A. D. Polyanin and A. V. Manzhirov Handbook of Integral Equations Second Edition, Updated, Revised and Extended Publisher: Chapman & Hall/CRC Press Publication Date: 14 February 2008 Number of Pages: 1144 Summary Preface Features Contents Index References

Index

A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z

A

• Abel equation
• — first kind, 10
• — generalized, 519, 527
• — Abel equation, generalized, first kind, 531
• — Abel equation, generalized, second kind, 141, 548
• — Abel equation, second kind, 138
• Abel problem, 520
• Abel type two-dimensional equation, 15
• absolutely continuous function, 529
• abstract Hilbert space, 873
• Airy equation, 1023
• Airy function, 1023
• — asymptotic expansions, 1023
• — definition, 1023
• — first kind, 1023
• — power series, 1023
• — second kind, 1023
• algebraic equations
• — linear, infinite system, 858, 861, 864, 868, 971
• — linear, infinite system, symmetric matrix, 850, 853
• alternating sums of powers of natural numbers, 920
• alternative, Fredholm, 637, 638, 643
• —  alternative, Fredholm, symmetric equations, 643
• alternative Fourier transform, 512
• amplitude, 1039
• analysis, functional, 1055
• analytic continuation theorem, 595, 714
• application of integral equations to differential equations, 875
• approach, Carleman–Vekua, 778
• approximate methods
• — nonlinear equations, constant integration limits, 826
• — nonlinear equations, variable integration limit, 811
• approximate solution, 688, 693
• approximate values of eigenvalues, Hilbert–Schmidt kernel, 845
• approximating a kernel, 687
• approximation
• — characteristic values, 646
• — eigenfunctions, Hilbert–Schmidt operator, 868, 872
• — eigenvalues, Hilbert–Schmidt operator, 868, 872
• — kernel, 687
• — Lanczos, 798
• — method, successive, 566, 579, 632, 633, 811, 826, 876
• — solution, 854
• arbitrary functions, 111, 191, 278, 357, 406, 410, 413, 437, 444, 456
• arbitrary parameters, 408, 411, 433, 453
• arbitrary powers, 12, 139, 223, 317, 939, 977
• arccosine, 176, 344
• arccotangent, 178, 347
• arcsine, 177, 345
• arctangent, 178, 346
• argument, complicated, 227, 346, 254
• Arutyunyan equation, 198
• associated Legendre functions, 107, 271, 1031, 1032, 1033
• — first kind, 1032
• — general case, 1032
• — integer indices, real argument, 1031
• — modified, 1033
• — second kind, 1032
• asymmetric form
• — Fourier cosine transform, 514
• — Fourier sine transform, 515
• — Fourier transform, 512
• asymptotic expansions, 509, 1017, 1022–1024, 1035
• — Airy functions, 1023
• — Bessel functions, 1017
• — modified Bessel functions, 1022
• — parabolic cylinder functions, 1035
• — Tricomi confluent hypergeometric functions, 1024
• asymptotic methods, 618
• — equations with logarithmic singularity, 618
• auxiliary conditions, 843, 845, 851, 856, 862, 869, 870
• auxiliary equation, 546, 550, 551, 527
• — application, 527
• — first kind, 550
• — second kind, 551
• auxiliary integral conditions, 841–843
• auxiliary results, 784
• axioms for addition, vectors, 1065
• axioms for addition and multiplication by scalars, vectors, 1065
• axis, real, 575, 713
• — Holder condition, axis, real, Holder condition, 575
• — Sokhotski–Plemelj formulas, 713

B

• Banach space, 1062, 1065, 1067
• base, Napierian, 906
• base of Napierian logarithm, 905
• base of natural logarithm, 905, 906
• basis
• — abstract space, 844, 863
• — Euclidean space, 857, 869
• — Hilbert space, 857, 867, 869
• — Hilbert space, special, 869
• — orthonormal, 855, 856
• Bateman method, 689
• — general scheme, 689
• — special cases, 690
• Bernoulli numbers, 1008
• Bernoulli polynomials, 1052
• Bessel's formula, 1018
• Bessel equation, 1016
• — modified, 1021
• Bessel function, 88, 187, 264, 269, 353, 958, 1016
• — asymptotic expansions, 1017
• — definitions, 1016
• — first kind, 261, 297, 1016
• — integral representations, 1017
• — modified, 97, 189, 269, 355, 1021
• — modified, first kind, 266, 1021
• — modified, second kind, 266, 1021
• — orthogonality properties, 1019
• — second kind, 264, 299, 1016
• — third kind, 1020
• — zeros, 1019
• beta function, 1012, 1014
• — incomplete, 1014, 1015
• bifurcation point, nonlinear integral equations, 834, 835
• bilinear series, 640
• — iterated kernels, 642
• binomial coefficients, 909, 920, 1007
• Boas transform, 250
• boundary conditions, 887
• boundary value problem,
• — first, 895, 896
• — Hilbert, 742
• — linear, representation, 892
• — nth-order differential equations 882
• — ODEs, 881
• — ODEs, reduction to Fredholm equations, 881
• — ODEs, reduction to Volterra equations, 877
• — Riemann, 595, 714
• — second, 895, 897
• — second-order differential equations, 883
• bounded closed domain, 839
• bounded set, 866
• — closed, 842
• bounded variation function, 1055, 1058
• — classes, 1056
• — criteria, 1057
• — definition, 1055
• — properties, 1056, 1057
• Boussinesq equation, 900
• Bubnov–Galerkin method, 697
• Buchholz transform, 274
• Bueckner equation, 801

C

• C(a, b), space of continuous functions, 1066
• C_α(0, 1), Holder space, 1066
• calculation of eigenvalues, 877
• canonical factorization, 680
• canonical form, 805–807
• — Hammerstein equation, 807
• canonical function, nonhomogeneous Riemann problem, 605
• Carleman equation, 243, 590
• Carleman method,
• — characteristic equations, 761
• — equation, convolution type, first kind, 606
• — equation, convolution type, second kind, 660
• — equation, difference kernels, 610
• Carleman–Vekua regularization, 778
• Cauchy criterion, 1067
• Cauchy integral, 708
• Cauchy kernel, 707, 757
• — characteristic equation, 761
• — complete singular equation, 757
• — equation on real axis, 743
• — general singular equation, first kind, 745
• — generalized, 783
• — integral equation, 743, 757
• Cauchy principal value, 709
• Cauchy problem,
• — first-order ODEs, 875, 876
• — ODEs, reduction to integral equations, 875
• — second-order ODEs, 876
• — special $n$th-order linear ODE, 876
• Cauchy residue theorem, 504
• Cauchy–Schwarz–Bunyakovsky inequality, 501
• Cauchy type and Fourier integrals, 592
• Cauchy type integral, 708
• Cauchy-type kernel, 751, 753
• characteristic equation, 758, 761
• — Cauchy kernel, 761
• — exceptional case, 767
• — Hilbert kernel, 769
• — real axis, 765
• — transposed, 758, 764
• characteristic operator, 758
• — transposed, 758
• characteristic value, 301, 625, 637, 639, 645, 697
• — approximation, 646
• — extremal properties, 644
• — system, 640
• Chebyshev formula, 535
• Chebyshev functions, 1049
• Chebyshev nodes, 748
• Chebyshev polynomial,
• — first kind, 109, 1048
• — second kind, 750, 1049
• closed-form solution,
• — case of constant coefficients, 770
• — general case, 771
• closed bounded set, 842
• closed domain, bounded, 839
• closed kernel, 578
• coefficient,
• — binomial, 909, 920, 1007
• — discontinuous, 739
• — rational, 601, 723
• — Riemann problem, 596, 718
• — undetermined, 692
• collocation method, 692, 693, 815
• — hypersingular integral equation, 755
• collocation points, 693
• combination,
• — elementary functions, 73, 255
• — hyperbolic functions, 39
• — trigonometric functions, 63, 252
• compact operator, 842, 843, 1069
• — self-adjoint, 843
• — self-adjoint positive, 873
• — self-adjoint positive definite, 1069
• compact self-adjoint operator, 843
• compact self-adjoint positive definite operator, 1069
• — eigenvalues, 1069
• compact self-adjoint positive operator, 873
• compactness of integral operator, sufficient condition, 842
• compatibility condition, 896, 897
• complementary error function, 1009, 1025
• complementary modulus, 1036, 1037
• complete elliptic integral,
• — first kind, 1035
• — second kind, 1035
• complete equation,
• — generalized Cauchy kernel, 783
• — Hilbert kernel, 780
• complete kernel, 578
• complete orthonormal system of functions, 844, 855
• complete singular integral equation, 757, 770, 772
• — Cauchy kernel, 757
• — Hilbert kernel, 759, 780
• — regularization method, 772
• — solution methods, 757
• complete space, 1067
• complete system, 1067
• complete system of eigenfunctions, 640
• complex linear space, 1065
• complicated argument, 227, 246, 254
• concentration, 890
• — integral equation, 890
• — integral equation, numerical method, 891
• condition,
• — auxiliary integral, 841–843, 845, 851, 856, 862, 869, 870
• — boundary, 887
• — compatibility, 896, 897
• — Holder, 709, 1066
• — Holder, real axis, 575
• — Lipschitz, 709, 1056, 1059
• — normality, 596
• — sufficient for compactness of integral operator, 842
• confluent hypergeometric equation, 1024
• confluent hypergeometric function, 107, 1024
• — Kummer, 1024
• — Tricomi, 1024, 1025
• — Tricomi, asymptotic expansions, 1024
• — Tricomi, integral representations, 1024
• — Whittaker, 1027
• — Wronskian, 1026
• conjugate kernels, 582
• connected domain, 731
• constant,
• — Euler, 533, 1013, 1017, 1026
• — Holder, 709
• continuation,
• — continuation, analytic, 714
• continuity, principle, 714
• continuous function of real argument,
• — values in Banach space, 840
• — values in Hilbert space, 840
• — values in space of functions square integrable over a closed bounded set, 842
• — values in space of functions square integrable over a ring-shaped domain, 841
• — values in space of square integrable functions, 840
• continuous operator, 1068
• contour, smooth, 708
• convergence,
• — almost everywhere, 1060
• — mean-square, 501
• convergent series, 509
• convolution theorem, 507, 513
• convolution type, 574, 606, 660, 669
• coordinate functions, 693, 697
• cosine, 46, 166, 246, 335, 558, 928
• — hyperbolic, 22, 154, 238, 327
• cosine integral, 87, 258, 1011
• cosine transform, 514
• cotangent, 62, 175, 252, 343
• — hyperbolic, 38, 162, 242, 333
• criterion, Cauchy, 1067
• Crum transform, 268
• curves, open, Riemann problem, 734
• cuspidal point, 708
• cylinder function, 1016
• cylindrical function, 1016
• — definitions, 1016

D

• De Moivre formulas, 911
• definite integrals, tables, 951
• definition,
• — Cauchy type integral, 708
• — hyperbolic functions, 913
• degenerate hypergeometric equation, 1024
• degenerate kernel, 111, 191, 278, 357, 519, 522, 539, 540–543, 569, 573, 589, 625, 627, 631, 810, 817
• — general, 523, 628
• — simplest, 627
• density, potential, 893
• derivative,
• — fractional, definition, 529
• — fractional, left-sided, 529
• — fractional, properties, 530
• — fractional, right-sided, 529
• — integrable, fractional, 531
• — logarithmic of gamma function, 1017, 1021
• — Riemann–Liouville, 529
• determinant, Fredholm, 636
• — Fredholm, method, 635
• difference kernel, 114, 203, 283, 372, 519, 524, 539, 544, 573, 574, 586, 625, 610, 626, 655, 683
• — entire axis, 655
• — finite interval, 683
• — weak singularity, 588
• differential equation
• — nth-order, boundary value problems, 882
• — ordinary, 527, 547, 686, 875, 877
• — ordinary, linear, 881
• — second-order, boundary value problems, 883
• differential equation and Volterra integral equations, 877
• differentiating, method for integral equations, 820
• differentiation,
• — fractional, method, 529
• — method, 564, 583, 810
• differentiation formulas, 910, 913, 916, 917
• diffusion flux, integral equations, 890
• digamma function, 1013, 1017
• direct sum of orthogonal subspaces, 845, 863, 869
• Dirichlet–Mehler integral, 1030
• Dirichlet problem,
• — exterior, 896
• — interior, 895
• — reduction to integral equations, 895, 896
• discontinuous coefficient, 739
• divisor transform, 269
• Dixon equation, 136
• domain,
• — bounded closed, 839
• — circular, 841
• — multidimensional, 839
• — one-dimensional, 839
• — ring-shaped, 841, 855, 862
• double layer potential, 893
• — Gauss formula, 894
• dual integral equation,
• — first kind, 295, 575, 610
• — first kind, exact solutions, 613
• — reduction to Fredholm equation, 615
• — second kind, 627
• — second kind, convolution type, 669

