# Methods of Mathematical Physics v1

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English Pages 598

Preface......Page 3
Contents......Page 4
1. Vectors......Page 22
1. Characterization of eigenvalues by a minimum-maximum problem......Page 24
3. Bessel's inequality. Completeness relation. Approximation in the mean......Page 26
4. Bilinear, quadratic, and Hermitian forms......Page 32
5. Orthogonal and unitary transformations......Page 35
§2. Linear transformations with a linear parameter......Page 38
1. Transformation to principal axes on the basis of a maximum principle......Page 44
2. Eigenvalues......Page 47
4. Inertial theorem for quadratic forms......Page 49
5. Representation of the resolvent of a form......Page 50
6. Solution of systems of linear equations associated with forms......Page 51
2. Applications. Constraints......Page 52
1. Linear independence and the Gram determinant......Page 55
5. Partial differential equations......Page 57
3. Generalized treatment of canonical transformations......Page 58
5. Infinitesimal linear transformations......Page 25
6. Perturbations......Page 63
7. Constraints......Page 65
8. Elementary divisors of a matrix or a bilinear form......Page 66
9. Spectrum of a unitary matrix......Page 67
References......Page 68
II. Series Expansions of Arbitrary Functions......Page 69
1. Definitions......Page 70
2. Orthogonalization of functions......Page 71
4. Orthogonal and unitary transformations with infinitely many variables......Page 76
6. Construction of complete systems of functions of several variables......Page 77
1. Convergence in function space......Page 78
§3. Measure of independence and dimension number......Page 82
1. Measure of independence......Page 27
2. Asymptotic dimension of a sequence of functions......Page 84
1. Weierstrass's approximation theorem......Page 86
4. Completeness of the trigonometric functions......Page 89
1. Proof of the fundamental theorem......Page 90
2. Multiple Fourier series......Page 94
5. Examples......Page 95
1. The fundamental theorem......Page 98
2. Extension of the result to several variables......Page 100
3. Reciprocity formulas......Page 101
§7. Examples of Fourier integrals......Page 102
1. Construction of the Legendre polynomials by orthogonaliza- of the powers 1, x, x2, .........Page 103
2. The generating function......Page 106
3. Other properties of the Legendre polynomials......Page 107
1. Generalization of the problem leading to Legendre polynomials......Page 108
2. Tchebycheff polynomials......Page 109
3. Jacobi polynomials......Page 111
4. Hermite polynomials......Page 112
5. Laguerre polynomials......Page 114
6. Completeness of the Laguerre and Hermite functions......Page 116
1. Hurwitz's solution of the isoperimetric problem......Page 118
3. The Fourier integral and convergence in the mean......Page 119
4. Spectral decomposition by Fourier series and integrals......Page 120
5. Dense systems of functions......Page 121
7. Fejer's summation theorem......Page 123
8. The Mellin inversion formulas......Page 124
9. The Gibbs phenomenon......Page 126
10. A theorem on Gram's determinant......Page 128
11. Application of the Lebesgue integral......Page 129
References......Page 132
1. Notation and basic concept......Page 133
2. Functions in integral representation......Page 134
3. Degenerate kernels......Page 135
§2. Fredholm's theorems for degenerate kernels......Page 136
§3. Fredholm's theorems for arbitrary kernels......Page 139
1. Existence of an eigenvalue of a symmetric kernel......Page 143
2. The totality of eigenfunctions and eigenvalues......Page 147
3. Maximum-minimum property of eigenvalues......Page 153
1. Expansion theorem......Page 155
2. Solution of the inhomogeneous linear integral equation......Page 157
3. Bilinear formula for iterated kernels......Page 158
4. Mercer's theorem......Page 159
§6. Neumann series and the reciprocal kernel......Page 161
§7. The Fredholm formulas......Page 163
1. A lemma......Page 168
2. Eigenfunctions of a symmetric kernel......Page 169
3. Unsymmetric kernels......Page 171
4. Continuous dependence of eigenvalues and eigenfunctions on the kernel......Page 172
§9. Extensions of the theory......Page 173
1. Problems......Page 174
2. Singular integral equations......Page 175
3. E. Schmidt's derivation of the Fredholm theorems......Page 176
5. Kellogg's method for the determination of eigenfunctions......Page 177
7. Example of an unsymmetric kernel without null solutions......Page 178
9. Abel's integral equation......Page 179
11. Integral equations of the first kind......Page 180
12. Method of infinitely many variables......Page 181
15. Symmetrizable kernels......Page 182
References......Page 183
1. Maxima and minima of functions......Page 185
