Handbook of Integral Equations

HANDBOOK OF INTEGRAL EQUATIONS......Page 1
HANDBOOK OF INTEGRAL EQUATIONS......Page 2
ANNOTATION......Page 4
FOREWORD......Page 5
SOME REMARKS AND NOTATION......Page 7
AUTHORS......Page 8
CONTENTS......Page 9
Part I: Exact Solutions of Integral Equations......Page 24
Table of Contents......Page 0
1.1-1. Kernels Linear in the Arguments x and t......Page 25
1.1-2. Kernels Quadratic in the Arguments x and t......Page 26
1.1-3. Kernels Cubic in the Arguments x and t......Page 27
1.1-4. Kernels Containing Higher-Order Polynomials in x and t......Page 28
1.1-5. Kernels Containing Rational Functions......Page 29
1.1-6. Kernels Containing Square Roots......Page 31
1.1-7. Kernels Containing Arbitrary Powers......Page 34
1.2-1. Kernels Containing Exponential Functions......Page 37
1.2-2. Kernels Containing Power-Law and Exponential Functions......Page 40
1.3-1. Kernels Containing Hyperbolic Cosine......Page 43
1.3-2. Kernels Containing Hyperbolic Sine......Page 48
1.3-3. Kernels Containing Hyperbolic Tangent......Page 54
1.3-4. Kernels Containing Hyperbolic Cotangent......Page 56
1.3-5. Kernels Containing Combinations of Hyperbolic Functions......Page 58
1.4-1. Kernels Containing Logarithmic Functions......Page 61
1.4-2. Kernels Containing Power-Law and Logarithmic Functions......Page 63
1.5-1. Kernels Containing Cosine......Page 64
1.5-2. Kernels Containing Sine......Page 69
1.5-3. Kernels Containing Tangent......Page 75
1.5-4. Kernels Containing Cotangent......Page 77
1.5-5. Kernels Containing Combinations of Trigonometric Functions......Page 78
1.6-1. Kernels Containing Arccosine......Page 81
1.6-2. Kernels Containing Arcsine......Page 83
1.6-3. Kernels Containing Arctangent......Page 84
1.6-4. Kernels Containing Arccotangent......Page 86
1.7-1. Kernels Containing Exponential and Hyperbolic Functions......Page 87
1.7-2. Kernels Containing Exponential and Logarithmic Functions......Page 91
1.7-3. Kernels Containing Exponential and Trigonometric Functions......Page 93
1.7-4. Kernels Containing Hyperbolic and Logarithmic Functions......Page 97
1.7-5. Kernels Containing Hyperbolic and Trigonometric Functions......Page 98
1.8-1. Kernels Containing Bessel Functions......Page 99
1.8-2. Kernels Containing Modified Bessel Functions......Page 106
1.8-3. Kernels Containing Associated Legendre Functions......Page 112
1.8-4. Kernels Containing Hypergeometric Functions......Page 113
1.9-1. Equations With Degenerate Kernel: K (•••• , t )=g 1(x )1(t )+g 2(x )2(t )......Page 114
1.9-2. Equations With Difference Kernel: K (•••• , t )=K (•••• – t )......Page 117
1.9-3. Other Equations......Page 124
1.10. Some Formulas and Transformations......Page 126
2.1-1. Kernels Linear in the Arguments x and t......Page 128
2.1-2. Kernels Quadratic in the Arguments x and t......Page 130
2.1-3. Kernels Cubic in the Arguments x and t......Page 133
2.1-4. Kernels Containing Higher-Order Polynomials in x and t......Page 134
2.1-5. Kernels Containing Rational Functions......Page 137
2.1-6. Kernels Containing Square Roots and Fractional Powers......Page 139
2.1-7. Kernels Containing Arbitrary Powers......Page 140
2.2-1. Kernels Containing Exponential Functions......Page 145
2.2-2. Kernels Containing Power-Law and Exponential Functions......Page 152
2.3-1. Kernels Containing Hyperbolic Cosine......Page 155
2.3-2. Kernels Containing Hyperbolic Sine......Page 157
2.3-3. Kernels Containing Hyperbolic Tangent......Page 162
2.3-4. Kernels Containing Hyperbolic Cotangent......Page 163
