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Ⅰ．The Algebra of Linear Transformations and Quadratic Forms1

1．Linear equations and linear transformations1

1．Vectors1

2．Orthogonal systems of vectors．Completeness3

3．Linear transformations．Matrices5

4．Bilinear,quadratic,and Hermitian forms11

5．Orthogonal and unitary transformations14

2．Linear transformations with a linear parameter17

3．Transformation to principal axes of quadratic and Hermitian forms23

1．Transformation to principal axes on the basis of a maximum principle23

2．Eigenvalues26

3．Generalization to Hermitian forms28

4．Inertial theorem for quadratic forms28

5．Representation of the resolvent of a form29

6．Solution of systems of linear equations associated with forms30

4．Minimum-maximum property of eigenvalues31

1．Characterization of eigenvalues by a minimum-maximum problem31

2．Applications．Constraints33

5．Supplement and problems34

1．Linear independence and the Gram determinant34

2．Hadamard＇s inequality for determinants36

3．Generalized treatment of canonical transformations37

4．Bilinear and quadratic forms of infinitely many variables41

5．Infinitesimal linear transformations41

6．Perturbations42

7．Constraints44

8．Elementary divisors of a matrix or a bilinear form45

9．Spectrum of a unitary matrix46

References47

Ⅱ．Series Expansions of Arbitrary Functions48

1．Orthogonal systems of functions49

1．Definitions49

2．Orthogonalization of functions50

3．Bessel＇s inequality．Completeness relation．Approximation in the mean51

4．Orthogonal and unitary transformations with infinitely many variables55

5．Validity of the results for several independent variables．More general assumptions56

6．Construction of complete systems of functions of several variables56

2．The accumulation principle for functions57

1．Convergence in function space57

3．Measure of independence and dimension number61

1．Measure of independence61

2．Asymptotic dimension of a sequence of functions63

4．Weierstrass＇s approximation theorem．Completeness of powers and of trigonometric functions65

1．Weierstrass＇s approximation theorem65

2．Extension to functions of several variables68

3．Simultaneous approximation of derivatives68

4．Completeness of the trigonometric functions68

5．Fourier series69

1．Proof of the fundamental theorem69

2．Multiple Fourier series73

3．Order of magnitude of Fourier coefficients74

4．Change in length of basic interval74

5．Examples74

6．The Fourier integral77

1．The fundamental theorem77

2．Extension of the result to several variables79

3．Reciprocity formulas80

7．Examples of Fourier integrals81

8．Legendre polynomials82

1．Construction of the Legendre polynomials by orthogonaliza of the powers 1,x,x282