2. Functionals......Page 188
3. Typical problems of the calculus of variations......Page 190
4. Characteristic difficulties of the calculus of variations......Page 194
1. The isoperimetric problem......Page 195
2. The Rayleigh-Ritz method. Minimizing sequences......Page 196
3. Other direct methods. Method of finite differences Infinitely many variables......Page 197
4. General remarks on direct methods of the calculus of variations......Page 203
§3. The Euler equations......Page 204
1. "Simplest problem" of the variational calculus......Page 205
2. Several unknown functions......Page 208
3. Higher derivatives......Page 211
4. Several independent variables......Page 212
5. Identical vanishing of the Euler differential expression......Page 214
6. Euler equations in homogeneous form......Page 217
7. Relaxing of conditions. Theorems of du Bois-Reymond and Haar......Page 220
8. Variational problems and functional equations......Page 226
§4. Integration of the Euler differential equation......Page 227
2. Geometrical problems. Transversality......Page 232
§6. The second variation and the Legendre condition......Page 235
1. Isoperimetric problems......Page 237
2. Finite subsidiary condition......Page 240
3. Differential equations as subsidiary conditions......Page 242
1. The Euler expression as a gradient in function space Invariance of the Euler expression......Page 243
2. Transformation of Au. Spherical coordinates......Page 245
3. Ellipsoidal coordinates......Page 247
1. Transformation of an ordinary minimum problem with subsidiary conditions......Page 252
2. Involutory transformation of the simplest variational problems......Page 254
3. Transformation of variational problems to canonical form......Page 259
4. Generalizations......Page 261
1. General remarks......Page 263
2. The vibrating string and the vibrating rod......Page 265
3. Membrane and plate......Page 267
§11. Reciprocal quadratic variational problems......Page 273
1. Variational problem for a given differential equation......Page 278
6. The indicatrix and applications......Page 279
7. Variable domains......Page 281
8. E. Noether's theorem on invariant variational problems. Integrals in particle mechanics......Page 283
9. Transversality for multiple integrals......Page 287
11. Thomson's principle in electrostatics......Page 288
12. Equilibrium problems for elastic bodies. Castigliano's principle......Page 289
13. The variational problem of buckling......Page 293
References......Page 295
1. Principle of superposition......Page 296
3. Formal relations. Adjoint differential expressions. Green's formulas......Page 298
4. Linear functional equations as limiting cases and analogues of systems of linear equations......Page 301
§2. Systems of a finite number of degrees of freedom......Page 302
1. Normal modes of vibration. Normal coordinates. General theory of motion......Page 303
2. General properties of vibrating systems......Page 306
§3. The vibrating string......Page 307
1. Free motion of the homogeneous string......Page 308
2. Forced motion......Page 310
3. The general nonhomogeneous string and the Sturm-Liouville eigenvalue problem......Page 312
§4. The vibrating rod......Page 316
1. General eigenvalue problem for the homogeneous membrane......Page 318
4. Rectangular membrane......Page 321
5. Circular membrane. Bessel functions......Page 323
6. Nonhomogeneous membrane......Page 327
2. Circular boundary......Page 328
1. Vibration and equilibrium problems......Page 329
2. Heat conduction and eigenvalue problems......Page 332
§8. Vibration of three-dimensional continua. Separation of variables......Page 334
1. Circle, sphere, and spherical shell......Page 336
3. The Lame problem......Page 340
1. Bessel functions......Page 345
2. Legendre functions of arbitrary order......Page 346
3. Jacobi and Tchebycheff polynomials......Page 348
4. Hermite and Laguerre polynomials......Page 349
1. Boundedness of the solution as the independent variable tends to infinity......Page 352
2. A sharper result. (Bessel functions)......Page 353
3. Boundedness as the parameter increases......Page 355
4. Asymptotic representation of the solutions......Page 356
5. Asymptotic representation of Sturm-Liouville eigenfunctions......Page 357
§ 12. Eigenvalue problems with a continuous spectrum......Page 360
2. Bessel functions......Page 361
4. The Schrodinger eigenvalue problem......Page 362
§13. Perturbation theory......Page 364
1. Simple eigenvalues......Page 365
2. Multiple eigenvalues......Page 367
3. An example......Page 369
1. Green's function and boundary value problem for ordinary differential equations......Page 372
2. Construction of Green's function; Green's function in the generalized sense......Page 375