2.4-1. Kernels Containing Logarithmic Functions......Page 165
2.4-2. Kernels Containing Power-Law and Logarithmic Functions......Page 166
2.5-1. Kernels Containing Cosine......Page 167
2.5-2. Kernels Containing Sine......Page 170
2.5-3. Kernels Containing Tangent......Page 175
2.5-4. Kernels Containing Cotangent......Page 176
2.6-1. Kernels Containing Arccosine......Page 177
2.6-2. Kernels Containing Arcsine......Page 178
2.6-4. Kernels Containing Arccotangent......Page 179
2.7-1. Kernels Containing Exponential and Hyperbolic Functions......Page 180
2.7-2. Kernels Containing Exponential and Logarithmic Functions......Page 181
2.7-3. Kernels Containing Exponential and Trigonometric Functions......Page 182
2.7-4. Kernels Containing Hyperbolic and Logarithmic Functions......Page 186
2.7-5. Kernels Containing Hyperbolic and Trigonometric Functions......Page 187
2.8-1. Kernels Containing Bessel Functions......Page 188
2.8-2. Kernels Containing Modified Bessel Functions......Page 190
2.9-1. Equations With Degenerate Kernel: K (•••• , t )=g 1(x )1(t )+ ···+ g n (•••• )n (•••••......Page 192
2.9-2. Equations With Difference Kernel: K (•••• , t )=K (•••• – t )......Page 204
2.9-3. Other Equations......Page 213
2.10. Some Formulas and Transformations......Page 216
3.1-1. Kernels Linear in the Arguments x and t......Page 218
3.1-2. Kernels Quadratic in the Arguments x and t......Page 220
3.1-3. Kernels Containing Integer Powers of x and t or Rational Functions......Page 221
3.1-4. Kernels Containing Square Roots......Page 223
3.1-5. Kernels Containing Arbitrary Powers......Page 224
3.1-6. Equation Containing the Unknown Function of a Complicated Argument......Page 228
3.1-7. Singular Equations......Page 229
3.2-1. Kernels Containing Exponential Functions......Page 230
3.2-2. Kernels Containing Power-Law and Exponential Functions......Page 232
3.3-1. Kernels Containing Hyperbolic Cosine......Page 233
3.3-2. Kernels Containing Hyperbolic Sine......Page 234
3.3-3. Kernels Containing Hyperbolic Tangent......Page 237
3.4-1. Kernels Containing Logarithmic Functions......Page 238
3.4-2. Kernels Containing Power-Law and Logarithmic Functions......Page 240
3.5-1. Kernels Containing Cosine......Page 241
3.5-2. Kernels Containing Sine......Page 242
3.5-4. Kernels Containing Cotangent......Page 245
3.5-6. Equations Containing the Unknown Function of a Complicated Argument......Page 246
3.6-1. Kernels Containing Hyperbolic and Logarithmic Functions......Page 247
3.6-2. Kernels Containing Logarithmic and Trigonometric Functions......Page 248
3.7-1. Kernels Containing Bessel Functions......Page 249
3.7-3. Other Kernels......Page 250
3.8-1. Equations With Degenerate Kernel......Page 251
3.8-2. Equations Containing Modulus......Page 252
3.8-3. Equations With Difference Kernel: K (•••• , t )=K (•••• – t )......Page 257
3.8-4. Other Equations of the Form b K (•••• , t )(••••••••••••••••••••......Page 258
3.8-5. Equations of the Form b K (•••• , t )( ) dt = F (•••• )......Page 260
4.1-1. Kernels Linear in the Arguments x and t......Page 268
4.1-2. Kernels Quadratic in the Arguments x and t......Page 271
4.1-3. Kernels Cubic in the Arguments x and t......Page 274
4.1-4. Kernels Containing Higher-Order Polynomials in x and t......Page 278
4.1-5. Kernels Containing Rational Functions......Page 281
4.1-6. Kernels Containing Arbitrary Powers......Page 284
4.1-7. Singular Equations......Page 286
4.2-1. Kernels Containing Exponential Functions......Page 287