E

• eigenfunctions, 301, 625, 639, 834, 867
• — construction, 696, 699
• — extremal properties, 644
• — Fredholm equation, second kind, 694
• — Hilbert–Schmidt kernel, 854, 856, 858, 861, 864, 865
• — Hilbert–Schmidt operator, 871
• — kernel, 844
• — linear operator, 1068
• — nonlinear equation, 834
• — nonlinear operator, 834
• — system, 640
• — system, complete, 640
• — system, incomplete, 640
• eigenvalues, 301, 625, 834
• — calculation, 877
• — compact self-adjoint positive definite operator, 1069
• — Hilbert–Schmidt kernel, 854, 856, 858, 861, 864, 865
• — Hilbert–Schmidt operator, 871
• — kernel, 844
• — linear operator, 1068
• — matrix, 845, 848, 856, 859, 861, 868, 872
• — operator, 867
• — positive, 648
• — self-adjoint operator, 1069
• eigenvectors of matrix, orthonormal, 845, 848, 856, 859, 868, 872
• eigenvectors of self-adjoint operator, 1069
• electrostatic problem, Roben, 897
• elementary functions, 73, 255, 257, 348, 905
• — combinations, 179
• — properties, 905
• elements,
• — linearly dependent, 1065
• — linearly independent, 843, 1065
• elliptic function, 1038
• — Jacobi, 1039
• — Weierstrass, 1042
• elliptic integral, 1035, 1036
• — complete, 1035
• — complete, first kind, 1035
• — complete, second kind, 1035
• — first kind, 1037
• — incomplete, 1037
• — second kind, 1037
• — third kind, 1037
• elliptic modulus, 1037
• elliptic theta functions, 1043
• entire axis, equation, 574, 586, 587, 626, 655
• equation,
• — Abel, first kind, 10
• — Abel, generalized, 519, 527
• — Abel, generalized, first kind, 531
• — Abel, generalized, second kind, 141, 548
• — Abel, second kind, 138
• — Abel type, first kind, 15
• — Abel type, two-dimensional, 15
• — Airy, 1023
• — Arutyunyan, 198
• — auxiliary, 546
• — auxiliary, application, 527
• — auxiliary, first kind, 550
• — auxiliary, second kind, 551
• — Bessel, 1016
• — Bessel, modified, 1021
• — Boussinesq, 900
• — Bueckner, 801
• — Carleman, 243, 590
• — Cauchy kernel, complete, 757
• — Cauchy kernel, first kind, 707
• — Cauchy kernel, first kind, real axis, 743
• — Cauchy kernel, general of first kind, 745
• — Cauchy kernel, simplest of first kind, 707, 743
• — Cauchy kernel, simplest of first kind, real axis, 743
• — characteristic, 758, 761
• — characteristic, Cauchy kernel, 761
• — characteristic, exceptional case, 767
• — characteristic, Hilbert kernel, 769
• — characteristic, real axis, 765
• — characteristic, transposed, 758, 764
• — compact self-adjoint and positive definite operator, 843
• — complete, generalized Cauchy kernels, 783
• — complete, Hilbert kernel, 780
• — complete singular, 757, 770, 772
• — complete singular, Cauchy kernel, 757
• — complete singular, regularization method, 772
• — confluent hypergeometric, 1024
• — contain arbitrary functions, 410, 413
• — contain arbitrary parameters, 408, 411
• — contain modulus, 278, 583
• — contain unknown function of complicated argument, 227, 254
• — convolution type, first kind, 574
• — convolution type, first kind, Carleman method, 606
• — convolution type, second kind, 626, 655, 657
• — convolution type, second kind, Carleman method, 660
• — degenerate kernel, 111, 191, 278, 357, 522, 540–543
• — degenerate kernel, nonlinear, method of differentiation, 810
• — difference kernel, 114, 203, 283, 372, 524, 544, 574, 586, 626, 685
• — difference kernel, Carleman method, 610
• — difference kernel, entire axis, 655
• — difference kernel, finite interval, 683, 685
• — difference kernel, weak singularity, 588
• — differential, 875, 877
• — differential, $n$th-order, boundary value problems, 882
• — differential, ordinary, 527, 547, 686
• — differential, ordinary, linear, 881
• — differential, second-order, boundary value problems, 883
• — diffusion flux, 890
• — Dixon, 136
• — dual, first kind, 295, 575, 610
• — dual, first kind, exact solutions, 613
• — dual, reduction to Fredholm equation, 615
• — dual, second kind, 627
• — dual, second kind, convolution type, 669
• — eigenfunctions, Fredholm equation, second kind, 694
• — elasticity, 621
• — entire axis, 574, 586, 587, 626, 655
• — exact methods, 588–592
• — exact solutions, 3–500
• — exponential nonlinearity, 411, 467
• — finite interval, 683, 685
• — finite interval, first kind, 744
• — first kind, 3, 519, 591, 624
• — first kind, reduction to equations of second kind, 591
• — first kind, weak singularity, 574
• — Fredholm, degenerate kernel, second kind, 627
• — Fredholm, first kind, 573, 623
• — Fredholm, second kind, 625, 685, 698, 701
• — Fredholm, second kind, system, 701
• — Fredholm, second kind on contour, 759
• — Fredholm, spectrum, 760
• — Fredholm, symmetric kernel, second kind, 639
• — Fredholm and dual equations, 615
• — Fredholm and Green's function, 881
• — function of complicated argument, 246
• — Gaussian hypergeometric, 1028
• — Gelfand–Levitan–Marchenko, 900
• — Gelfand–Levitan–Marchenko type, 898
• — general degenerate kernel, 523
• — generalized Abel, 519, 527
• — generalized Abel, first kind, 531
• — generalized Abel, second kind, 141, 548
• — generalized Cauchy kernel, complete, 783
• — generalized Schlomilch, equation, generalized Schl\"omilch, 254
• — Hammerstein, canonical form, 807
• — Hammerstein, first kind, 807
• — Hammerstein, second kind, 807
• — Hammerstein, second kind, degenerate kernel, 817
• — Hammerstein type, 807
• — Hilbert kernel, complete, 759, 780
• — Hilbert kernel, first kind, 707, 746
• — Hilbert kernel, general of first kind, 708, 747
• — Hilbert kernel, simplest of first kind, 707, 746
• — Hilbert kernel, simplest of first kind, complete, 759
• — Hilbert–Plessner, 255
• — homogeneous, 301, 502, 539, 625, 627, 637, 708, 751
• — hyperbolic nonlinearity, 414, 468
• — hypergeometric, 1028
• — hypergeometric, confluent, 1024
• — hypergeometric, degenerate, 1024
• — hypersingular, Cauchy-type kernel, first kind, 751
• — hypersingular, Cauchy-type kernel, general of first kind, 751
• — hypersingular, Cauchy-type kernel, simplest of first kind, 231, 751, 753
• — hypersingular, collocation method, 755
• — hypersingular, Hilbert-type kernel, first kind, 751
• — hypersingular, Hilbert-type kernel, general of first kind, 751
• — hypersingular, Hilbert-type kernel, simplest of first kind, 255, 754
• — hypersingular, numerical methods, 754
• — infinite integration limit, first kind, 537
• — infinite limits of integration, second kind, 702
• — Kadomtsev–Petviashvili, 901
• — kernel contains arbitrary functions, 111, 191, 278, 357
• — kernel contains arbitrary powers, 12
• — kernel contains combinations of elementary functions, 73, 179, 255, 348
• — kernel contains combinations of various functions, 565
• — kernel contains exponential functions, 15, 144, 231, 320
• — kernel contains higher-order polynomials in arguments, 6
• — kernel contains hyperbolic functions, 22, 154, 238, 327
• — kernel contains inverse trigonometric functions, 66, 176, 344
• — kernel contains logarithmic functions, 42, 45, 164, 242, 334
• — kernel contains power-law functions, 4, 45, 127, 217, 301
• — kernel contains rational functions, 7
• — kernel contains special functions, 86, 187, 258, 353
• — kernel contains square roots, 9
• — kernel contains sum of exponential functions, 564
• — kernel contains sum of hyperbolic functions, 564
• — kernel contains sum of trigonometric functions, 564
• — kernel contains trigonometric functions, 46, 166, 246, 335
• — kernel cubic in arguments, 5
• — kernel linear in arguments, 4
• — kernel quadratic in arguments, 4
• — Korteweg–de Vries, 899
• — Korteweg–de Vries, modified, 900
• — Krein's method, 588
• — Lalesco–Picard, 323
• — Laplace, 893
• — Laplace, potentials, properties, 892
• — Laplace, potentials, types, 892
• — Legendre, 1032
• — linear, constant integration limits, 502
• — linear, constant integration limits, first kind, 217, 502, 573
• — linear, constant integration limits, second kind, 301, 502, 625
• — linear, first kind, 502
• — linear, operator methods, 549
• — linear, second kind, 502
• — linear, solution methods, 519, 539, 573, 625
• — linear, structure of solutions, 502
• — linear, variable integration limit, first kind, 3, 502
• — linear, variable integration limit, second kind, 127, 502
• — linear and nonlinear PDEs, 898
• — logarithmic nonlinearity, 419, 472
• — logarithmic singularity, 618
• — logarithmic singularity, asymptotic methods, 618
• — Mathieu, 1045
• — Mathieu, modified, 1046
• — method of differentiating, 564, 583, 820
• — mixed multidimensional, bounded set, projection method, 866
• — mixed multidimensional, closed bounded set, 842
• — mixed multidimensional, Fredholm operator, 842
• — mixed multidimensional, Hilbert–Schmidt operator, 869
• — mixed multidimensional, integral operators of Volterra and Hilbert–Schmidt types, 866
• — mixed multidimensional, integral operators of Volterra and Schmidt types, 866
• — mixed multidimensional, methods of solving, 839–874
• — mixed multidimensional, Schmidt operator, 843
• — mixed multidimensional, Schmidt operator, equivalent form, 843
• — mixed multidimensional, symmetric Fredholm kernel, 842
• — mixed operator, 866, 869
• — mixed operator, auxiliary conditions, 869
• — mixed two-dimensional, circular domain, 841
• — mixed two-dimensional, finite interval, 840
• — mixed two-dimensional, finite interval, methods of solving, 843–854
• — mixed two-dimensional, Hilbert–Schmidt kernel and auxiliary conditions, finite interval, 845
• — mixed two-dimensional, Hilbert–Schmidt kernel and auxiliary conditions, ring-shaped domain, 856
• — mixed two-dimensional, Hilbert–Schmidt kernel and given right-hand side, finite interval, 843
• — mixed two-dimensional, Hilbert–Schmidt kernel and given right-hand side, ring-shaped domain, 855
• — mixed two-dimensional, ring-shaped domain, 841
• — mixed two-dimensional, ring-shaped domain, methods of solving, 855–866
• — mixed two-dimensional, Schmidt kernel, 841
• — mixed two-dimensional, Schmidt kernel, equivalent form, 842
• — mixed two-dimensional, Schmidt kernel and auxiliary conditions, ring-shaped domain, 862
• — mixed two-dimensional, Schmidt kernel and given right-hand side, finite interval, 848
• — modified Bessel, 1021
• — modified Korteweg–de Vries, 900
• — modified Mathieu, 1046
• — Nekrasov, 836
• — nonhomogeneous, 502, 539, 627, 708, 751
• — nonhomogeneous, positive solutions, 649
• — nonhomogeneous, solution, 642
• — nonlinear, 805, 807, 834, 899
• — nonlinear, bifurcation points, 834, 835
• — nonlinear, constant integration limits, 806, 829
• — nonlinear, constant integration limits, approximate methods, 826
• — nonlinear, constant integration limits, exact methods, 817
• — nonlinear, constant integration limits, first kind, 433
• — nonlinear, constant integration limits, numerical methods, 826
• — nonlinear, constant integration limits, second kind, 453
• — nonlinear, degenerate kernels, 817
• — nonlinear, eigenfunctions, 834
• — nonlinear, existence theorems, 830
• — nonlinear, uniqueness theorems, 830
• — nonlinear, variable integration limit, 805
• — nonlinear, variable integration limit, approximate methods, 811
• — nonlinear, variable integration limit, exact methods, 809
• — nonlinear, variable integration limit, first kind, 393
• — nonlinear, variable integration limit, numerical methods, 811
• — nonlinear, variable integration limit, second kind, 403
• — nonlinear, Volterra, 805
• — nonlinear, with parameter, local solutions, 835
• — nonlinearity, general form, 399, 425, 447, 477
• — nonnegative kernel, 648
• — nonsymmetric kernel, first kind, 580
• — one-sided, first kind, 574
• — one-sided, second kind, 626
• — operator, general projection problem, 873
• — operator, mixed, 866, 869
• — operator, mixed with auxiliary conditions, 869
• — operator, quadratic, 552
• — operator, solution, 553
• — ordinary differential, 527, 547, 686
• — parameter, 625
• — Picard–Goursat, 134
• — Poisson, 894
• — power-law nonlinearity, 408, 464
• — power-law nonlinearity that contains arbitrary functions, 444
• — quadratic nonlinearity, 819
• — quadratic nonlinearity that contains arbitrary functions, 397, 406, 437, 456
• — quadratic nonlinearity that contains arbitrary parameters, 393, 403, 433, 453
• — quadratic operator, 552
• — reducible to symmetric equation, 647
• — renewal, 203
• — right-hand side, 519, 539, 573, 625
• — right-hand side, special, 555
• — Schlomilch, 254, 452, 825
• — Schlomilch, generalized, 254
• — Schmidt integral operator, 843
• — Schmidt kernel, 843, 859, 863
• — Schmidt kernel and auxiliary conditions, finite interval, 851
• — Schmidt kernel and auxiliary conditions, ring-shaped domain, 862
• — Schmidt kernel and given right-hand side, finite interval, 848
• — Schmidt kernel and given right-hand side, ring-shaped domain, 859
• — Schmidt operator, 869
• — second kind, 591
• — second kind, operator method, 654
• — semiaxis, 574, 587, 626, 657
• — simplest hypersingular, Cauchy-type kernel, first kind, 231, 753
• — simplest hypersingular, Hilbert-type kernel, first kind, 255, 754
• — single kernel, first kind, 574, 626
• — singular, 228, 255, 319, 344, 707
• — singular, Bueckner type, 801
• — singular, complete, 757, 770, 772
• — singular, first kind, 707, 743
• — singular, generalized kernel, 792
• — singular, numerical solution, 799
• — singular, transposed, 758
• — singular, two-dimensional, 231
• — skew-symmetric, 647
• — solution methods, 501–901
• — special right-hand side, 555
• — surface concentration, 890
• — surface concentration, numerical method, 891
• — symmetric, 639, 647
• — symmetric, Fredholm alternative, 643
• — symmetric kernel, 639
• — symmetric kernel, first kind, 577
• — system, 701
• — transposed, 573, 575, 625, 627, 637
• — transposed of characteristic equation, 764
• — Tricomi, 319, 769, 769
• — Tricomi–Gellerstedt, 320
• — trigonometric nonlinearity, 420, 473
• — truncated first kind, 549
• — two kernels, first kind, 574, 607
• — two kernels, second kind, 626, 664
• — Urysohn, 806, 832
• — Urysohn, first kind, 806, 829
• — Urysohn, first kind, special, method, 821
• — Urysohn, second kind, 806
• — Urysohn, second kind, degenerate kernel, 818
• — Urysohn, second kind, special, method, 822
• — Urysohn type, 806
• — variable integration limit, 3
• — variable lower integration limit, first kind, 537
• — variable lower integration limit, second kind, 570
• — Volterra, 549, 805, 877
• — Volterra, first kind, 519, 524, 565
• — Volterra, first kind, connection with Volterra equations of second kind, 524
• — Volterra, first kind, existence of solution, 519
• — Volterra, first kind, Hammerstein form, 806
• — Volterra, first kind, problems, 520
• — Volterra, first kind, uniqueness of solution, 519
• — Volterra, first kind, Urysohn form, 805, 815
• — Volterra, Hammerstein form, 806
• — Volterra, nonlinear, 805
• — Volterra, quadratic nonlinearity, 809
• — Volterra, reduction to Wiener–Hopf equation, 528
• — Volterra, second kind, 524, 539, 565
• — Volterra, second kind, connection with Volterra equations of first kind, 524
• — Volterra, second kind, Hammerstein form, 816
• — Volterra, second kind, sequence, 855
• — Volterra, second kind, sequence of independent, 853, 865, 872
• — Volterra, second kind, Urysohn form, 805
• — Volterra, sequence, 844, 850, 862
• — Volterra, sequence of independent, 847, 858
• — Volterra, Urysohn form, 805, 811, 814, 816
• — weak singularity, 519
• — weak singularity, first kind, 532, 574
• — weak singularity, second kind, 625
• — weakly singular kernel, 532
• — Whittaker, 1027
• — Wiener–Hopf, 574, 626, 679
• — Wiener–Hopf, first kind, 285, 574, 538, 606
• — Wiener–Hopf, Krein's method, 679
• — Wiener–Hopf, second kind, 373, 547, 571, 626, 660, 679
• — Wiener–Hopf, second kind, exceptional case, 678
• — Wiener–Hopf, second kind, homogeneous, 672
• — Wiener–Hopf, second kind, index, 661
• — Wiener–Hopf, second kind, nonhomogeneous, 677
• — Wiener–Hopf, second kind, solution, 681
• — Wiener–Hopf, Volterra equation, 528
• equidistant surface, method, 891
• equilibrium potential, 897
• equivalent regularization, problem, 776
• Erdelyi–Kober operators, 532
• error function, 86, 258, 549, 1009, 1024
• — complementary, 1009, 1025
• estimates for spectral radius, 649
• Euclidean space, 845, 857, 863, 869, 1067
• — basis, 857, 869
• Euler constant, 533, 1013, 1017, 1026
• Euler formula, 911, 1013
• Euler numbers, 1008
• Euler polynomials, 1053
• exceptional case,
• — characteristic equation, 767
• — regularization, 779
• — Riemann problem, 605, 727
• — Wiener–Hopf equation, second kind, 678
• existence theorems, 875
• — nonlinear equations, 830
• — Stieltjes integral, 1058
• expansion, asymptotic, 509
• —  Airy functions, 1023
• —  Bessel functions, 1017
• —  modified Bessel functions, 1022
• —  parabolic cylinder functions, 1034
• —  Tricomi confluent hypergeometric functions, 1024
• expansion in power series, 910, 913, 916, 918
• exponent, growth, 505
• exponential form, 555
• exponential function, 15, 73, 77, 78, 144, 151, 179–181, 231, 234, 236, 257, 320, 326, 348, 349, 419, 564, 905, 940, 954, 963, 978, 984, 990, 998, 1002
• — properties, 905
• exponential integral, 86, 258, 1009, 1010, 1025
• exponential nonlinearity, 411, 467
• exponents, singularity, 787, 789
• expressions with,
• — arbitrary powers, 977
• — exponential functions, 963, 978, 984, 990, 998, 1002
• — hyperbolic functions, 964, 979, 985, 991
• — logarithmic functions, 965, 980, 985, 992, 999, 1002
• — power-law functions, 963, 983, 989, 998, 1001
• — rational functions, 971
• — special functions, 967, 981, 987, 993, 1000, 1004
• — square roots, 975
• — trigonometric functions, 966, 981, 986, 992, 999, 1003
• exterior Dirichlet problem, 896
• — reduction to integral equations, 896
• exterior Neumann problem, 897
• — reduction to integral equations, 896

