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Ⅰ．The Algebra of Linear Transformations and Quadratic Forms 1

1．Linear equations and linear transformations 1

1．Vectors 1

2．Orthogonal systems of vectors．Completeness 3

3．Linear transformations．Matrices 5

4．Bilinear,quadratic,and Hermitian forms 11

5．Orthogonal and unitary transformations 14

2．Linear transformations with a linear parameter 17

3．Transformation to principal axes of quadratic and Hermitian forms 23

1．Transformation to principal axes on the basis of a maximum principle 23

2．Eigenvalues 26

3．Generalization to Hermitian forms 28

4．Inertial theorem for quadratic forms 28

5．Representation of the resolvent of a form 29

6．Solution of systems of linear equations associated with forms 30

4．Minimum-maximum property of eigenvalues 31

1．Characterization of eigenvalues by a minimum-maximum problem 31

2．Applications．Constraints 33

5．Supplement and problems 34

1．Linear independence and the Gram determinant 34

2．Hadamard＇s inequality for determinants 36

3．Generalized treatment of canonical transformations 37

4．Bilinear and quadratic forms of infinitely many variables 41

5．Infinitesimal linear transformations 41

6．Perturbations 42

7．Constraints 44

8．Elementary divisors of a matrix or a bilinear form 45

9．Spectrum of a unitary matrix 46

References 47

Ⅱ．Series Expansions of Arbitrary Functions 48

1．Orthogonal systems of functions 49

1．Definitions 49

2．Orthogonalization of functions 50

3．Bessel＇s inequality．Completeness relation．Approximation in the mean 51

4．Orthogonal and unitary transformations with infinitely many variables 55

5．Validity of the results for several independent variables．More general assumptions 56

6．Construction of complete systems of functions of several variables 56

2．The accumulation principle for functions 57

1．Convergence in function space 57

3．Measure of independence and dimension number 61

1．Measure of independence 61

2．Asymptotic dimension of a sequence of functions 63

4．Weierstrass＇s approximation theorem．Completeness of powers and of trigonometric functions 65

1．Weierstrass＇s approximation theorem 65

2．Extension to functions of several variables 68

3．Simultaneous approximation of derivatives 68

4．Completeness of the trigonometric functions 68

5．Fourier series 69

1．Proof of the fundamental theorem 69

2．Multiple Fourier series 73

3．Order of magnitude of Fourier coefficients 74

4．Change in length of basic interval 74

5．Examples 74

6．The Fourier integral 77

1．The fundamental theorem 77

2．Extension of the result to several variables 79

3．Reciprocity formulas 80

7．Examples of Fourier integrals 81

8．Legendre polynomials 82

1．Construction of the Legendre polynomials by orthogonaliza of the powers 1,x,x2 82