4.2-2. Kernels Containing Power-Law and Exponential Functions......Page 292
4.3-1. Kernels Containing Hyperbolic Cosine......Page 294
4.3-2. Kernels Containing Hyperbolic Sine......Page 296
4.3-3. Kernels Containing Hyperbolic Tangent......Page 299
4.3-4. Kernels Containing Hyperbolic Cotangent......Page 300
4.4-1. Kernels Containing Logarithmic Functions......Page 301
4.5-1. Kernels Containing Cosine......Page 302
4.5-2. Kernels Containing Sine......Page 305
4.5-3. Kernels Containing Tangent......Page 308
4.5-4. Kernels Containing Cotangent......Page 309
4.5-5. Kernels Containing Combinations of Trigonometric Functions......Page 310
4.6-1. Kernels Containing Arccosine......Page 311
4.6-2. Kernels Containing Arcsine......Page 312
4.6-3. Kernels Containing Arctangent......Page 313
4.6-4. Kernels Containing Arccotangent......Page 314
4.7-2. Kernels Containing Exponential and Logarithmic Functions......Page 315
4.7-3. Kernels Containing Exponential and Trigonometric Functions......Page 316
4.7-4. Kernels Containing Hyperbolic and Logarithmic Functions......Page 317
4.7-5. Kernels Containing Hyperbolic and Trigonometric Functions......Page 318
4.7-6. Kernels Containing Logarithmic and Trigonometric Functions......Page 319
4.8-1. Kernels Containing Bessel Functions......Page 320
4.8-2. Kernels Containing Modified Bessel Functions......Page 322
4.9-1. Equations With Degenerate Kernel: K (•••• , t )=g 1(x )1(t )+ ···+ g n (•••• )n (•••••......Page 323
4.9-2. Equations With Difference Kernel: K (•••• , t )=K (•••• – t ) •......Page 338
4.9-3. Other Equations of the Form y (•••• )+ b K (•••• , t )(••••••••••••••••••••......Page 341
4.9-4. Equations of the Form y (•••• )+ b K (•••• , t )( ) dt = F (•••• )......Page 346
4.10. Some Formulas and Transformations......Page 355
5.1-1. Equations of the Form x y (••••••••••••••••••••••••••......Page 357
5.1-2. Equations of the Form x K (, t )•••• (••••••••••••••••••••••••••......Page 359
5.1-3. Equations of the Form x ( ) dt = F ()......Page 360
5.1-4. Equations of the Form y ()+ x a K (, t )•••• 2 (•••••••••••••••......Page 361
5.1-5. Equations of the Form y ()+ x a K (, t )•••• (••••••••••••••••••••••••••......Page 363
5.2-1. Equations of the Form x ( ) dt = F ()......Page 364
5.2-2. Equations of the Form y ()+ x a K (, t )•••• 2 (•••••••••••••••......Page 365
5.2-3. Equations of the Form y ()+ x ( ) dt = F ()......Page 366
5.3-1. Equations Containing Arbitrary Parameters......Page 367
5.3-2. Equations Containing Arbitrary Functions......Page 369
5.4-1. Equations Containing Arbitrary Parameters......Page 370
5.4-2. Equations Containing Arbitrary Functions......Page 372
5.5-1. Integrands With Nonlinearity of the Form cosh[•y (••••••......Page 373
5.5-2. Integrands With Nonlinearity of the Form sinh[•y (••••••......Page 374
5.5-3. Integrands With Nonlinearity of the Form tanh[•y (••••••......Page 375
5.5-4. Integrands With Nonlinearity of the Form coth[•y (••••••......Page 377
5.6-2. Integrands Containing Exponential Functions of x and t......Page 378
5.7-1. Integrands With Nonlinearity of the Form cos[•y (••••••••......Page 379
5.7-2. Integrands With Nonlinearity of the Form sin[•y (••••••......Page 381
5.7-3. Integrands With Nonlinearity of the Form tan[•y (••••••......Page 382
5.7-4. Integrands With Nonlinearity of the Form cot[•y (••••••......Page 383
5.8-1. Equations of the Form x ( ) dt = F ()......Page 384
5.8-2. Equations of the Form y ()+ x K (, t ) y (•••••••••••••••......Page 385
5.8-3. Equations of the Form y ()+ x a K (, t ) t , y (•••••••••••••••......Page 388