F

• factorization, 597, 674, 676, 677, 679, 720, 723
• — canonical, 680
• factorization problem, 676, 679
• Feller potential, 226
• Feller transform, 226
• field of scalars, 1065
• finite functional sums, 922
• finite interval, 683, 840, 843
• — equation, 683, 685
• — integrals, 951, 956
• — mixed equations, 840
• finite numerical sums, 919
• finite sums, 919
• finitely many singular points, 507
• first-order ODEs, 875, 876
• first boundary value problem, 895, 896
• Fischer–Riesz, theorem, 1062
• flow,
• — fluid, 888
• — nonisothermal in plane channel, 884
• fluid flow, 888
• flux, diffusion integral equations, 890
• form
• — canonical, 805–807
• — canonical of Hammerstein equation, 807
• — equivalent of mixed multidimensional equation with Schmidt operator, 843
• — equivalent of mixed two-dimensional equation with Schmidt kernel, 842
• — exponential, 555
• — Hammerstein, for Volterra equation, 806
• — Hammerstein, for Volterra equation of first kind, 806
• — Hammerstein, for Volterra equation of second kind, 816
• — polynomial, 553
• — quadratic, 644
• — Urysohn, for Volterra equation, 805, 811, 814, 816
• — Urysohn, for Volterra equation of first kind, 805, 815
• — Urysohn, for Volterra equation of second kind, 805
• form of infinite products, representation, 910, 916
• formula
• — Bessel's, 1018
• — Chebyshev, 535
• — Euler, 1013
• — Fourier inversion, 512
• — Gauss, 535
• — Gauss, for double layer potential, 894
• — Gauss, for volume potential, 894
• — Green's, 895
• — Hilbert inversion, 746
• — Hopf–Fock, 683
• — Kontorovich–Lebedev inversion, 516
• — Meijer inversion, 516
• — Poincare–Bertrand, 714
• — Poisson's, 1018
• — Post–Widder, 510
• — quadrature, 534, 815
• — Sokhotski–Plemelj, 713, 785
• — Stirling, 1013
• formulas
• — addition, 909, 915
• — calculation, 504
• — De Moivre, 911
• — differentiation, 910, 913, 916, 917
• — Euler, 911
• — integration, 910, 913, 916, 918
• — quadrature, 534, 793
• — reduction, 907, 939, 947
• — Sokhotski–Plemelj, for real axis, 713
• Fourier cosine transform, 514, 518
• — asymmetric form, 514
• — Parseval's relation, 514
• — tables, 983
• Fourier integral,
• — left, 594
• — one-sided, 593, 594
• — relationships with Cauchy type integral, 592
• — right, 594
• Fourier inversion formula, 512
• Fourier sine transform, 514, 518
• — asymmetric form, 515
• — Parseval's relation, 515
• — tables, 989
• Fourier transform, 235, 511, 512, 518, 658
• — alternative, 512
• — asymmetric form, 512
• — definition, 512
• — inverse, 512
• — inversion formula, 512
• — properties, 513
• — rational, 685
• fractional derivative, 529
• — definition, 529
• — integrable, 531
• — left-sided, 529
• — properties, 530
• — right-sided, 529
• fractional differentiation, method, 529
• fractional integral,
• — definition, 529
• — left-sided, 529
• — properties, 530
• — Riemann–Liouville, 529
• — right-sided, 529
• fractional integration, 548
• — by parts, 529
• — operator, 529
• — semigroup property, 529
• fractional order, integral, 529
• fractional powers, 138
• fracture mechanics, 791
• Fredholm alternative, 637, 638
• — symmetric equations, 643
• Fredholm determinant, 636
• — method, 635
• Fredholm equation, 615, 881
• — degenerate kernel, second kind, 627
• — first kind, 573, 623
• — second kind, 625, 685, 698, 701
• — second kind, on contour, 759
• — second kind, system, 701
• — spectrum, 760
• — symmetric kernel, second kind, 639
• Fredholm kernel, 573, 625, 839–841
• — positive definite, 840
• — positive definite, symmetric, 866
• — symmetric definite, 840
• — symmetric positive, 841
• — symmetric positive definite, 866
• Fredholm minor, 636
• Fredholm operator, 758, 842
• — symmetric kernel, generalization, 843
• Fredholm theorems, 637, 702, 777
• Fresnel cosine integral, 1012
• — generalized, 1012
• Fresnel integrals, 87, 258, 1011, 1012
• — generalized, 1012
• Fresnel sine integral, 1012
• — generalized, 1012
• Fubini theorem, 1064
• full measure, set, 1060
• function
• — absolutely continuous, 529
• — Airy, 1023
• — arccosine, 66
• — arccotangent, 71
• — arcsine, 68
• — arctangent, 70
• — associated Legendre, 107, 271, 1030–1033
• — associated Legendre, first kind, 1032
• — associated Legendre, general case, 1032
• — associated Legendre, integer indices and real argument, 1031
• — associated Legendre, second kind, 1032
• — Bessel, 88, 187, 264, 269, 353, 958, 1016
• — Bessel, asymptotic expansions, 1017
• — Bessel, definitions, 1016
• — Bessel, first kind, 261, 297, 1016
• — Bessel, integral representations, 1017
• — Bessel, modified, 97, 189, 269, 355, 1021
• — Bessel, modified, first kind, 266, 1021
• — Bessel, modified, second kind, 266, 1021
• — Bessel, orthogonality properties, 1019
• — Bessel, second kind, 264, 299, 1016
• — Bessel, third kind, 1020
• — Bessel, zeros, 1019
• — beta, 1012, 1014
• — beta, incomplete, 1014, 1015
• — canonical of nonhomogeneous Riemann problem, 605
• — Chebyshev, 1049
• — complementary error, 1009, 1025
• — confluent hypergeometric, 107, 1024
• — confluent hypergeometric, Kummer, 1024
• — confluent hypergeometric, Tricomi, 1024
• — confluent hypergeometric, Whittaker, 1027
• — confluent hypergeometric, Wronskian, 1026
• — cosine, 46
• — cotangent, 62
• — cylinder, 1016
• — cylindrical, 1016
• — digamma, 1013, 1017
• — elementary, 73, 179, 255, 257, 348
• — elementary, properties, 905
• — elliptic, 1038
• — elliptic, Jacobi, 1039
• — elliptic, Weierstrass, 1042
• — elliptic theta, 1043
• — error, 86, 258, 549, 1009, 1024
• — error, complementary, 1009, 1025
• — exponential, 15, 73, 77, 78, 144, 151, 179–181, 213, 234, 236, 257, 320, 326, 348, 349, 419, 564, 905, 940, 954, 963, 978, 984, 990, 998, 1002
• — exponential, properties, 905
• — gamma, 260, 1012
• — gamma, incomplete, 88, 260, 1014, 1024, 1025
• — gamma, logarithmic derivative, 1017, 1021
• — Gauss hypergeometric, 275, 1028
• — generalized Riemann zeta, 277
• — generating, 555, 580
• — generating, power-law, 557
• — generating contain cosines, 558
• — generating contain sines, 558
• — generating of exponential form, 555
• — Green's, 881–883
• — Hankel, 1020
• — Hankel, first kind, 265
• — Hankel, second kind, 265
• — harmonic, 893
• — Hermite, 1050
• — hyperbolic, 22, 73, 83, 84, 154, 164, 179, 185, 186, 238, 255, 327, 334, 348, 351, 352, 564, 911, 913, 922, 940, 955, 964, 979, 985, 991
• — hyperbolic, inverse, 917
• — hyperbolic, of half argument, 915
• — hyperbolic, of multiple argument, 915
• — hypergeometric, 1028
• — hypergeometric, confluent, 107, 1024
• — hypergeometric, confluent, Wronskian, 1026
• — hypergeometric, Gauss, 275, 1028
• — hypergeometric, Kummer confluent, 272
• — hypergeometric, Tricomi confluent, 273, 1025
• — hypergeometric, Whittaker confluent, 274, 1027
• — incomplete beta, 1014, 1015
• — incomplete gamma, 88, 260, 1014, 1024, 1025
• — index, 595
• — influence, 577, 882
• — integrable, 501, 502, 1058
• — integrable, Lebesgue, 1059, 1061
• — inverse hyperbolic, 917
• — inverse trigonometric, 66, 176, 344, 911, 948
• — irrational, 937
• — Jacobi elliptic, 1039
• — Jacobi elliptic, connection with Jacobi theta functions, 1044
• — Jacobi theta, 110, 1043
• — Jacobi theta, connection with Jacobi elliptic functions, 1044
• — Jacobi weight, 793
• — Kummer confluent hypergeometric, 272, 1024
• — Lebesgue integrable, 1059, 1061
• — left, 594
• — Legendre, 270, 1030
• — Legendre, associated, 107, 271, 1030–1033
• — Legendre, associated, first kind, 1032
• — Legendre, associated, second kind, 1032
• — Legendre, modified associated, 1033
• — Legendre, spherical of first kind, 299
• — Legendre, Wronskians, 1034
• — logarithmic, 42, 45, 77, 83, 85, 164, 165, 180, 185, 187, 242, 244, 255, 256, 334, 335, 349, 351, 353, 905, 943, 955, 965, 980, 985, 992, 999, 1002
• — logarithmic, properties, 906
• — MacDonald, 266, 1021
• — Mathieu, 1045, 1046
• — Mathieu, modified, 1046
• — measurable, 1060
• — modified associated Legendre, 1033
• — modified Bessel, 97, 189, 269, 355, 1021
• — modified Bessel, asymptotic expansions, 1022
• — modified Bessel, definitions, 1021
• — modified Bessel, first kind, 266, 1021
• — modified Bessel, integral representations, 1022
• — modified Bessel, second kind, 266, 1021
• — modified Mathieu, 1046
• — multivalued, 711
• — Neumann, 1016
• — of complicated argument, 227, 346, 254
• — one-sided, 594
• — parabolic cylinder, 276, 1034
• — parabolic cylinder, asymptotic expansions, 1035
• — parabolic cylinder, basic formulas, 1034
• — parabolic cylinder, definitions, 1034
• — parabolic cylinder, integral representations, 1035
• — parabolic cylinder, linear relations, 1035
• — parabolic cylinder, Weber, 1034
• — power, properties, 905
• — power-law, 4, 45, 127, 151, 165, 217, 236, 244, 301, 326, 335, 419, 951, 963, 983, 989, 998, 1001
• — power-law generating, 557
• — psi, 1012, 1013
• — rational, 7, 136, 220, 314, 933, 971
• — rational, inverse transforms, 506
• — Riemann zeta, generalized, 277
• — special, 86, 111, 187, 258, 277, 353, 967, 981, 987, 993, 1000, 1004
• — special, properties, 1007
• — spherical, Legendre of first kind, 299
• — square integrable, 501, 502
• — Struve, 264, 299, 516, 518
• — summable, 1059, 1061
• — summable, integral, 1061
• — tangent, 60
• — theta, Jacobi, 1043
• — total variation, 1055
• — Tricomi confluent hypergeometric, 273, 1024, 1025
• — Tricomi confluent hypergeometric, asymptotic expansions, 1024
• — Tricomi confluent hypergeometric, integral representations, 1024
• — trigonometric, 78, 84, 85, 166, 176, 181, 186, 187, 246, 252, 256, 295, 335, 344, 349, 352, 353, 564, 907, 922, 944, 956, 966, 981, 986, 992, 999, 1003
• — trigonometric, inverse, 176, 344, 911, 948
• — trigonometric, of half argument, 909
• — trigonometric, of multiple arguments, 909
• — trigonometric, of single argument, relations, 908
• — trigonometric, powers, 908
• — Weber, 88
• — Weber parabolic cylinder, 1034
• — Weierstrass elliptic, 1042
• — weight, Jacobi, 793
• — Whittaker, 1027
• — Whittaker confluent hypergeometric, 274, 1027
• function of bounded variation, 1055
• function of real argument,
• — values in Banach space, continuous, 840
• — values in Hilbert space, continuous, 840
• — values in space of functions square integrable functions, continuous, 841
• — values in space of functions square integrable over closed bounded set, continuous, 842
• — values in space of functions square integrable over ring-shaped domain, continuous, 841
• function of several variables, 839
• functional analysis, some notions, 1055
• functional series, infinite, 925
• functional sums, finite, 922
• functions
• — coordinate, 693, 697
• — measurable, 1060
• — of bounded variation, 1055, 1058, 1066
• — orthogonal, 582
• — power, 905
• — real-valued, multidimensional, 839
• — with finitely many singular points, 507
• fundamental solution, 881