2．The generating function 85

3．Other properties of the Legendre polynomials 86

9．Examples of other orthogonal systems 87

1．Generalization of the problem leading to Legendre polynomials 87

2．Tchebycheff polynomials 88

3．Jacobi polynomials 90

4．Hermite polynomials 91

5．Laguerre polynomials 93

6．Completeness of the Laguerre and Hermite functions 95

10．Supplement and problems 97

1．Hurwitz＇s solution of the isoperimetric problem 97

2．Reciprocity formulas 98

3．The Fourier integral and convergence in the mean 98

4．Spectral decomposition by Fourier series and integrals 99

5．Dense systems of functions 100

6．A Theorem of H．Müntz on the completeness of powers 102

7．Fejér＇s summation theorem 102

8．The Mellin inversion formulas 103

9．The Gibbs phenomenon 105

10．A theorem on Gram＇s determinant 107

11．Application of the Lebesgue integral 108

References 111

Ⅲ．Linear Integral Equations 112

1．Introduction 112

1．Notation and basic concepts 112

2．Functions in integral representation 113

3．Degenerate kernels 114

2．Fredholm＇s theorems for degenerate kernels 115

3．Fredholm＇s theorems for arbitrary kernels 118

4．Symmetric kernels and their eigenvalues 122

1．Existence of an eigenvalue of a symmetric kernel 122

2．The totality of eigenfunctions and eigenvalues 126

3．Maximum-minimum property of eigenvalues 132

5．The expansion theorem and its applications 134

1．Expansion theorem 134

2．Solution of the inhomogeneous linear integral equation 136

3．Bilinear formula for iterated kernels 137

4．Mercer＇s theorem 138

6．Neumann series and the reciprocal kernel 140

7．The Fredholm formulas 142

8．Another derivation of the theory 147

1．A lemma 147

2．Eigenfunctions of a symmetric kernel 148

3．Unsymmetric kernels 150

4．Continuous dependence of eigenvalues and eigenfunctions on the kernel 151

9．Extensions of the theory 152

10．Supplement and problems for Chapter Ⅲ 153

1．Problems 153

2．Singular integral equations 154

3．E．Schmidt＇s derivation of the Fredholm theorems 155

4．Enskog＇s method for solving symmetric integral equations 156

5．Kellogg＇s method for the determination of eigenfunctions 156

6．Symbolic functions of a kernel and their eigenvalues 157

7．Example of an unsymmetric kernel without null solutions 157

8．Volterra integral equation 158

9．Abel＇s integral equation 158

10．Adjoint orthogonal systems belonging to an unsymmetric kernel 159

11．Integral equations of the first kind 159

12．Method of infinitely many variables 160

13．Minimum properties of eigenfunctions 161

14．Polar integral equations 161

15．Symmetrizable kernels 161

16．Determination of the resolvent kernel by functional equations 162

17．Continuity of definite kernels 162

18．Hammerstein＇s theorem 162

References 162

Ⅳ．The Calculus of Variations 164

1．Problems of the calculus of variations 164

1．Maxima and minima of functions 164

2．Functionals 167

3．Typical problems of the calculus of variations 169

4．Characteristic difficulties of the calculus of variations 173

2．Direct solutions 174

1．The isoperimetric problem 174

2．The Rayleigh-Ritz method．Minimizing sequences 175

3．Other direct methods．Method of finite differences．Infinitely many variables 176

4．General remarks on direct methods of the calculus of variations 182

3．The Euler equations 183

1．＂Simplest problem＂ of the variational calculus 184

2．Several unknown functions 187

3．Higher derivatives 190

4．Several independent variables 191

5．Identical vanishing of the Euler differential expression 193

6．Euler equations in homogeneous form 196

7．Relaxing of conditions．Theorems of du Bois-Reymond and Haar 199

8．Variational problems and functional equations 205

4．Integration of the Euler differential equation 206

5．Boundary conditions 208

1．Natural boundary conditions for free boundaries 208

2．Geometrical problems．Transversality 211

6．The second variation and the Legendre condition 214

7．Variational problems with subsidiary conditions 216

1．Isoperimetric problems 216

2．Finite subsidiary conditions 219

3．Differential equations as subsidiary conditions 221

8．Invariant character of the Euler equations 222

1．The Euler expression as a gradient in function space．Invariance of the Euler expression 222

2．Transformation of ？u．Spherical coordinates 224

3．Ellipsoidal coordinates 226

9．Transformation of variational problems to canonical and involutory form 231

1．Transformation of an ordinary minimum problem with subsidiary conditions 231

2．Involutory transformation of the simplest variational problems 233

3．Transformation of variational problems to canonical form 238

4．Generalizations 240

10．Variational calculus and the differential equations of mathematical physics 242

1．General remarks 242

2．The vibrating string and the vibrating rod 244

3．Membrane and plate 246

11．Reciprocal quadratic variational problems 252

12．Supplementary remarks and exercises 257

1．Variational problem for a given differential equation 257

2．Reciprocity for isoperimetric problems 258

3．Circular light rays 258

4．The problem of Dido 258

5．Examples of problems in space 258

6．The indicatrix and applications 258

7．Variable domains 260

8．E．Noether＇s theorem on invariant variational problems．Integrals in particle mechanics 262