2．The generating function85

3．Other properties of the Legendre polynomials86

9．Examples of other orthogonal systems87

1．Generalization of the problem leading to Legendre polynomials87

2．Tchebycheff polynomials88

3．Jacobi polynomials90

4．Hermite polynomials91

5．Laguerre polynomials93

6．Completeness of the Laguerre and Hermite functions95

10．Supplement and problems97

1．Hurwitz＇s solution of the isoperimetric problem97

2．Reciprocity formulas98

3．The Fourier integral and convergence in the mean98

4．Spectral decomposition by Fourier series and integrals99

5．Dense systems of functions100

6．A Theorem of H．Müntz on the completeness of powers102

7．Fejér＇s summation theorem102

8．The Mellin inversion formulas103

9．The Gibbs phenomenon105

10．A theorem on Gram＇s determinant107

11．Application of the Lebesgue integral108

References111

Ⅲ．Linear Integral Equations112

1．Introduction112

1．Notation and basic concepts112

2．Functions in integral representation113

3．Degenerate kernels114

2．Fredholm＇s theorems for degenerate kernels115

3．Fredholm＇s theorems for arbitrary kernels118

4．Symmetric kernels and their eigenvalues122

1．Existence of an eigenvalue of a symmetric kernel122

2．The totality of eigenfunctions and eigenvalues126

3．Maximum-minimum property of eigenvalues132

5．The expansion theorem and its applications134

1．Expansion theorem134

2．Solution of the inhomogeneous linear integral equation136

3．Bilinear formula for iterated kernels137

4．Mercer＇s theorem138

6．Neumann series and the reciprocal kernel140

7．The Fredholm formulas142

8．Another derivation of the theory147

1．A lemma147

2．Eigenfunctions of a symmetric kernel148

3．Unsymmetric kernels150

4．Continuous dependence of eigenvalues and eigenfunctions on the kernel151

9．Extensions of the theory152

10．Supplement and problems for Chapter Ⅲ153

1．Problems153

2．Singular integral equations154

3．E．Schmidt＇s derivation of the Fredholm theorems155

4．Enskog＇s method for solving symmetric integral equations156

5．Kellogg＇s method for the determination of eigenfunctions156

6．Symbolic functions of a kernel and their eigenvalues157

7．Example of an unsymmetric kernel without null solutions157

8．Volterra integral equation158

9．Abel＇s integral equation158

10．Adjoint orthogonal systems belonging to an unsymmetric kernel159

11．Integral equations of the first kind159

12．Method of infinitely many variables160

13．Minimum properties of eigenfunctions161

14．Polar integral equations161

15．Symmetrizable kernels161

16．Determination of the resolvent kernel by functional equations162

17．Continuity of definite kernels162

18．Hammerstein＇s theorem162

References162

Ⅳ．The Calculus of Variations164

1．Problems of the calculus of variations164

1．Maxima and minima of functions164

2．Functionals167

3．Typical problems of the calculus of variations169

4．Characteristic difficulties of the calculus of variations173

2．Direct solutions174

1．The isoperimetric problem174

2．The Rayleigh-Ritz method．Minimizing sequences175

3．Other direct methods．Method of finite differences．Infinitely many variables176

4．General remarks on direct methods of the calculus of variations182

3．The Euler equations183

1．＂Simplest problem＂ of the variational calculus184

2．Several unknown functions187

3．Higher derivatives190

4．Several independent variables191

5．Identical vanishing of the Euler differential expression193

6．Euler equations in homogeneous form196

7．Relaxing of conditions．Theorems of du Bois-Reymond and Haar199

8．Variational problems and functional equations205

4．Integration of the Euler differential equation206

5．Boundary conditions208

1．Natural boundary conditions for free boundaries208

2．Geometrical problems．Transversality211

6．The second variation and the Legendre condition214

7．Variational problems with subsidiary conditions216

1．Isoperimetric problems216

2．Finite subsidiary conditions219

3．Differential equations as subsidiary conditions221

8．Invariant character of the Euler equations222

1．The Euler expression as a gradient in function space．Invariance of the Euler expression222

2．Transformation of ？u．Spherical coordinates224

3．Ellipsoidal coordinates226

9．Transformation of variational problems to canonical and involutory form231

1．Transformation of an ordinary minimum problem with subsidiary conditions231

2．Involutory transformation of the simplest variational problems233

3．Transformation of variational problems to canonical form238

4．Generalizations240

10．Variational calculus and the differential equations of mathematical physics242

1．General remarks242

2．The vibrating string and the vibrating rod244

3．Membrane and plate246

11．Reciprocal quadratic variational problems252

12．Supplementary remarks and exercises257

1．Variational problem for a given differential equation257

2．Reciprocity for isoperimetric problems258

3．Circular light rays258

4．The problem of Dido258

5．Examples of problems in space258

6．The indicatrix and applications258

7．Variable domains260

8．E．Noether＇s theorem on invariant variational problems．Integrals in particle mechanics262

9．Transversality for multiple integrals266

10．Euler＇s differential expressions on surfaces267

11．Thomson＇s principle in electrostatics267

12．Equilibrium problems for elastic bodies．Castigliano＇s principle268

13．The variational problem of buckling272

References274

Ⅴ．Vibration and Eigenvalue Problems275

1．Preliminary remarks about linear differential equations275

1．Principle of superposition275

2．Homogeneous and nonhomogeneous problems．Boundary conditions277

3．Formal relations．Adjoint differential expressions．Green＇s formulas277

4．Linear functional equations as limiting cases and analogues of systems of linear equations280

2．Systems of a finite number of degrees of freedom281

1．Normal modes of vibration．Normal coordinates．General theory of motion282

2．General properties of vibrating systems285

3．The vibrating string286

1．Free motion of the homogeneous string287

2．Forced motion289

3．The general nonhomogeneous string and the Sturm-Liouville eigenvalue problem291

4．The vibrating rod295

5．The vibrating membrane297

1．General eigenvalue problem for the homogeneous membrane297

2．Forced motion300

3．Nodal lines300

4．Rectangular membrane300

5．Circular membrane．Bessel functions302

6．Nonhomogeneous membrane306

6．The vibrating plate307

1．General remarks307

2．Circular boundary307

7．General remarks on the eigenfunction method308

1．Vibration and equilibrium problems308

2．Heat conduction and eigenvalue problems311

8．Vibration of three-dimensional continua．Separation of variables313

9．Eigenfunctions and the boundary value problem of potential theory315

1．Circle,sphere,and spherical shell315

2．Cylindrical domain319

3．The Lamé problem319

10．Problems of the Sturm-Liouville type．Singular boundary points324

1．Bessel functions324

2．Legendre functions of arbitrary order325

3．Jacobi and Tchebycheff polynomials327

4．Hermite and Laguerre polynomials328

11．The asymptotic behavior of the solutions of Sturm-Liouville equations331

1．Boundedness of the solution as the independent variable tends to infinity331

2．A sharper result．(Bessel functions)332

3．Boundedness as the parameter increases334

4．Asymptotic representation of the solutions335

5．Asymptotic representation of Sturm-Liouville eigenfunctions336

12．Eigenvalue problems with a continuous spectrum339

1．Trigonometric functions340

2．Bessel functions340

3．Eigenvalue problem of the membrane equation for the infinite plane341

4．The Schrodinger eigenvalue problem341

13．Perturbation theory343

1．Simple eigenvalues344

2．Multiple eigenvalues346

3．An example348

14．Green＇s function (influence function) and reduction of differen tial equations to integral equations351