G

• Galerkin method, 582
• gamma function, 260, 1012
• — incomplete, 88, 260, 1014, 1024, 1025
• — logarithmic derivative, 1017, 1021
• Gauss formula, 535
• — for double layer potential, 894
• — for volume potential, 894
• Gauss hypergeometric functions, 275, 1028
• Gauss transform, 237
• Gaussian hypergeometric equation, 1028
• Gegenbauer polynomials, 1051
• Gelfand–Levitan–Marchenko equation, 900
• general degenerate kernel, 523
• general equation of first kind with Cauchy kernel, 745
• general hypersingular equation of first kind with Cauchy-type kernel, finite interval, 751
• general hypersingular equation of first kind with Hilbert-type kernel, 751
• general projection problem, 873
• — special case, 846, 852, 857, 870
• general scheme,
• — Bateman method, 689
• — method of quadratures, 568
• — quadrature method for Fredholm equations of second kind, 698
• — solving of dual integral equations, 611
• — successive approximation method, 566
• general singular equation of first kind with Hilbert kernel, 708, 747
• generalization of Fredholm integral operator with symmetric kernel, 843
• generalized Abel equation, 519, 527
• — first kind, 531
• — second kind, 141, 548
• generalized Cauchy kernel, 783
• generalized Fresnel cosine integral, 1012
• generalized Fresnel integral, 1012
• generalized Fresnel sine integral, 1012
• generalized Jentzch theorem, 648
• generalized kernel of integral equation, 783
• generalized Laguerre polynomials, 1047
• generalized Liouville theorem, 595, 714
• generalized Mehler–Fock transform, 271
• generalized Riemann zeta function, 277
• generalized Schlomilch equation, 254
• generating function, 555, 580
• — containing cosines, 558
• — containing sines, 558
• — exponential form, 555
• — power-law, 557
• Green's formula, 895
• Green's function, 881–883
• growth exponent, 505

H

• Hammerstein equation, 807, 817, 830
• — canonical form, 807
• — degenerate kernel, second kind, 817
• — first kind, 807
• — second kind, 807
• Hammerstein form, Volterra equation, 806
• — — first kind, 806
• — — second kind, 816
• Hankel function, 1020
• — first kind, 265
• — second kind, 265
• Hankel transform, 261, 515, 518
• — Parseval's relation, 515, 516
• Hardy transform, 264
• harmonic function, 893
• Hartley transform, 252, 518
• Hermite functions, 1050
• Hermite interpolation polynomial, 716
• Hermite polynomial, 108, 1024, 1025, 1050
• Hilbert boundary value problem, 742
• Hilbert inversion formula, 746
• Hilbert kernel, 707, 780
• — characteristic equation, 769
• — complete singular equation, 759, 780
• — equation, 759
• — equations of first kind, 746
• Hilbert Plessner equation, 255
• Hilbert problem, 742
• Hilbert–Schmidt kernel, 841, 843, 845, 853, 855, 856, 860
• Hilbert–Schmidt kernel, approximate values of eigenvalues, 845
• Hilbert–Schmidt kernel, eigenfunctions, 854, 856, 858, 861, 864, 865
• Hilbert–Schmidt operator, 842, 843, 866, 871
• — approximation for eigenfunctions, 868, 872
• — approximation for eigenvalues, 868, 872
• — eigenfunctions, 871
• — eigenvalues, 871
• Hilbert–Schmidt theorem, 641, 1069
• Hilbert–Schmidt theory, 843
• Hilbert space, 839, 845, 857, 863, 867, 869, 1067
• — abstract, 873
• — basis, 857, 867, 869
• — linear operators, 1067, 1068
• — special basis, 869
• Hilbert transform, 228, 255, 518, 743
• Hilbert transform on semiaxis, 229
• Hilbert type kernel, 751, 754
• Holder condition, 709, 1066
• Holder condition on real axis, 575
• Holder constant, 709
• Holder inequality, 1065
• Holder space, 1066
• homogeneous integral equation, 301, 502, 539, 625, 627, 637, 708, 751
• homogeneous problem, 596, 602, 742
• homogeneous problem solution, 720
• homogeneous Wiener–Hopf equation, second kind, 672
• Hopf–Fock formula, 683
• hyperbolic cosine, 22, 154, 238, 327
• hyperbolic cotangent, 38, 162, 242, 333
• hyperbolic function, 22, 73, 83, 84, 154, 179, 185, 186, 238, 255, 327, 334, 348, 351, 352, 564, 911, 913, 922, 940, 955, 964, 979, 985, 991
• — combinations, 164
• — half argument, 915
• — multiple argument, 915
• hyperbolic nonlinearity, 414, 468
• hyperbolic sine, 28, 156, 238, 329
• hyperbolic tangent, 36, 161, 241, 332
• hypergeometric equation, 1028
• — confluent, 1024
• — degenerate, 1024
• hypergeometric function, 1028
• — confluent, 107, 1024
• — confluent, Kummer, 272, 1024
• — confluent, Tricomi, 1024, 1025
• — confluent, Whittaker, 274, 1027
• — confluent, Wronskian, 1026
• — Gauss, 275, 1028
• — Gauss, basic properties, 1028
• — Kummer confluent, 272, 1024
• — Tricomi confluent, 273
• — Whittaker confluent, 274, 1027
• hypergeometric series, 1028
• hypersingular equation, 751
• — Cauchy-type kernel, 751, 753
• — collocation method, 755
• — first kind, Cauchy-type kernel on finite interval, 751
• — first kind, Hilbert-type kernel, 751
• — Hilbert-type kernel, 751, 754
• — numerical methods, 754
• — simplest of first kind, Cauchy-type kernel, 231, 753
• — simplest of first kind, Hilbert-type kernel, 255, 754
• hypersingular integral,
• — definition, 751
• — in sense of Hadamard principal value, 752

I

• identities, integral, 895
• identity operator, 842, 873
• ill-posed problem, 623
• — general notions, 623
• incomplete beta function, 1014, 1015
• incomplete elliptic integrals, 1036
• incomplete gamma function, 88, 260, 1014, 1024, 1025
• incomplete kernel, 578
• incomplete system of eigenfunctions, 640
• indefinite integrals, tables, 933
• independent elements, linearly, 843, 1063
• index, 603, 661, 664
• — notion, 716
• index of function, 595
• index of Riemann problem, 596, 731
• index of Wiener–Hopf equation, 661
• inequality,
• — Cauchy–Schwarz–Bunyakovsky, 501
• — Holder, 1063
• — triangle, 501
• infinite functional series, 925
• infinite numerical series, 924
• infinite products, 910, 916
• infinite system of linear algebraic equations, 858, 861, 864, 868, 971
• infinite system of linear algebraic equations with symmetric matrix, 850, 853
• influence function, 577, 882
• inner product, 501, 644
• integrable fractional derivative, 531
• integrable function, 501, 502, 1056
• — Lebesgue, 1059
• integral,
• — Cauchy, 708
• — Cauchy type, 708
• — Cauchy type, relationships with Fourier integral, 592
• — complete elliptic, 1035
• — complete elliptic, first kind, 1035
• — complete elliptic, second kind, 1035
• — cosine, 87, 258, 1011
• — definite, tables, 951
• — Dirichlet–Mehler, 1030
• — elliptic, 1035, 1036
• — elliptic, complete, 1035
• — elliptic, first kind, 1036
• — elliptic, incomplete, 1036
• — elliptic, second kind, 1036
• — elliptic, third kind, 1036
• — exponential, 86, 258, 1009, 1010, 1025
• — Fourier, left, 594
• — Fourier, one-sided, 593, 594
• — Fourier, relationships with Cauchy type integral, 592
• — Fourier, right, 594
• — fractional, definition, 529
• — fractional, left-sided, 529
• — fractional, properties, 530
• — fractional, Riemann–Liouville, 529
• — fractional, right-sided, 529
• — fractional order, 529
• — Fresnel, 87, 258, 1011, 1012
• — Fresnel, generalized, 1012
• — Fresnel cosine, 1012
• — Fresnel cosine, generalized, 1012
• — Fresnel sine, 1012
• — Fresnel sine, generalized, 1012
• — hypersingular, definition, 751
• — hypersingular, in sense of Hadamard principal value, 752
• — incomplete elliptic, 1036
• — indefinite, tables, 933
• — involving arbitrary powers, 939
• — involving Bessel functions, 958
• — involving exponential functions, 940, 954
• — involving hyperbolic functions, 940, 955
• — involving inverse trigonometric functions, 948
• — involving irrational functions, 937
• — involving logarithmic functions, 943, 955
• — involving power-law functions, 951
• — involving rational functions, 933
• — involving trigonometric functions, 944, 956
• — Jacobi weight function, 793
• — Laplace, 1030
• — Lebesgue, 1057
• — Lebesgue, definition, 1059
• — Lebesgue, properties, 1059
• — left Fourier, 594
• — logarithmic, 258, 1009, 1010, 1025
• — Mehler, 299, 615
• — one-sided Fourier, 593, 594
• — probability, 1009
• — Riemann, 1057
• — Riemann–Liouville fractional, 529
• — right-sided fractional, 529
• — right Fourier, 594
• — sine, 87, 258, 1011
• — singular, 709
• — singular, principal value, 709
• — step-function, 1059
• — Stieltjes, 1055, 1056
• — Stieltjes, basic definitions, 1055
• — Stieltjes, existence theorems, 1056
• — Stieltjes, properties, 1056
• — summable function, 1059
• integral conditions, auxiliary, 841–843
• integral equation, {\it see\/ equation
• integral identities, 895
• integral operator,
• — compactness, sufficient condition, 842
• — Fredholm, 842
• — Fredholm, symmetric kernel, 843
• — Hilbert–Schmidt, 842, 843, 866
• — positive definite, 842
• — positive definite kernel, 843
• — Schmidt, 843, 866
• — self-adjoint, 842, 843
• — spectral radius, 649
• — symmetric kernel, 843
• — Volterra, 842
• integral representations,
• — Bessel functions, 1017
• — modified Bessel functions, 1022
• — parabolic cylinder functions, 1034
• — Tricomi confluent hypergeometric functions, 1024
• integral sum, Stieltjes, 1055
• integral transform, {\it see\/ transform
• integrand,
• — contain exponential functions, 419
• — contain power-law functions, 419
• — nonlinearity, 414–416, 418, 420, 422–424, 467–470, 472–475
• integration,
• — fractional, 548
• — fractional, by parts, 529
• — fractional, operator, 529
• — fractional, semigroup property, 529
• interior Dirichlet problem, 895
• — reduction to integral equations, 895
• interior Neumann problem, 895
• — reduction to integral equations, 895
• interpolation nodes, 534
• interpolation polynomial,
• — Hermite, 716
• — Lagrange, 748
• inverse Fourier transform, 512
• inverse hyperbolic functions, 917
• inverse Laplace transforms, tables, 969
• inverse Mellin transform, 510, 1001
• inverse transform,
• — rational functions, 506
• — representation as asymptotic expansions, 509
• — representation as convergent series, 509
• inverse trigonometric function, 66, 176, 344, 911, 948
• inversion formula,
• — Hilbert, 746
• — Kontorovich–Lebedev, 516
• — Meijer, 516
• inversion of functions with finitely many singular points, 507
• investigation of differential equations, 875
• irrational functions, 937
• iterated kernel, 566, 632
• — bilinear series, 642
• iteration process, 811, 814