5.8-4. Other Equations......Page 390
6.1-1. Equations of the Form b K (•••••••••• ) dt = F ()......Page 391
6.1-2. Equations of the Form b G ( ) dt = F ()......Page 393
6.1-3. Equations of the Form y ()+ b K (, t )2 (•••••••••••••••......Page 395
6.1-5. Equations of the Form y ()+ b G ( ) dt = F ()......Page 396
6.2-1. Equations of the Form b G ( ) dt = F ()......Page 398
6.2-2. Equations of the Form y ()+ b K (, t )2 (•••••••••••••••......Page 404
6.2-3. Equations of the Form y ()+ b K (, t )n ()m (••••••••••••••••••••••......Page 405
6.2-4. Equations of the Form y ()+ b G ( ) dt = F ()......Page 406
6.3-2. Equations of the Form y ()+ b K (, t )• (•••••••••••••••......Page 410
6.3-3. Equations of the Form y ()+ b G ( ) dt = F ()......Page 412
6.4-2. Other Integrands......Page 413
6.5-2. Integrands With Nonlinearity of the Form sinh[•y (••••••......Page 414
6.5-3. Integrands With Nonlinearity of the Form tanh[•y (••••••......Page 415
6.5-4. Integrands With Nonlinearity of the Form coth[•y (••••••......Page 416
6.5-5. Other Integrands......Page 417
6.6-2. Other Integrands......Page 418
6.7-2. Integrands With Nonlinearity of the Form sin[•y (••••••......Page 419
6.7-3. Integrands With Nonlinearity of the Form tan[•y (••••••......Page 420
6.7-4. Integrands With Nonlinearity of the Form cot[•y (••••••......Page 421
6.7-5. Other Integrands......Page 422
6.8-1. Equations of the Form b G ( ) dt = F ()......Page 423
6.8-2. Equations of the Form y ()+ b K (, t )y (•••••••••••••••......Page 426
6.8-3. Equations of the Form y ()+ b K (, t )t , y (•••••••••••••••......Page 428
6.8-5. Equations of the Form F x , y () + b a G x , t , y (), y (•••••••••••......Page 432
6.8-6. Other Equations......Page 433
Part II: Methods for Solving Integral Equations......Page 445
7.1-1. Some Definitions......Page 446
7.1-2. The Structure of Solutions to Linear Integral Equations......Page 447
7.1-4. Residues. Calculation Formulas......Page 448
7.2-1. Definition. The Inversion Formula......Page 449
7.2-2. The Inverse Transforms of Rational Functions......Page 450
7.2-6. The Post–Widder Formula......Page 451
7.3-1. Definition. The Inversion Formula......Page 452
7.3-3. The Relation Among the Mellin, Laplace, and Fourier Transforms......Page 453
7.4-3. The Alternative Fourier Transform......Page 454
7.5-1. The Fourier Cosine Transform......Page 455
7.6-1. The Hankel Transform......Page 456
7.6-3. The Kontorovich–Lebedev Transform and Other Transforms......Page 457
8.1-1. Equations of the First Kind. Function and Kernel Classes......Page 459
8.2-1. Equations With Kernel of the Form K (•••• , t )=g 1(x )1(t )+g 2(x )2(t )......Page 460
8.2-2. Equations With General Degenerate Kernel......Page 461
8.3-1. The First Method......Page 462
8.4-1. A Solution Method Based on the Laplace Transform......Page 463
8.4-3. Convolution Representation of a Solution......Page 464
8.4-4. Application of an Auxiliary Equation......Page 465
8.4-6. Reduction of a Volterra Equation to a Wiener–Hopf Equation......Page 466
8.5-2. The Definition of Fractional Derivatives......Page 467
8.5-3. Main Properties......Page 468
8.5-4. The Solution of the Generalized Abel Equation......Page 469
8.6-1. A Method of Transformation of the Kernel......Page 470
8.6-2. Kernel With Logarithmic Singularity......Page 471
8.7-1. Quadrature Formulas......Page 472
8.7-2. The General Scheme of the Method......Page 473
8.7-4. An Algorithm for an Equation With Degenerate Kernel......Page 474
