    [*] The Exceptional Character of the Monge-Ampere Equation
    [*] Transition from the Hypprbolic tothp Elliptic Case Through Complex Domains
    [*] The Analyticity of the Solutions in the Elliptic Case
    [LIST]
    [*] Function-Theoretic Remark
    [*] Analyticity of the Solutions of \Delta u = f(x,y,u,p,q)
    [*] Remark on the General Differential Equation F(x, y, u, p, q, r, s, t) = 0
    [/LIST]
    [*] Use of Complex Quantities for the Continuation of Solutions
    [/LIST]
    [*] Appendix: Transient Problems and Heaviside Operational Calculus
    [LIST]
    [*] Solution of Transient Problems by Integral Representation
    [LIST]
    [*] Explicit Example. The Wave Equation
    [*] General Formulation of the Problem
    [*] The Integral of Duhamel
    [*] Method of Superposition of Exponential Solutions
    [/LIST]
    [*] The Heaviside Method of Operators
    [LIST]
    [*] The Simplest Operators
    [*] Examples of Operators and Applications
    [*] Applications to Heat Conduction
    [*] Wave Equation
    [*] Justification of the Operational Calculus Interpretation of Further Operators
    [/LIST]
    [*] General Theory of Transient Problems
    [LIST]
    [*] The Laplace Transformation
    [*] Solution of Transient Problems by the Laplace Transformation
    [*] Example. The Wave and Telegraph Equations
    [/LIST]
    [/LIST]
    [/LIST]
    [*] Hyperbolic Differential Equations in More Than Two Independent Variables
    [LIST]
    [*] Introduction
    [*] Uniqueness, Construction, and Geometry of Solutions
    [LIST]
    [*] Differential Equations of Second Order. Geometry of Characteristics
    [LIST]
    [*] Quasi-Linear Differential Equations of Second Order
    [*] Linear Differential Equations
    [*] Rays or Bicharacteristics
    [*] Characteristics as Wave Fronts
    [*] Invariance of Characteristics
    [*] Ray Cone, Normal Cone, and Ray Conoid
    [*] Connection with a Riemann Metric
    [*] Reciprocal Transformations
    [*] Huyghens' Construction of Wave Fronts
    [*] Space-Like Surfaces. Time-Like Directions
    [/LIST]
    [*] Second Order Equations. The Role of Characteristics
    [LIST]
    [*] Discontinuities of Second Order
    [*] The Differential Equation along a Characteristic Surface
    [*] Propagation of Discontinuities along Rays
    [*] Illustration. Solution of Cauchy's Problem for the Wave Equation in Three Space Dimensions
    [/LIST]
    [*] Geometry of Characteristics for Higher Order Operators
    [LIST]
    [*] Notation
    [*] Characteristic Surfaces, Forms, and Matrices
    [*] Interpretation of the Characteristic Condition in Time and Space. Normal Cone and Normal Surface. Characteristic Nullvectors and Eigenvalues
    [*] Construction of Characteristic Surfaces or Fronts. Hays, Ray Cone, Ray Conoid
    [*] Wave Fronts and Huyghens' Construction. Ray Surface and Normal Surfaces
    [LIST]
    [*] Example
    [/LIST]
    [*] Invariance Properties
    [*] Hyperbolicity. Space-Like Manifolds, Time-Like Directions
    [*] Symmetric Hyperbolic Operators
    [*] Symmetric Hyperbolic Equations of Higher Order
    [*] Multiple Characteristic Sheets and Reducibility
    [*] Lemma on Bicharacteristic Directions
    [/LIST]
    [*] Examples. Hydrodynamics, Crystal Optics, Magnetohydrodynamics
    [LIST]
    [*] Introduction
    [*] The Differential Equation System of Hydrodynamics
    [*] Crystal Optics
    [*] The Shapes of the Normal and Ray Surfaces
    [*] Cauchy's Problem for Crystal Optics
    [*] Magnetohydrodynamics
    [/LIST]
    [*] Propagation of Discontinuities and Cauchy's Problem
    [LIST]
    [*] Introduction
    [*] Discontinuities of First Derivatives for Systems of First Order. Transport Equation.