J

• Jacobi elliptic function, 1038
• — connection with Jacobi theta functions, 1042
• Jacobi polynomials, 1049
• Jacobi theta function, 110, 1042
• — connection with Jacobi elliptic functions, 1042
• — properties, 1042
• — relations and formulas, 1042
• — series representation, 1042
• Jacobi weight function, 793
• Jentzch theorem, generalized, 648
• Jordan lemma, 505
• jump problem, 596

K

• K-transform, 518
• Kadomtsev–Petviashvili equation, 901
• Kellog's method for finding characteristic values in case of symmetric kernel, 645
• kernel
• — approximation, 687
• — Cauchy, 707, 757
• — Cauchy, characteristic equation, 761
• — Cauchy, complete singular integral equation, 757
• — Cauchy, generalized, 783
• — Cauchy, integral equations, 757
• — Cauchy-type, 751, 753
• — closed, 578
• — complete, 578
• — conjugate, 582
• — containing arbitrary functions, 111, 191, 278, 357
• — containing arbitrary powers, 12, 139, 223, 317
• — containing arccosine, 66, 176, 344
• — containing arccotangent, 71, 178, 347
• — containing arcsine, 68, 177, 345
• — containing arctangent, 70, 178, 346
• — containing associated Legendre functions, 107, 271
• — containing Bessel functions, 88, 187, 353
• — containing Bessel functions of first kind, 261, 297
• — containing Bessel functions of second kind, 264, 299
• — containing Chebyshev polynomials, 109
• — containing combination of Bessel and modified Bessel functions, 269
• — containing combination of Bessel functions, 264
• — containing combination of elementary functions, 179, 255, 348
• — containing combination of hyperbolic functions, 39, 164, 334
• — containing combination of trigonometric functions, 63, 176, 252, 344
• — containing combination of various functions, 565
• — containing confluent hypergeometric functions, 107
• — containing cosine, 46, 166, 246, 335
• — containing cosine integral, 87
• — containing cosine integrals, 258
• — containing cotangent, 62, 175, 252, 343
• — containing elementary functions, 257
• — containing error function, 86, 258
• — containing exponential function, 15, 19, 73, 77, 78, 144, 151, 179–181, 231, 234, 236, 257, 320, 326, 348, 349
• — containing exponential integral, 86, 258
• — containing fractional powers, 138
• — containing Fresnel integral, 87, 258
• — containing gamma function, 260
• — containing Gauss hypergeometric function, 275
• — containing Hermite polynomial, 108
• — containing higher-order polynomial in arguments, 6, 133, 311
• — containing hyperbolic cosine, 22, 154, 238, 237
• — containing hyperbolic cotangent, 38, 162, 242, 333
• — containing hyperbolic function, 22, 73, 83, 84, 154, 179, 185, 186, 238, 255, 327, 348, 351, 352
• — containing hyperbolic sine, 28, 156, 238, 329
• — containing hyperbolic tangent, 36, 161, 241, 332
• — containing incomplete gamma function, 88, 260
• — containing integer powers of arguments, 220
• — containing inverse trigonometric function, 66, 176, 344
• — containing Jacobi theta functions, 110
• — containing Kummer confluent hypergeometric function, 272
• — containing Laguerre polynomial, 110
• — containing Legendre function, 270
• — containing Legendre polynomial, 105
• — containing Legendre spherical function of first kind, 299
• — containing logarithmic function, 42, 45, 77, 83, 85, 164, 165, 180, 185, 187, 242, 244, 255, 256, 334, 335, 349, 351, 353
• — containing logarithmic integral, 258
• — containing modified Bessel function, 97, 189, 355
• — containing modified Bessel function of first kind, 266
• — containing modified Bessel function of second kind, 266
• — containing other special function, 111, 277
• — containing parabolic cylinder function, 276
• — containing power-law function, 4, 19, 45, 127, 151, 165, 217, 236, 244, 301, 326, 335
• — containing rational function, 7, 136, 220, 314
• — containing sine, 52, 169, 247, 337
• — containing sine integral, 87, 258
• — containing special function, 86, 187, 258, 353
• — containing square roots, 9, 222
• — containing square roots powers, 138
• — containing sum of exponential functions, 564
• — containing sum of hyperbolic functions, 564
• — containing sum of trigonometric functions, 564
• — containing tangent, 60, 174, 251, 342
• — containing Tricomi confluent hypergeometric function, 273
• — containing trigonometric function, 46, 78, 84, 85, 166, 181, 186, 187, 246, 256, 295, 335, 349, 352, 353
• — containing Whittaker confluent hypergeometric function, 274
• — cubic in arguments, 5, 132, 307
• — degenerate, 111, 191, 278, 357, 519, 522, 539, 540–543, 569, 573, 589, 625, 627, 631, 810, 817
• — degenerate, general, 523
• — degenerate, general case, 628
• — degenerate, simplest, 627
• — difference, 114, 203, 283, 372, 519, 524, 539, 544, 573, 574, 586, 610, 625, 626, 655, 683
• — difference, on entire axis, 655
• — difference, with weak singularity, 588
• — eigenfunction, 844
• — eigenvalue, 844
• — Fredholm, 573, 625, 839–841
• — Fredholm, positive definite, 840
• — Fredholm, positive definite, symmetric, 866
• — Fredholm, symmetric definite, 840, 841
• — general degenerate, 523
• — generalized, 783
• — Hilbert, 707, 780
• — Hilbert, characteristic equation, 769
• — Hilbert, complete singular integral equation, 759, 780
• — Hilbert, integral equations, 759
• — Hilbert–Schmidt, 841, 843, 845, 853, 855, 856, 860
• — Hilbert–Schmidt, approximate values of eigenvalues, 845
• — Hilbert–Schmidt, eigenfunctions, 854, 856, 858, 861, 864, 865
• — Hilbert–Schmidt, eigenvalues, 854, 856, 858, 864, 865
• — Hilbert-type, 751, 754
• — incomplete, 578
• — iterated, 566, 632
• — iterated, bilinear series, 642
• — linear in arguments, 4, 127, 217, 301
• — logarithmic, 519, 588
• — nondegenerate, 589, 631
• — nonnegative, 648
• — nonsymmetric, 580, 647
• — of integral equation, 519, 573, 625
• — of integral transform, 503
• — orthogonal, 634
• — oscillation, 651
• — oscillation, definition, 651
• — oscillation, theorems, 651
• — polar, 519, 532, 574, 588
• — positive definite, 641
• — quadratic in arguments, 4, 129, 219, 304
• — resolvent, 844
• — Schmidt, 582, 841, 848, 851, 859, 860, 862
• — simplest degenerate, 627
• — singular, weakly, 532
• — spectral radius, 649
• — stochastic, 654
• — symmetric, 573, 577, 625, 639, 645
• — symmetric, resolvent, 644
• — trace, 646
• — transformation, method, 532
• — Volterra, 839
• — weakly singular, 532
• — with logarithmic singularity, 533
• — with rational Fourier transforms, 685
• — with weak singularity, 519, 532, 574, 588, 625
• Kontorovich–Lebedev inversion formula, 516
• Kontorovich–Lebedev transform, 267, 516, 518
• Korteweg–de Vries equation, 899
• — modified, 900
• Krein's method, 588, 683
• — for integral equations, 588
• — for Wiener–Hopf equations, 679
• Kummer confluent hypergeometric function, 272, 1024
• Kummer series, 1024
• Kummer transformation, 1025

L

• L₂-norm, 501
• Lagrange interpolation polynomial, 748
• Laguerre polynomial, 110, 1024, 1045
• — generalized, 1045
• Lalesco–Picard equation, 323
• Lanczos approximation, 798
• Laplace equation, 893
• — potentials, properties, 892
• Laplace integral, 1030
• Laplace transform, 235, 505, 511, 518, 524, 544, 658, 809
• — definition, 505
• — inverse, tables, 969
• — inversion formula, 505
• — properties, 507
• — solution method, 524
• — tables, 961
• — two-side, 234, 518
• Lavrentiev regularization method, 621
• layer potential, single, 893
• least squares method, 695
• — description, 695
• — normal system, 695
• Lebedev transform, 269
• Lebesgue integrable function, 1059
• Lebesgue integral, 1057
• — definition, 1059
• — properties, 1059
• Lebesgue space L_p(a, b), 1064
• Lebesgue theorem on dominated convergence, 1060
• left-sided fractional derivative, 529
• left-sided fractional integral, 529
• left Fourier integral, 594
• left function, 594
• left regularization, 775
• — method, 775
• left regularizer, 703
• Legendre equation, 1032
• Legendre functions, 270, 1030
• — associated, 107, 271, 1030
• — associated, first kind, 1032
• — associated, modified, 1033
• — associated, second kind, 1032
• — modified associated, 1033
• — Wronskians, 1034
• Legendre polynomials, 105, 856, 1030
• — orthonormal, 844
• Legendre spherical functions, first kind, 299
• lemma, Jordan, 505
• limit theorems, 507
• linear algebraic equations,
• — infinite system, 858, 861, 864, 868, 971
• — infinite system with symmetric matrix, 850, 853
• linear boundary value problems, representation, 892
• linear equation, 898
• — constant integration limits, 502
• — first kind, 502
• — first kind, constant integration limits, 217, 573
• — first kind, variable integration limit, 3
• — operator methods, 549
• — second kind, 502
• — second kind, constant integration limits, 301, 625
• — second kind, variable integration limit, 127
• — solution methods, 519, 539, 573, 625
• — structure of solutions, 502
• — variable integration limit, 502
• linear normed spaces, 1063
• linear operator, 502, 1066
• — eigenfunction, 1066
• — eigenvalue, 1066
• linear operators in Hilbert spaces, 1065, 1066
• linear ordinary differential equations, 881
• linear relations of parabolic cylinder functions, 1034
• linear space, 1063
• — complex, 1063
• — real, 1063
• linear superposition principle, 502
• linearly dependent elements, 1063
• linearly independent elements, 843, 1063
• Liouville theorem, generalized, 714
• Lipschitz condition, 709, 1054, 1057
• local solutions of nonlinear integral equation with parameter, 835
• logarithm,
• — Napierian, base, 905
• — natural, base, 905
• logarithmic derivative of gamma function, 1017, 1021
• logarithmic function, 42, 45, 77, 83, 85, 164, 165, 180, 185, 187, 242, 244, 255, 256, 334, 335, 349, 351, 353, 905, 943, 955, 965, 980, 985, 992, 999, 1002
• — properties, 906
• logarithmic integral, 258, 1009, 1010, 1025
• logarithmic kernel, 519, 588
• logarithmic nonlinearity, 419, 472
• logarithmic singularity, 533, 618
• — kernel, 533