8.8-1. An Equation of the First Kind With Variable Lower Limit of Integration......Page 475
8.8-2. Reduction to a Wiener–Hopf Equation of the First Kind......Page 476
9.1-1. Preliminary Remarks. Equations for the Resolvent......Page 477
9.2-1. Equations With Kernel of the Form K (••••••••••••••• )+• (•••••••†•••••......Page 478
9.2-2. Equations With Kernel of the Form K (•••••••••••••••••••••†•••••......Page 479
9.2-3. Equations With Kernel of the Form K (••••••••••••••••••••••••• )(x – t ) m –1......Page 480
9.2-5. Equations With Degenerate Kernel of the General Form......Page 481
9.3-1. A Solution Method Based on the Laplace Transform......Page 482
9.3-3. Reduction to Ordinary Differential Equations......Page 484
9.3-5. Method of Fractional Integration for the Generalized Abel Equation......Page 485
9.4-1. Application of a Solution of a “Truncated” Equation of the First Kind......Page 487
9.4-2. Application of the Auxiliary Equation of the Second Kind......Page 488
9.4-3. A Method for Solving “Quadratic” Operator Equations......Page 489
9.4-5. A Generalization......Page 491
9.5-1. The General Scheme......Page 492
9.5-2. A Generating Function of Exponential Form......Page 493
9.5-3. Power-Law Generating Function......Page 495
9.5-4. Generating Function Containing Sines and Cosines......Page 496
9.6-2. Description of the Method......Page 497
9.6-3. The Model Solution in the Case of an Exponential Right-Hand Side......Page 498
9.6-4. The Model Solution in the Case of a Power-Law Right-Hand Side......Page 499
9.6-6. The Model Solution in the Case of a Cosine-Shaped Right-Hand Side......Page 500
9.7-1. Equations With Kernel Containing a Sum of Exponential Functions......Page 501
9.7-4. Equations Whose Kernels Contain Combinations of Various Functions......Page 502
9.8-2. The Second Method......Page 503
9.9-1. The General Scheme......Page 504
9.10-1. The General Scheme of the Method......Page 505
9.10-3. The Case of a Degenerate Kernel......Page 506
9.11-1. An Equation of the Second Kind With Variable Lower Integration Limit......Page 507
9.11-2. Reduction to a Wiener–Hopf Equation of the Second Kind......Page 508
10.1-2. Integral Equations of the First Kind With Weak Singularity......Page 509
10.1-3. Integral Equations of Convolution Type......Page 510
10.2-1. The Main Equation and the Auxiliary Equation......Page 511
10.3-1. Equation With Difference Kernel on the Entire Axis......Page 512
10.3-3. Equation With Kernel K (, t )=K () and Some Generalizations......Page 513
10.4-1. Relationships Between the Fourier Integral and the Cauchy Type Integral......Page 514
10.4-2. One-Sided Fourier Integrals......Page 515
10.4-4. The Riemann Boundary Value Problem......Page 517
10.4-5. Problems With Rational Coefficients......Page 523
10.4-6. Exceptional Cases. The Homogeneous Problem......Page 524
10.4-7. Exceptional Cases. The Nonhomogeneous Problem......Page 526
10.5-2. Integral Equations of the First Kind With Two Kernels......Page 529
10.6-1. The Carleman Method for Equations With Difference Kernels......Page 532
10.6-2. Exact Solutions of Some Dual Equations of the First Kind......Page 534
10.6-3. Reduction of Dual Equations to a Fredholm Equation......Page 535
10.7-2. The Solution for Large •......Page 539
10.7-3. The Solution for Small •......Page 540
10.8-1. The Lavrentiev Regularization Method......Page 542
10.8-2. The Tikhonov Regularization Method......Page 543
11.1-1. Fredholm Equations and Equations With Weak Singularity of the Second Kind......Page 544