3. Equivalence of integral and differential equations......Page 379
4. Ordinary differential equations of higher order......Page 383
1. Ordinary differential equations......Page 392
3. Green's function and conformal mapping......Page 398
5. Green's function for Au = 0 in a rectangular parallelepiped.......Page 399
6. Green's function for Au in the interior of a rectangle......Page 405
7. Green's function for a circular ring......Page 407
1. Examples for the vibrating string......Page 409
2. Vibrations of a freely suspended rope; Bessel functions......Page 411
3. Examples for the explicit solution of the vibration equation. Mathieu functions......Page 412
4. Boundary conditions with parameters......Page 413
5. Green's tensors for systems of differential equations......Page 414
9. Limits for the validity of the expansion theorems......Page 416
References......Page 417
VI. Applicationof the Calculus of Variations to Eigenvalue Problems......Page 418
1. Classical extremum properties......Page 419
2. Generalizations......Page 423
4. The maximum-minimum property of eigenvalues......Page 426
1. General theorems......Page 428
2. Infinite growth of the eigenvalues......Page 433
3. Asymptotic behavior of the eigenvalues in the Sturm-Liouville problem......Page 435
4. Singular differential equations......Page 436
5. Further remarks concerning the growth of eigenvalues. Occurrence of negative eigenvalues......Page 437
6. Continuity of eigenvalues......Page 439
1. Completeness of the eigenfunctions......Page 445
2. The expansion theorem......Page 447
3. Generalization of the expansion theorem......Page 448
1. The equation Au +Xu= 0 for a rectangle......Page 450
2. The equation Au + Xu = 0 for domains consisting of a finite number of squares or cubes......Page 452
3. Extension to the general differential equation L[u] + Xpu =0......Page 455
4. Asymptotic distribution of eigenvalues for an arbitrary domain......Page 457
5. Sharper form of the laws of asymptotic distribution of eigenvalues for the differential equation Au + Xu = 0......Page 464
§5. Eigenvalue problems of the Schrodinger type......Page 466
§6. Nodes of eigenfunctions......Page 472
1. Minimizing properties of eigenvalues. Derivation from completeness......Page 476
2. Characterization of the first eigenfunction by absence of nodes......Page 479
3. Further minimizing properties of eigenvalues......Page 480
5. Parameter eigenvalue problems......Page 481
8. Estimates of eigenvalues when singular points occur......Page 482
11. Nodal points for the Sturm -Liouville problem. Maximum-minimum principle......Page 484
References......Page 485
§1. Preliminary discussion of linear second order differential equations......Page 487
§2. Bessel functions......Page 488
1. Application of the integral transformation......Page 489
2. Hankel functions......Page 490
3. Bessel and Neumann functions......Page 492
4. Integral representations of Bessel functions......Page 495
5. Another integral representation of the Hankel and Bessel functions......Page 497
6. Power series expansion of Bessel functions......Page 503
7. Relations between Bessel functions......Page 506
8. Zeros of Bessel functions......Page 513
9. Neumann functions......Page 517
1. Schlafli's integral......Page 522
2. Integral representations of Laplace......Page 524
3. Legendre functions of the second kind......Page 525
4. Associated Legendre functions. (Legendre functions of higher order.)......Page 526
1. Legendre functions......Page 527
2. Tchebycheff functions......Page 528
3. Hermite functions......Page 529
4. Laguerre functions......Page 530
§5. Laplace spherical harmonics......Page 531
1. Determination of 2n + 1 spherical harmonics of n-th order......Page 532
2. Completeness of the system of functions......Page 533
4. The Poisson integral......Page 534
5. The Maxwell-Sylvester representation of spherical harmonics......Page 535
§6. Asymptotic expansions......Page 43
1. Stirling's formula......Page 543
2. Asymptotic calculation of Hankel and Bessel functions for large values of the arguments......Page 545
3. The saddle point method......Page 547
4. Application of the saddle point method to the calculation of Hankel and Bessel functions for large parameter and large argument......Page 548
5. General remarks on the saddle point method......Page 553
6. The Darboux method......Page 53
7. Application of the Darboux method to the asymptotic expansion of Legendre polynomials......Page 554
1. Introduction and notation......Page 556
2. Orthogonal transformations......Page 557
3. A generating function for spherical harmonics......Page 560
4. Transformation formula......Page 563
5. Expressions in terms of angular coordinates......Page 564