M

• MacDonald function, 266, 1021
• mass transfer to particle in fluid flow complicated by surface reaction, 888
• Mathieu equation, 1043
• — modified, 1045
• Mathieu function, 1043, 1044
• — modified, 1043, 1045
• matrix,
• — eigenvalues, 845, 848, 856, 859, 861, 868, 872
• — eigenvectors, orthonormal, 845, 848, 856, 859, 868, 872
• — orthonormal eigenvectors, 845, 848, 856, 859, 868, 872
• mean-square convergence, 501
• measurable function, 1058
• measurable set, 1060
• — integration, 1061
• measure,
• — full, set, 1058
• — zero, set, 1058
• measure of set, 1061
• mechanics, fracture, 791
• Mehler–Fock transform, 270, 518
• — generalized, 271
• Mehler integral, 299, 615
• Meijer inversion formula, 516
• Meijer transform, 266, 516, 517
• Mellin transform, 510, 511, 518, 587, 657, 658
• — definition, 510
• — inverse, 510
• — inverse, tables, 1001
• — inversion formula, 510
• — properties, 511
• — tables, 997
• method,
• — approximation, successive, 566
• — Bateman, 689
• — Bateman, general scheme, 689
• — Bateman, special cases, 690
• — Bubnov–Galerkin, 697
• — Bubnov–Galerkin, description, 697
• — Carleman, for characteristic equations, 761
• — Carleman, for equations of convolution type of first kind, 606
• — Carleman, for equations with difference kernels, 610
• — Carleman, for integral equations of convolution type of second kind, 660
• — collocation, 692, 693, 815
• — collocation, for solving hypersingular integral equation, 755
• — exact, 588
• — Galerkin, 582
• — Kellog's, for finding characteristic values in case of symmetric kernel, 645
• — Krein's, 588, 683
• — Krein's, for integral equations, 588
• — Krein's, for Wiener–Hopf equations, 679
• — Multhopp–Kalandiya, 747
• — Newton–Kantorovich, 813, 814, 827
• — Newton–Kantorovich, modified, 814, 827
• — nonlinear equations with constant integration limits, exact, 817
• — nonlinear equations with variable integration limit, exact, 809
• — operator, 549, 654
• — operator, for solving integral equations of second kind, 654
• — Picard, 876
• — projection, for solving mixed equations on bounded set, 866
• — quadrature, 698, 816 829
• — quadrature, general scheme, 698
• — regularization, 704
• — regularization, for complete singular integral equations, 772
• — regularization, for equations with infinite limits of integration, 702
• — regularization, Lavrentiev, 621
• — regularization, Tikhonov, 622, 829
• — solution, Laplace transform, 524
• — successive approximation, 566, 811, 826
• — successive approximation, general scheme, 566
• — successive approximation, resolvent, 566
• — Tikhonov regularization, 829
• — trace, for approximation of characteristic values, 646
• — Wiener–Hopf, 671
• — Wiener–Hopf, scheme, 676
• — Zakharov–Shabat, 898
• method based on solution of auxiliary equation, 546
• method for,
• — solving quadratic operator equations, 552
• — special Urysohn equations of first kind, 821
• — special Urysohn equations of second kind, 822
• method of,
• — approximating kernel by degenerate one, 687
• — differentiating, for integral equations, 820, 564, 583
• — differentiation, 564, 583, 810
• — differentiation, for nonlinear equations with degenerate kernel, 810
• — equidistant surface, 891
• — fractional differentiation, 529
• — fractional integration, for generalized Abel equation, 548
• — Fredholm determinants, 635
• — Fredholm determinants, 635
• — integral transforms, 586, 655, 809, 819
• — least squares, 695
• — least squares, description, 695
• — least squares, normal system, 695
• — left regularization, 775
• — model solutions, 559, 655, 659
• — model solutions, description, 560
• — numerical integration of equation for surface concentration, 891
• — quadratures, 534, 568, 698
• — quadratures, algorithm based on trapezoidal rule, 536
• — quadratures, general scheme, 535, 568
• — quadratures, trapezoidal rule, 568
• — replacing kernel by degenerate kernel, 687
• — right regularization, 775
• — successive approximations, 579, 632, 633, 811, 876
• — successive approximations, for ODEs, 876
• — transformation of kernel, 532
• methods
• — approximate, for nonlinear equations with constant integration limits, 826
• — approximate, for nonlinear equations with variable integration limit, 811
• — asymptotic, 618
• — asymptotic, for solving equations with logarithmic singularity, 618
• — exact, for integral equations, 588
• — exact, for nonlinear equations with constant integration limits, 817
• — exact, for nonlinear equations with variable integration limit, 809
• — for solving complete singular integral equations, 757
• — for solving equations with difference kernels on finite interval, 683
• — for solving integral equations, 499
• — for solving linear equations, 519, 539, 573, 625
• — for solving multidimensional mixed integral equations, 839
• — for solving nonlinear integral equations, 805
• — for solving singular integral equations of first kind, 707
• — integral equations of first kind, 707
• — numerical, for hypersingular equations, 754
• — numerical, for nonlinear equations with constant integration limits, 826
• — numerical, for nonlinear equations with variable integration limit, 811
• — of solving mixed integral equations on finite interval, 843
• — of solving mixed integral equations on ring-shaped domain, 855
• — operator, for solving linear integral equations, 549
• — regularization, 621
• minor, Fredholm, 636
• mixed equation, 839
• — bounded set, projection method, 866
• — circular domain, 841
• — closed bounded set, 842
• — finite interval, 840
• — Hilbert–Schmidt kernel, finite interval, 843
• — Hilbert–Schmidt kernel, ring-shaped domain and given right-hand side, 855
• — multidimensional, 839
• — multidimensional, solution methods, 839
• — on finite interval, methods of solving, 843
• — on ring-shaped domain, methods of solving, 855
• — ring-shaped domain, 841
• — Schmidt kernel and auxiliary conditions on ring-shaped domain, 862
• — Schmidt Kernel and given right-hand side on interval, 848
• mixed multidimensional equation,
• — Fredholm operator, 842
• — Schmidt operator, 843
• — Schmidt operator, equivalent form, 843
• — symmetric Fredholm kernel, 842
• — Volterra and Hilbert–Schmidt types operators, 866
• — Volterra and Schmidt types operators, 866
• mixed operator equation, 866, 869
• — with given right-hand side, 866
• mixed operator equations with auxiliary conditions, 869
• mixed two-dimensional equation, Schmidt kernel, 841
• — equivalent form, 842
• model solution,
• — cosine-shaped right-hand side, 563
• — exponential right-hand side, 561
• — power-law right-hand side, 562
• — sine-shaped right-hand side, 562
• model solutions, method, 559, 655, 659
• modified associated Legendre functions, 1033
• modified Bessel equation, 1021
• modified Bessel function, 97, 189, 269, 355, 1021
• — asymptotic expansions, 1022
• — definitions, 1021
• — first kind, 266, 1021
• — integral representations, 1022
• — second kind, 266, 1021
• modified Korteweg–de Vries equation, 900
• modified Mathieu function, 1043, 1045
• modified Newton–Kantorovich method, 814, 827
• modulus, 278, 583
• — complementary, 1036
• — elliptic, 1036
• Multhopp–Kalandiya method, 747
• multidimensional domain, 839
• — integration, 839
• multidimensional equation, mixed, 839
• — Fredholm operator, 842
• — integral operators of Volterra and Hilbert–Schmidt types, 866
• — integral operators of Volterra and Schmidt types, 866
• — Schmidt operator, 843
• — solution methods, 839
• — symmetric Fredholm kernel, 842
• multidimensional real-valued functions, 839
• multiply connected domain, 731
• multivalued functions, 711

N

• Napierian base, 906
• Napierian logarithms, base, 905
• natural logarithms, base, 905
• natural numbers, powers, sums, 919
• Nekrasov equation, 836
• Neumann function, 1016
• Neumann problem,
• — exterior, reduction to integral equations, 896
• — interior, 895
• — interior, reduction to integral equations, 895
• Neumann series, 567, 633
• Newton–Kantorovich method, 813, 814, 827
• — modified, 814, 827
• nodes,
• — Chebyshev, 748
• — interpolation, 534
• — quadrature, 534
• nondegenerate kernel, 589, 631
• nonhomogeneous equation, 502, 539, 627, 708, 751
• — positive solutions, 649
• — solution, 642
• nonhomogeneous problem, 604, 742
• — solution, 721
• nonhomogeneous Riemann problem, canonical function, 605
• nonhomogeneous Wiener–Hopf equation of second kind, 677
• nonisothermal flow in plane channel, 884
• nonlinear equation, 807, 834, 899
• — bifurcation points, 834, 835
• — constant integration limits, 806, 829
• — constant integration limits, approximate methods, 826
• — constant integration limits, exact methods, 817
• — constant integration limits, numerical methods, 826
• — degenerate kernel, 817
• — degenerate kernel, method of differentiation, 810
• — eigenfunctions, 834
• — existence theorems, 830
• — first kind with constant limits of integration, 433
• — parameter, local solutions, 835
• — second kind with variable limit of integration, 403
• — second kind with constant limits of integration, 453
• — solution methods, 805
• — uniqueness theorems, 830
• — variable limit of integration, 805
• — variable limit of integration, approximate methods, 811
• — variable limit of integration, exact methods, 809
• — variable limit of integration, numerical methods, 811
• nonlinear operator, eigenfunctions, 834
• nonlinear PDEs, 898
• nonlinear problem of nonisothermal flow in plane channel, 884
• nonlinear Volterra integral equation, 805
• nonlinearity, 414–416, 418, 467–470, 472–475
• — exponential, 411, 467
• — general form, 399, 425, 447, 477
• — hyperbolic, 414, 468
• — logarithmic, 419, 472
• — power-law, 408, 444, 464
• — quadratic, 393, 397, 403, 406, 437, 453, 456
• — trigonometric, 420, 473
• nonnegative kernels, 648
• nonorthogonal polynomials, 1050
• nonsymmetric kernel, 580, 647
• norm, 501, 644, 839
• — operator, 1066
• normal system of method of least squares, 695
• normality condition, 596
• normed space, 1063
• — linear, 1063
• notion of almost everywhere, 1058
• notion of index, 716
• nth-order differential equations, boundary value problems, 882
• nth-order linear ODE, 876
• numbers, 1007
• — Bernoulli, 1008
• — Euler, 1008
• — natural, powers, sums, 919
• numerical integration, method, 891
• numerical methods for hypersingular equations, 754
• numerical methods for nonlinear equations with constant integration limits, 826
• numerical methods for nonlinear equations with variable limit of integration, 811
• numerical series, 924
• — infinite, 924
• numerical solution, singular equations, 799
• — generalized kernels, 792
• numerical sums, 921
• — finite, 919

O

• ODE,
• — first-order, 875, 876
• — method of successive approximations, 876
• — nth-order, linear, 876
• — second-order, 876
• Olevskii transform, 276
• one-dimensional domain, 839
• — integration, 839
• one-sided equation, 574, 626
• one-sided Fourier integrals, 593, 594
• one-sided function, 594
• open curves, 734
• — Riemann problem, 734
• operator
• — compact, 842, 843, 1067
• — compact, self-adjoint, 843
• — compact, self-adjoint positive, 873
• — compact, self-adjoint positive definite, 1067
• — compact, self-adjoint positive definite, eigenvalues, 1067
• — Erdelyi–Kober, 532
• — Fredholm, 758, 842
• — Fredholm, symmetric kernel, generalization, 843
• — Hilbert–Schmidt, 842, 843, 866, 871
• — Hilbert–Schmidt, approximation for eigenfunctions, 868, 872
• — Hilbert–Schmidt, approximation for eigenvalues, 868, 872
• — Hilbert–Schmidt, eigenfunction, 871
• — Hilbert–Schmidt, eigenvalues, 871
• — identity, 842, 873
• — integral, characteristic, 758
• — integral, characteristic, transposed, 758
• — integral, compactness, sufficient condition, 842
• — integral, continuous, 1066
• — integral, domain, 1066
• — integral, domain of definition, 1066
• — integral, eigenvalues, 867
• — integral, positive definite, 842
• — integral, self-adjoint, 842, 843, 1067
• — integral, self-adjoint, eigenvalues, 1067
• — integral, self-adjoint, eigenvectors, 1067
• — integral, spectral radius, 649
• — integral, spectrum, 1066
• — integral, transposed, 758
• — integral, transposed characteristic, 758
• — integral with positive definite kernel, 843
• — integral with symmetric kernel, 843
• — linear, 502, 1066
• — linear, eigenfunction, 1066
• — linear, eigenvalue, 1066
• — linear in Hilbert spaces, 1065, 1066
• — nonlinear, eigenfunctions, 834
• — norm, 1066
• — orthogonal projection, 1067
• — point, continuous, 1066
• — positive definite, 842, 1067
• — regular, 758
• — regularizing, 703
• — Schmidt, 843, 866
• — singular, 758
• — singular, certain properties, 772
• — Volterra, 842, 873
• operator equation,
• — general projection problem, 873
• — general projection problem, 873
• — mixed, 866, 869
• — mixed with auxiliary conditions, 869
• — quadratic, 552
• — solution, 553
• operator method, 549, 654
• operator method for solving integral equations of second kind, 654
• operator of fractional integration, 529
• operator of orthogonal projection, 846, 852, 857, 563, 870
• order, fractional, integral, 529
• ordinary differential equations, 527, 547, 686
• — linear, 881
• orthogonal function, 582
• orthogonal kernels, 634
• orthogonal polynomials, 1045
• — system, 795
• orthogonal projection, operator, 846, 852, 857, 563, 870, 1067
• orthogonal projector, 1067
• orthogonal subspaces, 873
• — direct sum, 845, 863, 869
• orthogonal system, 1065
• orthogonal vectors, 1065
• orthogonality properties of Bessel functions, 1019
• orthonormal basis, 855, 856
• orthonormal eigenvectors of matrix, 845, 848, 856, 859, 868, 872
• orthonormal Legendre polynomials, 844
• orthonormal system, 1065
• — complete, 844, 855
• oscillation kernel, 651
• — definition, 651
• — theorems, 651