11.1-3. Integral Equations of Convolution Type of the Second Kind......Page 545
11.2-1. The Simplest Degenerate Kernel......Page 546
11.2-2. Degenerate Kernel in the General Case......Page 547
11.3-2. Method of Successive Approximations......Page 550
11.3-3. Construction of the Resolvent......Page 551
11.3-4. Orthogonal Kernels......Page 552
11.4-1. A Formula for the Resolvent......Page 553
11.4-2. Recurrent Relations......Page 554
11.6-1. Characteristic Values and Eigenfunctions......Page 555
11.6-2. Bilinear Series......Page 557
11.6-4. Bilinear Series of Iterated Kernels......Page 558
11.6-5. Solution of the Nonhomogeneous Equation......Page 559
11.6-7. The Resolvent of a Symmetric Kernel......Page 560
11.6-9. Integral Equations Reducible to Symmetric Equations......Page 561
11.7-2. Solution of Equations of the Second Kind on the Semiaxis......Page 562
11.8-1. Equation With Difference Kernel on the Entire Axis......Page 563
11.8-2. An Equation With the Kernel K (, t )=t –1 Q (t ) on the Semiaxis......Page 565
11.8-3. Equation With the Kernel K (, t )=t • Q () on the Semiaxis......Page 566
11.8-4. The Method of Model Solutions for Equations on the Entire Axis......Page 567
11.9-1. The Wiener–Hopf Equation of the Second Kind......Page 568
11.9-2. An Integral Equation of the Second Kind With Two Kernels......Page 572
11.9-3. Equations of Convolution Type With Variable Integration Limit......Page 575
11.9-4. Dual Equation of Convolution Type of the Second Kind......Page 577
11.10-1. Some Remarks......Page 579
11.10-2. The Homogeneous Wiener–Hopf Equation of the Second Kind......Page 580
11.10-3. The General Scheme of the Method. The Factorization Problem......Page 584
11.10-4. The Nonhomogeneous Wiener–Hopf Equation of the Second Kind......Page 585
11.10-5. The Exceptional Case of a Wiener–Hopf Equation of the Second Kind......Page 586
11.11-1. Some Remarks. The Factorization Problem......Page 587
11.11-2. The Solution of the Wiener–Hopf Equations of the Second Kind......Page 589
11.12-1. Krein’s Method......Page 591
11.12-2. Kernels With Rational Fourier Transforms......Page 593
11.12-3. Reduction to Ordinary Differential Equations......Page 594
11.13-1. Approximation of the Kernel......Page 595
11.13-2. The Approximate Solution......Page 596
11.14-1. The General Scheme of the Method......Page 597
11.14-2. Some Special Cases......Page 598
11.15-1. General Remarks......Page 600
11.15-2. The Approximate Solution......Page 601
11.15-3. The Eigenfunctions of the Equation......Page 602
11.16-1. Description of the Method......Page 603
11.16-2. The Construction of Eigenfunctions......Page 604
11.17-2. Characteristic Values......Page 605
11.18-1. The General Scheme for Fredholm Equations of the Second Kind......Page 606
11.18-2. Construction of the Eigenfunctions......Page 607
11.18-3. Specific Features of the Application of Quadrature Formulas......Page 608
11.19-2. The Method of Reducing a System of Equations to a Single Equation......Page 609
11.20-1. Basic Equation and Fredholm Theorems......Page 610
11.20-2. Regularizing Operators......Page 611
11.20-3. The Regularization Method......Page 612
12.1-2. Integral Equations of the First Kind With Hilbert Kernel......Page 614
12.2-1. Definition of the Cauchy Type Integral......Page 615
12.2-3. The Principal Value of a Singular Integral......Page 616
12.2-4. Multivalued Functions......Page 618
12.2-5. The Principal Value of a Singular Curvilinear Integral......Page 619
12.3-1. The Principle of Argument. The Generalized Liouville Theorem......Page 621