1．Green＇s function and boundary value problem for ordinary differential equations351

2．Construction of Green＇s function;Green＇s function in the generalized sense354

3．Equivalence of integral and differential equations358

4．Ordinary differential equations of higher order362

5．Partial differential equations363

15．Examples of Green＇s function371

1．Ordinary differential equations371

2．Green＇s function for ？u:circle and sphere377

3．Green＇s function and conformal mapping377

4．Green＇s function for the potential equation on the surface of a sphere378

5．Green＇s function for ？u = 0 in a rectangular parallelepiped378

6．Green＇s function for ？u in the interior of a rectangle384

7．Green＇s function for a circular ring386

16．Supplement to Chapter V388

1．Examples for the vibrating string388

2．Vibrations of a freely suspended rope;Bessel functions390

3．Examples for the explicit solution of the vibration equation．Mathieu functions391

4．Boundary conditions with parameters392

5．Green＇s tensors for systems of differential equations393

6．Analytic continuation of the solutions of the equation ？u + λu = 0395

7．A theorem on the nodal curves of the solutions of ？u + λu = 0395

8．An example of eigenvalues of infinite multiplicity395

9．Limits for the validity of the expansion theorems395

References396

Ⅵ．Application of the Calculus of Variations to Eigenvalue Problems397

1．Extremum properties of eigenvalues398

1．Classical extremum properties398

2．Generalizations402

3．Eigenvalue problems for regions with separate components405

4．The maximum-minimum property of eigenvalues405

2．General consequences of the extremum properties of the eigenvalues407

1．General theorems407

2．Infinite growth of the eigenvalues412

3．Asymptotic behavior of the eigenvalues in the Sturm-Liouville problem414

4．Singular differential equations415

5．Further remarks concerning the growth of eigenvalues．Occurrence of negative eigenvalues416

6．Continuity of eigenvalues418

3．Completeness and expansion theorems424

1．Completeness of the eigenfunctions424

2．The expansion theorem426

3．Generalization of the expansion theorem427

4．Asymptotic distribution of eigenvalues429

1．The equation ？u + λu = 0 for a rectangle429

2．The equation ？u + λu = 0 for domains consisting of a finite number of squares or cubes431

3．Extension to the general differential equation L[u] + λρu = 0434

4．Asymptotic distribution of eigenvalues for an arbitrary domain436

5．Sharper form of the laws of asymptotic distribution of eigenvalues for the differential equation ？u + λu = 0．443

5．Eigenvalue problems of the Schrodinger type445

6．Nodes of eigenfunctions451

7．Supplementary remarks and problems455

1．Minimizing properties of eigenvalues．Derivation from completeness455

2．Characterization of the first eigenfunction by absence of nodes458

3．Further minimizing properties of eigenvalues459

4．Asymptotic distribution of eigenvalues460

5．Parameter eigenvalue problems460

6．Boundary conditions containing parameters461

7．Eigenvalue problems for closed surfaces461

8．Estimates of eigenvalues when singular points occur461

9．Minimum theorems for the membrane and plate463

10．Minimum problems for variable mass distribution463

11．Nodal points for the Sturm-Liouville problem．Maximum minimum principle463

References464

Ⅶ．Special Functions Defined by Eigenvalue Problems466

1．Preliminary discussion of linear second order differential equations466

2．Bessel functions467

1．Application of the integral transformation468

2．Hankel．functions469

3．Bessel and Neumann functions471

4．Integral representations of Bessel functions474

5．Another integral representation of the Hankel and Bessel functions476

6．Power series expansion of Bessel functions482

7．Relations between Bessel functions485

8．Zeros of Bessel functions492

9．Neumann functions496

3．Legendre functions501

1．Schlafli＇s integral501

2．Integral representations of Laplace503

3．Legendre functions of the second kind504

4．Associated Legendre functions．(Legendre functions of higher order．)505

4．Application of the method of integral transformation to Legendre,Tchebycheff,Hermite,and Laguerre equations506

1．Legendre functions506

2．Tchebycheff functions507

3．Hermite functions508

4．Laguerre function＇s509

5．Laplace spherical harmonics510

1．Determination of 2n + 1 spherical harmonics of n-th order．511

2．Completeness of the system of functions512

3．Expansion theorem513

4．The Poisson integral513

5．The Maxwell-Sylvester representation of spherical harmonics514

6．Asymptotic expansions522

1．Stirling＇s formula522

2．Asymptotic calculation of Hankel and Bessel functions for large values of the arguments524

3．The saddle point method526

4．Application of the saddle point method to the calculation of Hankel and Bessel functions for large parameter and large argument527

5．General remarks on the Baddle point method532

6．The Darboux method532

7．Application of the Darboux method to the asymptotic expansion of Legendre polynomials533

7．Appendix to Chapter Ⅶ．Transformation of Spherical Harmonics535

1．Introduction and notation535

2．Orthogonal transformations536

3．A generating function for spherical harmonics539

4．Transformation formula542

5．Expressions in terms of angular coordinates543

Additional Bibliography547

Index551