P

• Paley–Wiener transform, 260
• parabolic cylinder function, 276, 1034
• — asymptotic expansions, 1034
• — basic formulas, 1034
• — definitions, 1034
• — integral representations, 1034
• — linear relations, 1034
• — Weber, 1034
• parameter of integral equation, 625
• parameters, arbitrary, 408, 411, 433, 453
• Parseval's relation,
• — Fourier cosine transform, 514
• — Fourier sine transform, 515
• — Hankel transform, 515, 516
• particular solutions of PDEs, 887
• PDEs, nonlinear, 898
• PDEs with boundary conditions,
• — third kind, 887
• — third kind, reduction to integral equations, 887
• permutator, 654
• Picard–Goursat equation, 134
• Picard method, 876
• Pochhammer symbol, 1007
• Poincare–Bertrand formula, 714
• point
• — bifurcation, 835
• — bifurcation of nonlinear integral equations, 834, 835
• — collocation, 693
• — cuspidal, 708
• — regular, 1066
• — singular, 507
• point operator, continuous, 1066
• Poisson's formula, 1018
• Poisson equation, 894
• polar kernel, 519, 532, 574, 588
• polynomial
• — Bernoulli, 1050
• — Chebyshev, 109, 1047
• — Chebyshev, second kind, 750
• — Euler, 1051
• — Gegenbauer, 1050
• — generalized Laguerre, 1045
• — Hermite, 108, 1024, 1025, 1048
• — higher-order in arguments, 6, 133, 311
• — interpolation, Hermite, 716
• — interpolation, Lagrange, 748
• — Jacobi, 1049
• — Lagrange interpolation, 748
• — Laguerre, 110, 1024, 1045
• — Laguerre, generalized, 1045
• — Legendre, 105, 856, 1030
• — Legendre, orthonormal, 844
• — nonorthogonal, 1050
• — orthogonal, 1045
• — orthogonal, system, 795
• — orthonormal Legendre, 844
• — ultraspherical, 1050
• polynomial form, 553
• positive definite Fredholm kernel, 840
• — symmetric, 866
• positive definite integral operator, 842
• positive definite kernel, 641
• positive definite operator, 1067
• positive eigenvalue, 648
• positive Fredholm kernel, symmetric, 841
• positive solutions of nonhomogeneous integral equation, 649
• Post–Widder formula, 510
• potential,
• — density, 893
• — double layer, 893
• — double layer, Gauss formula, 894
• — equilibrium, 897
• — Feller, 226
• — Laplace equation, 892
• — Laplace equation, properties, 892
• — layer, single, 893
• — Riesz, 226
• — Roben, 897
• — single layer, 893
• — volume, 893
• — volume, Gauss formula, 894
• power-law functions, 4, 45, 127, 151, 165, 217, 236, 244, 301, 326, 335, 419, 951, 963, 983, 989, 998, 1001
• power-law generating function, 557
• power-law nonlinearity, 408, 464
• power-law nonlinearity that contain arbitrary functions, 444
• power function, 905
• — properties, 905
• power series, 925
• — expansion, 910, 913, 916, 918
• power series in parameter, 632
• power series of Airy functions, 1023
• powers, arbitrary, 139, 223, 317, 939, 977
• powers, fractional, 138
• powers of natural numbers, sums, 919
• principal value,
• — curvilinear integral, 712
• — singular curvilinear integral, 712
• — singular integral, 709
• principle
• — linear superposition, 502
• principle of argument, 714
• principle of continuity, 714
• probability integral, 1009
• problem
• — Abel, 520
• — boundary value, first, 895, 896
• — boundary value, for nth-order differential equations, 882
• — boundary value, for ODEs, 877, 881
• — boundary value, for second-order differential equations, 883
• — boundary value, linear, representation, 892
• — boundary value, Riemann, 595
• — boundary value, second, 895, 897
• — Cauchy, for ODEs, reduction to integral equations, 875
• — Cauchy, for second-order ODEs, 876
• — Cauchy, for special nth-order linear ODE, 876
• — Dirichlet, exterior, reduction to integral equations, 896
• — Dirichlet, interior, 895
• — Dirichlet, interior, reduction to integral equations, 895
• — electrostatic, Roben, 897
• — factorization, 676, 679
• — general projection, 873
• — general projection, for operator equation, 873
• — general projection, special case, 846, 852, 857, 870
• — Hilbert, 742
• — Hilbert, boundary value, 742
• — homogeneous, 596, 602, 742
• — homogeneous, solution, 720
• — ill-posed, 623, 624
• — ill-posed, general notions, 623
• — interior Dirichlet, 895
• — interior Dirichlet, reduction to integral equations, 895
• — interior Neumann, 895
• — interior Neumann, reduction to integral equations, 895
• — jump, 596
• — linear boundary value, representation, 892
• — Neumann, exterior, reduction to integral equations, 896
• — Neumann, interior, 895
• — Neumann, interior, reduction to integral equations, 895
• — nonhomogeneous, 604, 742
• — nonhomogeneous, solution, 721
• — nonhomogeneous Riemann, canonical function, 605
• — nonlinear of nonisothermal flow in plane channel, 884
• — projection, general, for operator equation, 873
• — projection, general, special case, 846, 852, 857, 870
• — Riemann, 596, 685, 714
• — Riemann, boundary value, 595
• — Riemann, coefficient, 596, 718
• — Riemann, discontinuous coefficient, 739
• — Riemann, exceptional cases, 727
• — Riemann, for half-plane, 725
• — Riemann, for open curves, 734
• — Riemann, for real axis, 592
• — Riemann, general case, 741
• — Riemann, index, 596, 731
• — Riemann, multiply connected domain, 731
• — Riemann, nonhomogeneous, canonical function, 605
• — Riemann, open curves, 734
• — Riemann, right-hand side, 596, 718
• — Riemann, statement, 718
• — Riemann, with discontinuous coefficient, 739
• — Riemann, with rational coefficients, 723
• — Roben electrostatic, 897
• — second boundary value, 895, 897
• — tautochrone, 520
• — well-posed, 623
• problem of equivalent regularization, 776
• problem with rational coefficients, 601
• process, iteration, 811, 814
• product
• — infinite, 910, 916
• — inner, 501, 644
• — scalar, 839
• progressions, 919, 924
• projection, orthogonal, operator, 846, 852, 857, 563, 870
• projection method for solving mixed equations on bounded set, 866
• projection problem
• — general, for operator equation, 873
• — general, special case, 846, 852, 857, 870
• projector, orthogonal, 1067
• properties
• — basic of Gauss hypergeometric functions, 1028
• — certain of singular operators, 772
• — orthogonality of Bessel functions, 1019
• property, semigroup of fractional integration, 529
• psi function, 1012, 1013

Q

• quadratic form, 644
• quadratic nonlinearity, 393, 397, 403, 406
• — containing arbitrary functions, 437, 456
• — containing arbitrary parameters, 433, 453
• quadrature formula, 534, 793, 815
• quadrature method, 698, 816, 829
• — general scheme, 698
• quadrature nodes, 534
• quadratures, method, 534, 568, 698
• — algorithm based on trapezoidal rule, 536
• — general scheme, 535

R

• radius,
• — estimates, 649
• — of integral operator, 649
• — of kernel, 649
• rational coefficients, 601, 723
• rational Fourier transforms, 685
• rational functions, 7, 136, 220, 314, 933, 971
• — inverse transforms, 506
• reaction, surface, 888
• real-valued functions, multidimensional, classes, 839
• real axis
• — Holder condition, 575
• — Sokhotski–Plemelj formulas, 713
• real linear space, 1063
• rectangle rule, 534
• recurrent relations, 636
• reduction formulas, 907, 939, 947
• regular operator, 758
• regular points, 1066
• regular value, 301, 625, 637
• regularization, 774
• — Carleman–Vekua, 778
• — equivalent, problem, 776
• — left, 775
• — left, method, 775
• — right, 776
• — right, method, 775
• regularization in exceptional cases, 779
• regularization method, 621, 704
• — complete singular integral equations, 772
• — equations with infinite limits of integration, 702
• — Lavrentiev, 621
• — Tikhonov, 622, 829
• regularizer, 774
• — left, 703
• — right, 704
• regularizing operators, 703
• relation,
• — linear of parabolic cylinder functions, 1034
• — Parseval's, Fourier cosine transform, 514
• — Parseval's, Fourier sine transform, 515
• — Parseval's, Hankel transform, 515, 516
• recurrent, 636
• relations between Mellin, Laplace, and Fourier transforms, 511
• remainder, 534
• renewal equation, 203
• representation,
• — Bessel functions, 1017
• — form of infinite products, 910, 916
• — Gauss hypergeometric functions, 1028
• — inverse transforms as asymptotic expansions, 509
• — inverse transforms as convergent series, 509
• — modified Bessel functions, 1022
• — parabolic cylinder functions, 1034
• — series of Jacobi theta functions, 1042
• — Tricomi confluent hypergeometric functions, 1024
• residual, 692
• residue theorem, Cauchy, 504
• residues, 504
• resolvent, 539, 567, 626, 633, 635
• — construction, 633
• — kernel, 844
• — symmetric kernel, 644
• results, auxiliary, 784
• Riemann boundary value problem, 595, 714
• Riemann integral, 1057
• Riemann–Liouville derivatives, 529
• Riemann–Liouville fractional integrals, 529
• Riemann problem, 596, 685, 714
• — coefficient, 596, 718
• — exceptional cases, 727
• — for half-plane, 725
• — for multiply connected domain, 731
• — for open curves, 734
• — for real axis, 592
• — general case, 741
• — index, 596, 731
• — nonhomogeneous, canonical function, 605
• — right-hand side, 596, 718
• — statement, 718
• — with discontinuous coefficient, 739
• — with rational coefficients, 723
• Riemann zeta function, generalized, 277
• Riesz potential, 226
• Riesz Schauder theory, Riesz–Schauder theory, 843
• Riesz transform, 226
• right-hand side, 757
• — equation, 519, 573, 625
• — integral equation, 539
• — Riemann problem, 596, 718
• — special, 555
• right-sided fractional derivative, 529
• right-sided fractional integral, 529
• right Fourier integral, 594
• right function, 594
• right regularization, 776
• — method, 775
• right regularizer, 704
• ring-shaped domain, 841, 855, 862
• Roben electrostatic problem, 897
• Roben potential, 897
• roots, square, 138, 222, 975
• rule
• — rectangle, 534
• — Simpson's, 534
• — trapezoidal, 534, 568