    [*] Discontinuities of Initial Values. Introduction of Ideal Functions. Progressing Waves
    [*] Propagation of Discontinuities for Systems of First Order
    [*] Characteristics with Constant Multiplicity
    [LIST]
    [*] Examples for Propagation of Discontinuities Along Manifolds of More Than One Dimension. Conical Refraction
    [/LIST]
    [*] Resolution of Initial Discontinuities and Solution of Cauchy's Problem
    [LIST]
    [*] Characteristic Surfaces as Wave Fronts
    [/LIST]
    [*] Solution of Cauchy's Problem by Convergent Wave Expansions
    [*] Systems of Second and Higher Order
    [*] Supplementary Remarks. Weak Solutions. Shocks
    [/LIST]
    [*] Oscillatory Initial Values. Asymptotic Expansion of the Solution. Transition to Geometrical Optics
    [LIST]
    [*] Preliminary Remarks. Progressing Waves of Higher Order
    [*] Construction of Asymptotic Solutions
    [*] Geometrical Optics
    [/LIST]
    [*] Examples of Uniqueness Theorems and Domain of Dependence for Initial Value Problems
    [LIST]
    [*] The Wave Equation
    [*] The Differential Equation u_{tt} - \Delta u + \frac{\lambda}{t} u_t = 0 (Darboux Equation)
    [*] Maxwell's Equations in Vacuum
    [/LIST]
    [*] Domains of Dependence for Hyperbolic Problems
    [LIST]
    [*] Introduction
    [*] Description of the Domain of Dependence
    [/LIST]
    [*] Energy Integrals and Uniqueness for Linear Symmetric Hyperbolic Systems of First Order
    [LIST]
    [*] Energy Integrals and Uniqueness for the Cauchy Problem
    [*] Energy Integrals of First and Higher Order
    [*] Energy Inequalities for Mixed Initial and Boundary Value Problems
    [*] Energy Integrals for Single Second Order Equations
    [/LIST]
    [*] Energy Estimates for Equations of Higher Order
    [LIST]
    [*] Introduction
    [*] Energy Identities and Inequalities for Solutions of Higher Order Hyperbolic Operators. Method of Leray and Garding
    [*] Other Methods
    [/LIST]
    [*] The Existence Theorem
    [LIST]
    [*] Introduction
    [*] The Existence Theorem
    [*] Remarks on Persistence of Properties of Initial Values and on Corresponding Semigroups. Huyghens' Minor Principle
    [*] Focussing. Example of Nonpersistence of Differentiability
    [*] Remarks about Quasi-Linear Systems
    [*] Remarks about Problems of Higher Order or Nonsymmetric Systems
    [/LIST]
    [/LIST]
    [*] Representation of Solution
    [LIST]
    [*] Introduction
    [LIST]
    [*] Outline. Notations
    [*] Some Integral Formulas. Decomposition of Functions into Plane Waves
    [/LIST]
    [*] Equations of Second Order with Constant Coefficients
    [LIST]
    [*] Cauchy's Problem
    [*] Construction of the Solution for the Wave Equation
    [*] Method of Descent
    [*] Further Discussion of the Solution. Huyghens' Principle
    [*] The Nonhomogeneous Equation. Duhamel's Integral
    [*] Cauchy's Problem for the General Linear Equation of Second Order
    [*] The Radiation Problem
    [/LIST]
    [*] Method of Spherical Means. The Wave Equation and the Darboux Equation
    [LIST]
    [*] Darboux's Differential Equation for Mean Values
    [*] Connection with the Wave Equation
    [*] The Radiation Problem for the Wave Equation
    [*] Generalized Progressing Spherical Waves
    [/LIST]
    [*] The Initial Value Problem for Elastic Waves Solved by Spherical Means
    [*] Method of Plane Mean Values. Application to General Hyperbolic Equations with Constant Coefficients
    [LIST]
    [*] General Method
    [*] Application to the Solution of the Wave Equation
    [/LIST]
    [*] Application to the Equations of Crystal Optics and Other Equations of Fourth Order.