12.3-3. Notion of the Index......Page 623
12.3-4. Statement of the Riemann Problem......Page 625
12.3-5. The Solution of the Homogeneous Problem......Page 627
12.3-6. The Solution of the Nonhomogeneous Problem......Page 628
12.3-7. The Riemann Problem With Rational Coefficients......Page 630
12.3-8. The Riemann Problem for a Half-Plane......Page 632
12.3-9. Exceptional Cases of the Riemann Problem......Page 634
12.3-10. The Riemann Problem for a Multiply Connected Domain......Page 638
12.3-12. The Hilbert Boundary Value Problem......Page 641
12.4-2. An Equation With Cauchy Kernel on the Real Axis......Page 642
12.4-3. An Equation of the First Kind on a Finite Interval......Page 643
12.4-4. The General Equation of the First Kind With Cauchy Kernel......Page 644
12.4-5. Equations of the First Kind With Hilbert Kernel......Page 645
12.5-1. A Solution That is Unbounded at the Endpoints of the Interval......Page 646
12.5-2. A Solution Bounded at One Endpoint of the Interval......Page 648
12.5-3. Solution Bounded at Both Endpoints of the Interval......Page 649
13.1-1. Integral Equations With Cauchy Kernel......Page 650
13.1-3. Fredholm Equations of the Second Kind on a Contour......Page 652
13.2-1. A Characteristic Equation With Cauchy Kernel......Page 654
13.2-2. The Transposed Equation of a Characteristic Equation......Page 657
13.2-3. The Characteristic Equation on the Real Axis......Page 658
13.2-4. The Exceptional Case of a Characteristic Equation......Page 660
13.2-6. The Tricomi Equation......Page 662
13.3-1. Closed-Form Solutions in the Case of Constant Coefficients......Page 663
13.3-2. Closed-Form Solutions in the General Case......Page 664
13.4-1. Certain Properties of Singular Operators......Page 665
13.4-2. The Regularizer......Page 667
13.4-3. The Methods of Left and Right Regularization......Page 668
13.4-4. The Problem of Equivalent Regularization......Page 669
13.4-5. Fredholm Theorems......Page 670
13.4-6. The Carleman–Vekua Approach to the Regularization......Page 671
13.4-7. Regularization in Exceptional Cases......Page 672
13.4-8. The Complete Equation With Hilbert Kernel......Page 673
14.1-1. Nonlinear Volterra Integral Equations......Page 676
14.1-2. Nonlinear Equations With Constant Integration Limits......Page 677
14.2-1. The Method of Integral Transforms......Page 678
14.2-2. The Method of Differentiation for Integral Equations......Page 679
14.2-3. The Successive Approximation Method......Page 680
14.2-4. The Newton–Kantorovich Method......Page 682
14.2-5. The Collocation Method......Page 683
14.2-6. The Quadrature Method......Page 684
14.3-1. Nonlinear Equations With Degenerate Kernels......Page 685
14.3-2. The Method of Integral Transforms......Page 687
14.3-3. The Method of Differentiating for Integral Equations......Page 688
14.3-4. The Successive Approximation Method......Page 689
14.3-5. The Newton–Kantorovich Method......Page 690
14.3-7. The Tikhonov Regularization Method......Page 692
Supplements......Page 694
Reduction formulas......Page 695
Powers of trigonometric functions......Page 696
Expansion into power series......Page 697
Products of hyperbolic functions......Page 698
Expansion into power series......Page 699
Addition and subtraction of inverse trigonometric functions......Page 700
Differentiation formulas......Page 701
Expansion into power series......Page 702
Integrals containing a + x and b + x .......Page 703
Integrals containing a 2 + x 2 .......Page 704
Integrals containing a 2 – x 2 .......Page 705
