# Methods of Mathematical Physics by Hilbert and Courant

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1. Jan 28, 2013

### micromass

Staff Emeritus

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
2. Feb 3, 2013

### micromass

Staff Emeritus
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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