    [LIST]
    [*] Solution of Cauchy's Problem
    [*] Further Discussion of the Solution. Domain of Dependence. Gaps
    [/LIST]
    [*] The Solution of Cauchy's Problem as Linear Functional of the Data. Fundamental Solutions
    [LIST]
    [*] Description. Notations
    [*] Construction of the Radiation Function by Decomposition of the Delta Function
    [*] Regularity of the Radiation Matrix
    [LIST]
    [*] The Generalized Huyghens Principle
    [/LIST]
    [*] Example. Special Linear Systems with Constant Coefficients Theorem on Gaps
    [*] Example. The Wave Equation
    [*] Example. Hadamard's Theory for Single Equations of Second Order
    [*] Further Examples. Two Independent Variables. Remarks
    [/LIST]
    [*] U1trahyperbolic Differential Equations and General Differential Equations of Second Order with Constant Coefficients
    [LIST]
    [*] The General Mean Value Theorem of Asgeirsson
    [*] Another Proof of the Mean Value Theorem
    [*] Application to the Wave Equation
    [*] Solutions of the Characteristic Initial Value Problem for the Wave Equation
    [*] Other Applications. The Mean Value Theorem for Confocal Ellipsoids
    [/LIST]
    [*] Initial Value Problcms for Non-Space-Like Initial Manifolds
    [LIST]
    [*] Functions Determined by Mean Values over Spheres with Centers in a Plane
    [*] Applications to the Initial Value Problem
    [/LIST]
    [*] Remarks About Progressing Waves, Transmission of Signals and Huyghens' Principle
    [LIST]
    [*] Distortion-Free Progressing Waves.
    [*] Spherical Waves
    [*] Radiation and Huyghens' Principle
    [/LIST]
    [/LIST]
    [*] Appendix: Ideal Functions or Distributions
    [LIST]
    [*] Underlying Definitions and Concepts
    [LIST]
    [*] Introduction
    [*] Ideal Elements
    [*] Notations and Definitions
    [*] Iterated Integration
    [*] Linear Functionals and Operators - Bilinear Form
    [*] Continuity of Functionals. Support of Test Functions
    [*] Lemma About r-Continuity
    [*] Some Auxiliary Functions
    [*] Examples
    [/LIST]
    [*] Ideal Functions
    [LIST]
    [*] Introduction
    [*] Definition by Linear Differential Operators
    [*] Definition by Weak Limits
    [*] Definition by Linear Functionals
    [*] Equivalence. Representation of Functionals
    [*] Some Conclusions
    [*] Example. The Delta-Function
    [*] Identification of Ideal with Ordinary Functions
    [*] Definite Integrals. Finite Parts
    [/LIST]
    [*] Calculus with Ideal Functions
    [LIST]
    [*] Linear Processes
    [*] Change of Independent Variables
    [*] Examples. Transformations of the Delta-Function
    [*] Multiplication and Convolution of Ideal Functions
    [/LIST]
    [*] Additional Remarks. Modifications of the Theory
    [LIST]
    [*] Introduction
    [*] Different Spaces of Test Functions. The Space S. Fourier Transforms
    [*] Periodic Functions
    [*] Ideal Functions and Hilbert Spaces. Negative Norms Strong Definitions
    [*] Remark on Other Classes of Ideal Functions
    [/LIST]
    [/LIST]
    [/LIST]
    [*] Bibliography
    [*] Index
    [/LIST]
     
     
    Last edited: Feb 3, 2013
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