Integrals containing a 3 – x 3 .......Page 706
Integrals containing x 1•••• 2 .......Page 707
Integrals containing (•••• 2 – a 2 ) 1•••• 2 .......Page 708
2.3. Integrals Containing Exponential Functions......Page 709
Integrals containing cosh x .......Page 710
Integrals containing sinh x .......Page 711
2.5. Integrals Containing Logarithmic Functions......Page 712
Integrals containing cos x .......Page 713
Integrals containing sin x .......Page 714
Reduction formulas.......Page 716
2.7 Integrals Containing Inverse Trigonometric Functions......Page 717
3.1. Integrals Containing Power-Law Functions......Page 719
3.2. Integrals Containing Exponential Functions......Page 721
3.3. Integrals Containing Hyperbolic Functions......Page 722
3.4. Integrals Containing Logarithmic Functions......Page 723
3.5. Integrals Containing Trigonometric Functions......Page 724
4.1. General Formulas......Page 727
4.3. Expressions With Exponential Functions......Page 729
4.4. Expressions With Hyperbolic Functions......Page 730
4.5. Expressions With Logarithmic Functions......Page 731
4.6. Expressions With Trigonometric Functions......Page 732
4.7. Expressions With Special Functions......Page 733
5.1. General Formulas......Page 735
5.2. Expressions With Rational Functions......Page 737
5.3. Expressions With Square Roots......Page 741
5.4. Expressions With Arbitrary Powers......Page 743
5.5. Expressions With Exponential Functions......Page 744
5.6. Expressions With Hyperbolic Functions......Page 745
5.7. Expressions With Logarithmic Functions......Page 746
5.9. Expressions With Special Functions......Page 747
6.2. Expressions With Power-Law Functions......Page 749
6.3. Expressions With Exponential Functions......Page 750
6.5. Expressions With Logarithmic Functions......Page 751
6.6. Expressions With Trigonometric Functions......Page 752
6.7. Expressions With Special Functions......Page 753
7.2. Expressions With Power-Law Functions......Page 755
7.3. Expressions With Exponential Functions......Page 756
7.4. Expressions With Hyperbolic Functions......Page 757
7.6. Expressions With Trigonometric Functions......Page 758
7.7. Expressions With Special Functions......Page 759
8.1. General Formulas......Page 762
8.3. Expressions With Exponential Functions......Page 763
8.5. Expressions With Trigonometric Functions......Page 764
8.6. Expressions With Special Functions......Page 765
9.1. Expressions With Power-Law Functions......Page 766
9.2. Expressions With Exponential and Logarithmic Functions......Page 767
9.3. Expressions With Trigonometric Functions......Page 768
9.4. Expressions With Special Functions......Page 769
Binomial coefficients......Page 772
Integral exponent......Page 773
Integral sine......Page 774
Fresnel integrals......Page 775
Asymptotic expansion (Stirling formula)......Page 776
Definitions. Integral representations......Page 777
Definition and basic formulas......Page 778
The Bessel functions for • = ±n ; n =0,1,2,.........Page 779
Integrals with Bessel functions......Page 780
Definitions. Basic formulas......Page 781
Wronskians and similar formulas.......Page 782
Definitions. Basic Formulas......Page 783
Some transformations and linear relations......Page 784
Degenerate hypergeometric functions for n =0,1,.........Page 785
Definition......Page 786
Definitions. Basic formulas......Page 788
Legendre polynomials......Page 789
Legendre polynomials......Page 790
Chebyshev polynomials......Page 791
Jacobi polynomials......Page 792
References......Page 793