9．Transversality for multiple integrals 266

10．Euler＇s differential expressions on surfaces 267

11．Thomson＇s principle in electrostatics 267

12．Equilibrium problems for elastic bodies．Castigliano＇s principle 268

13．The variational problem of buckling 272

References 274

Ⅴ．Vibration and Eigenvalue Problems 275

1．Preliminary remarks about linear differential equations 275

1．Principle of superposition 275

2．Homogeneous and nonhomogeneous problems．Boundary conditions 277

3．Formal relations．Adjoint differential expressions．Green＇s formulas 277

4．Linear functional equations as limiting cases and analogues of systems of linear equations 280

2．Systems of a finite number of degrees of freedom 281

1．Normal modes of vibration．Normal coordinates．General theory of motion 282

2．General properties of vibrating systems 285

3．The vibrating string 286

1．Free motion of the homogeneous string 287

2．Forced motion 289

3．The general nonhomogeneous string and the Sturm-Liouville eigenvalue problem 291

4．The vibrating rod 295

5．The vibrating membrane 297

1．General eigenvalue problem for the homogeneous membrane 297

2．Forced motion 300

3．Nodal lines 300

4．Rectangular membrane 300

5．Circular membrane．Bessel functions 302

6．Nonhomogeneous membrane 306

6．The vibrating plate 307

1．General remarks 307

2．Circular boundary 307

7．General remarks on the eigenfunction method 308

1．Vibration and equilibrium problems 308

2．Heat conduction and eigenvalue problems 311

8．Vibration of three-dimensional continua．Separation of variables 313

9．Eigenfunctions and the boundary value problem of potential theory 315

1．Circle,sphere,and spherical shell 315

2．Cylindrical domain 319

3．The Lamé problem 319

10．Problems of the Sturm-Liouville type．Singular boundary points 324

1．Bessel functions 324

2．Legendre functions of arbitrary order 325

3．Jacobi and Tchebycheff polynomials 327

4．Hermite and Laguerre polynomials 328

11．The asymptotic behavior of the solutions of Sturm-Liouville equations 331

1．Boundedness of the solution as the independent variable tends to infinity 331

2．A sharper result．(Bessel functions) 332

3．Boundedness as the parameter increases 334

4．Asymptotic representation of the solutions 335

5．Asymptotic representation of Sturm-Liouville eigenfunctions 336

12．Eigenvalue problems with a continuous spectrum 339

1．Trigonometric functions 340

2．Bessel functions 340

3．Eigenvalue problem of the membrane equation for the infinite plane 341

4．The Schrodinger eigenvalue problem 341

13．Perturbation theory 343

1．Simple eigenvalues 344

2．Multiple eigenvalues 346

3．An example 348

14．Green＇s function (influence function) and reduction of differen tial equations to integral equations 351

1．Green＇s function and boundary value problem for ordinary differential equations 351

2．Construction of Green＇s function;Green＇s function in the generalized sense 354

3．Equivalence of integral and differential equations 358

4．Ordinary differential equations of higher order 362

5．Partial differential equations 363

15．Examples of Green＇s function 371

1．Ordinary differential equations 371

2．Green＇s function for ？u:circle and sphere 377

3．Green＇s function and conformal mapping 377

4．Green＇s function for the potential equation on the surface of a sphere 378

5．Green＇s function for ？u = 0 in a rectangular parallelepiped 378

6．Green＇s function for ？u in the interior of a rectangle 384

7．Green＇s function for a circular ring 386

16．Supplement to Chapter V 388

1．Examples for the vibrating string 388

2．Vibrations of a freely suspended rope;Bessel functions 390

3．Examples for the explicit solution of the vibration equation．Mathieu functions 391

4．Boundary conditions with parameters 392

5．Green＇s tensors for systems of differential equations 393

6．Analytic continuation of the solutions of the equation ？u + λu = 0 395

7．A theorem on the nodal curves of the solutions of ？u + λu = 0 395

8．An example of eigenvalues of infinite multiplicity 395

9．Limits for the validity of the expansion theorems 395

References 396

Ⅵ．Application of the Calculus of Variations to Eigenvalue Problems 397

1．Extremum properties of eigenvalues 398

1．Classical extremum properties 398

2．Generalizations 402

3．Eigenvalue problems for regions with separate components 405

4．The maximum-minimum property of eigenvalues 405

2．General consequences of the extremum properties of the eigenvalues 407

1．General theorems 407

2．Infinite growth of the eigenvalues 412

3．Asymptotic behavior of the eigenvalues in the Sturm-Liouville problem 414

4．Singular differential equations 415

5．Further remarks concerning the growth of eigenvalues．Occurrence of negative eigenvalues 416

6．Continuity of eigenvalues 418

3．Completeness and expansion theorems 424

1．Completeness of the eigenfunctions 424

2．The expansion theorem 426

3．Generalization of the expansion theorem 427

4．Asymptotic distribution of eigenvalues 429

1．The equation ？u + λu = 0 for a rectangle 429

2．The equation ？u + λu = 0 for domains consisting of a finite number of squares or cubes 431

3．Extension to the general differential equation L[u] + λρu = 0 434

4．Asymptotic distribution of eigenvalues for an arbitrary domain 436

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 type 445

6．Nodes of eigenfunctions 451

7．Supplementary remarks and problems 455

1．Minimizing properties of eigenvalues．Derivation from completeness 455

2．Characterization of the first eigenfunction by absence of nodes 458

3．Further minimizing properties of eigenvalues 459

4．Asymptotic distribution of eigenvalues 460

5．Parameter eigenvalue problems 460

6．Boundary conditions containing parameters 461

7．Eigenvalue problems for closed surfaces 461

8．Estimates of eigenvalues when singular points occur 461

9．Minimum theorems for the membrane and plate 463

10．Minimum problems for variable mass distribution 463

11．Nodal points for the Sturm-Liouville problem．Maximum minimum principle 463

References 464

Ⅶ．Special Functions Defined by Eigenvalue Problems 466

1．Preliminary discussion of linear second order differential equations 466

2．Bessel functions 467

1．Application of the integral transformation 468

2．Hankel．functions 469

3．Bessel and Neumann functions 471

4．Integral representations of Bessel functions 474

5．Another integral representation of the Hankel and Bessel functions 476

6．Power series expansion of Bessel functions 482

7．Relations between Bessel functions 485

8．Zeros of Bessel functions 492

9．Neumann functions 496

3．Legendre functions 501

1．Schlafli＇s integral 501

2．Integral representations of Laplace 503

3．Legendre functions of the second kind 504

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

4．Application of the method of integral transformation to Legendre,Tchebycheff,Hermite,and Laguerre equations 506

1．Legendre functions 506

2．Tchebycheff functions 507

3．Hermite functions 508

4．Laguerre function＇s 509

5．Laplace spherical harmonics 510

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

2．Completeness of the system of functions 512

3．Expansion theorem 513

4．The Poisson integral 513

5．The Maxwell-Sylvester representation of spherical harmonics 514

6．Asymptotic expansions 522

1．Stirling＇s formula 522

2．Asymptotic calculation of Hankel and Bessel functions for large values of the arguments 524

3．The saddle point method 526

4．Application of the saddle point method to the calculation of Hankel and Bessel functions for large parameter and large argument 527

5．General remarks on the Baddle point method 532

6．The Darboux method 532

7．Application of the Darboux method to the asymptotic expansion of Legendre polynomials 533

7．Appendix to Chapter Ⅶ．Transformation of Spherical Harmonics 535

1．Introduction and notation 535

2．Orthogonal transformations 536

3．A generating function for spherical harmonics 539

4．Transformation formula 542

5．Expressions in terms of angular coordinates 543

Additional Bibliography 547

Index 551