S

• scalar, 1063
• scalar product, 839
• scalars, field, 1063
• scheme,
• — general, Bateman method, 689
• — general, method of quadratures, 568
• — general, successive approximation method, 566
• Schlomilch equation, 254, 452, 825
• Schlomilch equation, generalized, 254
• Schmidt integral operator, 843, 866
• Schmidt kernel, 582, 841, 848, 851, 859, 860, 862
• Schmidt operator, 866
• second-order differential equations, boundary value problems, 883
• second-order ODEs, 876
• second boundary value problem, 895, 897
• segment, finite, equation, 683, 685
• self-adjoint operator, 842, 843, 1067
• — eigenvalues, 1067
• — eigenvectors, 1067
• semiaxis,
• — equation, 574, 587, 626, 657
• — Hilbert transform, 229
• semigroup property of fractional integration, 529
• sequence of independent Volterra equations, 847, 858
• sequence of independent Volterra equations of second kind, 853, 865, 872
• sequence of Volterra equations, 844, 850, 862
• sequence of Volterra equations of second kind, 855
• series,
• — bilinear, 640
• — bilinear, iterated kernels, 642
• — convergent, 509
• — functional, infinite, 925
• — hypergeometric, 1028
• — infinite, 919
• — infinite functional, 925
• — infinite numerical, 924
• — Kummer, 1024
• — Neumann, 567, 633
• — numerical, 924
• — numerical, infinite, 924
• — power, 913, 925
• — power, expansion, 910, 916, 918
• — power in parameter, 632
• — power of Airy functions, 1023
• — trigonometric, in one variable, involving cosine, 928
• — trigonometric, in one variable, involving sine, 927
• — trigonometric, in two variables, 930
• series representation of Jacobi theta functions, 1042
• set, 866
• — bounded, closed, 842
• — closed bounded, 842
• — measurable, 1060
• — measure, 1061
• set of full measure, 1058
• set of zero measure, 1058
• sets, measurable, 1060
• — measurable, integration, 1061
• — zero measure, 1058
• several variables, function, 839
• side, right-hand
• — of equation, 519, 573, 625
• — of integral equation, 539
• — of Riemann problem, 596
• — of Riemann problem, 718
• — special, 555
• simple hypersingular equation of first kind with Cauchy-type kernel, 231
• simple hypersingular equation of first kind with Hilbert-type kernel, 255
• simplest degenerate kernel, 627
• simplest equation with Cauchy kernel, 743
• simplest hypersingular equation for first kind with Hilbert-type kernel, 754
• simplest singular equation of first kind with Hilbert kernel, 707, 746
• Simpson's rule, 534
• sine, 52, 169, 247, 337, 558, 927
• — hyperbolic, 28, 156, 238, 329
• sine integral, 87, 258, 1011
• sine transform, Fourier, Parseval's relation, 515
• single layer potential, 893
• singular curvilinear integral, principal value, 712
• — 228, 255, 319, 344
• — Bueckner type, 801
• — Cauchy kernel, complete, 757
• — Cauchy kernel, first kind, 707
• — complete, 757, 770, 772
• — first kind, 743
• — generalized kernels, 792
• — generalized kernels, direct numerical solution, 792
• — Hilbert kernel, 759
• — Hilbert kernel, complete, 759, 780
• — numerical solution, 799
• — simplest of first kind with Hilbert kernel, 707, 746
• — transposed, 758
• — two-dimensional, 231
• singular equations of first kind, 707
• singular integral, 709
• — principal value, 709, 712
• singular kernel, weakly, 532
• singular operator, 758
• singular operators, certain properties, 772
• singular points, 507
• singularities, solutions, 783
• singularity
• — logarithmic, 533, 618
• — logarithmic, kernel, 533
• — weak, 574, 588, 625
• — weak, kernel, 519, 532, 574, 588, 625
• singularity exponents, 787, 789
• skew-symmetric integral equation, 647
• smooth contour, 708
• Sokhotski–Plemelj formula, 713, 785
• Sokhotski–Plemelj formulas for real axis, 713
• solution
• — approximate, 688, 693
• — approximation, 854
• — convolution representation, 526
• — direct numerical of singular integral equations with generalized kernels, 792
• — exact of simple hypersingular equation with Cauchy-type kernel, 753
• — exact of simple hypersingular equation with Hilbert-type kernel, 754
• — fundamental, 881
• — homogeneous problem, 720
• — integral equations, exact, 1–500
• — model, cosine-shaped right-hand side, 563
• — model, exponential right-hand side, 561
• — model, power-law right-hand side, 562
• — model, sine-shaped right-hand side, 562
• — nonhomogeneous problem, 721
• — numerical, of singular integral equations, 799
• — simple hypersingular equation with Cauchy-type kernel, exact, 753
• — simple hypersingular equation with Hilbert-type kernel, exact, 754
• — stable, 623
• — trivial, 502
• solution method, Laplace transform, 524
• solution method based on Laplace transform, 544
• solution of auxiliary equation, method, 546
• solution of generalized Abel equation, 531
• solution of operator equations of polynomial form, 553
• solutions
• — closed-form, case of constant coefficients, 770
• — closed-form, general case, 771
• — fundamental, 881
• — local of nonlinear integral equation with parameter, 835
• — model, method, 559, 655, 659
• — particular of PDEs, 887
• — positive of nonhomogeneous integral equation, 649
• solutions of dual integral equations, general scheme, 611
• solutions of nonlinear PDEs, representation in terms of solutions of linear integral equations, 898
• solutions singularities, 783
• solving linear equations, methods, 519, 539
• solving quadratic operator equations, 552
• Sonine transform, 114
• space
• — Banach, 1065
• — basis, 844, 863
• — complete, 1065
• — complex linear, 1063
• — Euclidean, 845, 857, 863, 869, 1065
• — Euclidean, basis, 857, 869
• — Hilbert, 839, 845, 857, 863, 867, 869, 1065
• — Hilbert, abstract, 873
• — Hilbert, basis, 857, 867, 869
• — Hilbert, linear operators, 1065, 1066
• — Hilbert, special basis, 869
• — Holder, 1064
• — Lebesgue, 1064
• — linear, 1063
• — linear, complex, 1063
• — linear, normed, 1063
• — linear, real, 1063
• — normed, 1063
• — normed linear, 1063
• — real linear, 1063
• — vector, 1063
• space of continuous functions, 1064
• space of functions of bounded variation, 1064
• special basis of Hilbert space, 869
• special case of general projection problem, 846, 852, 857, 870
• special functions, 86, 111, 187, 258, 277, 353, 967, 981, 987, 993, 1000, 1004
• special right-hand side, 555
• special Urysohn equations of first kind, method, 821
• special Urysohn equations of second kind, method, 822
• spectral radius, estimates, 649
• spectral radius of integral operator, 649
• spectral radius of kernel, 649
• spectrum of Fredholm integral equation, 760
• spectrum of operator, 1066
• spherical functions, Legendre of first kind, 299
• square integrable function, 501, 502
• square root, 9, 138, 222, 975
• stable solution, 623
• statement of Riemann problem, 718
• step-function, 1058
• — integral, 1059
• Stieltjes integral, 1055, 1056
• — basic definitions, 1055
• — existence theorems, 1056
• — properties, 1056
• Stieltjes integral sum, 1055
• Stieltjes transform, 221
• Stirling formula, 1013
• stochastic kernel, 654
• structure of solutions to linear integral equations, 502
• Struve function, 264, 299, 516, 518
• subspace, 1063
• — orthogonal, 873
• — orthogonal, direct sum, 845, 863, 869
• successive approximation method, 566, 579, 632, 633, 811, 826, 876
• — for ODEs, 876
• — general scheme, 566
• — resolvent, 566
• sufficient condition for compactness of integral operator, 842
• sum
• — contain binomial coefficients, 920
• — contain integers, 920
• — finite, 919
• — finite functional, 922
• — finite numerical, 919
• — functional, finite, 922
• — integral, Stieltjes, 1055
• — involving hyperbolic functions, 922
• — involving trigonometric functions, 922
• — numerical, 921
• — numerical, finite, 919
• — of exponential functions, 564
• — of hyperbolic functions, 564
• — of orthogonal subspaces, direct, 845, 863, 869
• — of powers of natural numbers, 919, 920
• — of powers of natural numbers, alternating, 920
• — of trigonometric functions, 564
• — Stieltjes integral, 1055
• summable function, 1059
• — integral, 1059
• superposition principle, linear, 502
• surface, equidistant, method, 891
• surface concentration,
• — equation, method of numerical integration, 891
• — integral equation, 890
• surface reaction, 888
• symbol, Pochhammer, 1007
• symbols, 1007
• symmetric definite Fredholm kernel, 840
• symmetric equation, 639, 647
• — Fredholm alternative, 643
• symmetric kernel, 573, 577, 625, 639, 645
• — resolvent, 644
• symmetric positive definite Fredholm kernel, 866
• symmetric positive Fredholm kernel, 841
• system
• — complete, 1065
• — complete orthonormal, 855
• — Fredholm integral equations of second kind, 701
• — infinite of linear algebraic equations, 858, 861, 864, 868, 971
• — infinite of linear algebraic equations with symmetric matrix, 850, 853
• — normal of method of least squares, 695
• — orthogonal, 1065
• — orthonormal, 1065
• — orthonormal, complete, 855
• — Volterra integral equations, 549
• system of characteristic values, 640
• system of eigenfunctions, 640
• — complete, 640
• — incomplete, 640
• system of equations, 701
• — reduction to single equation, 701
• system of Fredholm equations of second kind, 701
• system of functions,
• — complete orthonormal, 844
• — orthonormal, complete, 844
• system of orthogonal polynomials, 795

T

• tables of definite integrals, 951
• tables of Fourier cosine transforms, 983
• tables of Fourier sine transforms, 989
• tables of indefinite integrals, 933
• tables of inverse Laplace transforms, 969
• tables of inverse Mellin transforms, 1001
• tables of Laplace transforms, 961
• tables of Mellin transforms, 997
• tangent, 60, 174, 251, 342
• — hyperbolic, 36, 161, 241, 332
• tautochrone problem, 520
• terms of potentials, 892
• theorem
• — analytic continuation, 595, 714
• — Cauchy residue, 504
• — convolution, 507, 513
• — existence, 875
• — existence, for nonlinear equations, 830
• — existence, for Stieltjes integral, 1056
• — Fischer–Riesz, 1060
• — Fredholm, 637, 702, 777
• — Fubini, 1062
• — generalized Jentzch, 648
• — generalized Liouville, 595, 714
• — Hilbert–Schmidt, 641, 1067
• — Jentzch, generalized, 648
• — Lebesgue on dominated convergence, 1060
• — limit, 507
• — residue, Cauchy, 504
• — uniqueness, 875
• — uniqueness, for nonlinear equations, 830
• theory,
• — Hilbert–Schmidt, 843
• — Riesz–Schauder, 843
• theta functions, Jacobi, 110, 1042
• Tikhonov regularization method, 622, 829
• total variation of function, 1053
• trace method for approximation of characteristic values, 646
• trace of kernel, 646
• transform
• — alternative Fourier, 512
• — Boas, 250
• — Bochner, 263, 518
• — Buchholz, 274
• — cosine, Fourier, Parseval's relation, 514
• — Crum, 268
• — divisor, 269
• — Feller, 226
• — Fourier, 235, 511, 512, 518, 658
• — Fourier, alternative, 512
• — Fourier, asymmetric form, 512
• — Fourier, definition, 512
• — Fourier, inverse, 512
• — Fourier, inversion formula, 512
• — Fourier, properties, 513
• — Fourier, rational, 685
• — Fourier cosine, 514, 518
• — Fourier cosine, asymmetric form, 514
• — Fourier cosine, Parseval's relation, 514
• — Fourier cosine, tables, 983
• — Fourier sine, 514, 518
• — Fourier sine, asymmetric form, 515
• — Fourier sine, Parseval's relation, 515
• — Fourier sine, tables, 989
• — Gauss, 237
• — generalized Mehler–Fock, 271
• — Hankel, 261, 515, 518
• — Hankel, Parseval's relation, 515, 516
• — Hardy, 264
• — Hartley, 252, 518
• — Hilbert, 228, 255, 518, 743
• — Hilbert, on semiaxis, 229
• — integral, 503, 515
• — integral, kernel, 503
• — integral, method, 586, 655, 809, 819
• — integral, table, 517
• — inverse, 503
• — inverse, representation as asymptotic expansions, 509
• — inverse, representation as convergent series, 509
• — inverse Fourier, 512
• — inverse Laplace, tables, 969
• — inverse Mellin, 510
• — inverse Mellin, tables, 1001
• — inverse of rational functions, 506
• — kernel, 503, 586, 655, 809, 819
• — Kontorovich–Lebedev, 267, 516, 518
• — Laplace, 235, 505, 511, 518, 524, 544, 658, 809
• — Laplace, definition, 505
• — Laplace, inverse, tables, 969
• — Laplace, inversion formula, 505
• — Laplace, properties, 507
• — Laplace, solution method, 524
• — Laplace, tables, 961
• — Laplace, two-side, 234, 518
• — Lebedev, 269
• — Mehler–Fock, 270, 518
• — Mehler–Fock, generalized, 271
• — Meijer, 516, 517
• — Mellin, 510, 511, 518, 587, 657, 658
• — Mellin, definition, 510
• — Mellin, inverse, 510
• — Mellin, inverse, tables, 1001
• — Mellin, inversion formula, 510
• — Mellin, properties, 511
• — Mellin, tables, 997
• — Olevskii, 276
• — Paley–Wiener, 260
• — rational Fourier, 685
• — Riesz, 226
• — sine, Fourier, Parseval's relation, 515
• — Sonine, 114
• — Stieltjes, 221
• — table, 517
• — two-side Laplace, 234, 518
• — Weber, 265, 518
• — Weierstrass, 237, 518
• transformation, Kummer, 1025
• transformation of kernel, method, 532
• transposed characteristic equation, 758
• transposed characteristic operator, 758
• transposed equation, 573, 575, 625, 627, 637
• transposed equation of characteristic equation, 764
• transposed operator, 758
• transposed singular equation, 758
• trapezoidal rule, 534, 568
• triangle inequality, 501
• Tricomi confluent hypergeometric function, 273, 1024, 1025
• — asymptotic expansions, 1024
• — integral representations, 1024
• Tricomi equation, 319, 769
• Tricomi–Gellerstedt equation, 320
• trigonometric functions, 46, 78, 84, 85, 166, 181, 186, 187, 246, 252, 256, 295, 335, 344, 349, 352, 353, 564, 907, 922, 944, 956, 966, 981, 986, 992, 999, 1003
• — addition, 908
• — combinations, 176
• — inverse, 176, 344, 911, 948
• — inverse, addition, 912
• — inverse, relations, 912
• — inverse, subtraction, 912
• — of half argument, 909
• — of multiple arguments, 909
• — of single argument, relations, 908
• — powers, 908
• — products, 908
• — relationship, 916
• — subtraction, 908
• — sum, 564
• trigonometric nonlinearity, 420, 473
• trigonometric series
• — in one variable, involving cosine, 928
• — in one variable, involving sine, 927
• — in two variables, 930
• trivial solution, 502
• two-dimensional equation of Abel type, 15
• two-dimensional integral equation, mixed with Schmidt kernel, 841
• two-dimensional singular equation, 231
• two-side Laplace transform, 234, 518
• type, convolution, 574, 606, 660, 669

U

• ultraspherical polynomials, 1050
• undetermined coefficients, 692
• uniqueness theorems, 875
• uniqueness theorems for nonlinear equations, 830
• unknown function of complicated argument, 227, 246, 254
• Urysohn equation, 806, 832
• — first kind, 806, 829
• — second kind, 806
• — second kind with degenerate kernel, 818
• — special of first kind, method, 821
• — special of second kind, method, 822
• Urysohn form
• — Volterra equation, 805, 811, 814, 816
• — Volterra equation, first kind, 805, 815
• — Volterra equation, second kind, 805

V

• value
• — approximate of eigenvalues of Hilbert–Schmidt kernel, 845
• — Cauchy principal, 709
• — characteristic, 301, 625, 637, 639, 645, 697
• — characteristic, approximation, 646
• — characteristic, extremal properties, 644
• — characteristic, system, 640
• — in Banach space, continuous function of real argument, 840
• — in Hilbert space, continuous function of real argument, 840
• — in space of functions square integrable over closed bounded set, continuous function of real argument, 842
• — in space of functions square integrable over ring-shaped domain, continuous function of real argument, 841
• — in space of square integrable functions, continuous function of real argument, 840
• — regular, 301, 625, 637
• variable integration limit, 3, 805, 809, 811
• variable limit of integration, 3, 805, 809, 811
• variable lower integration limit, 537, 570
• variable lower limit of integration, 537, 570
• variables, several, function, 839
• variation, total, of function, 1053
• variation function, bounded, 1056
• vector, 1063
• — axioms for addition, 1063
• — axioms relating addition of vectors with their multiplication by scalars, 1063
• — orthogonal, 1065
• vector space, 1063
• Volterra equation, 549, 805, 877
• — first kind, 519, 524, 565
• — first kind, connection with Volterra equations of second kind, 524
• — first kind, existence of solution, 519
• — first kind, in Hammerstein form, 806
• — first kind, in Urysohn form, 805, 815
• — first kind, problems, 520
• — first kind, uniqueness of solution, 519
• — Hammerstein form, 806
• — nonlinear, 805
• — quadratic nonlinearity, 809
• — reduction to Wiener–Hopf equation, 528
• — second kind, 524, 539, 565
• — second kind, connection with Volterra equations of first kind, 524
• — second kind, in Urysohn form, 805
• — second kind, of Hammerstein form, 816
• — second kind, reduction to Volterra equations of first kind, 565
• — second kind, sequence, 855
• — second kind, sequence of independent, 853, 865, 872
• — sequence, 844, 850, 862
• — sequence of independent, 847, 858
• — systems, 549
• — Urysohn form, 805, 811, 814, 816
• Volterra integral operator, 842
• Volterra kernel, 839
• Volterra operator, 873
• volume potential, 893
• — Gauss formula, 894

W

• weak singularity, 574, 588, 625
• — kernel, 519, 532, 574, 588, 625
• weakly singular kernel, 532
• Weber function, 88
• Weber parabolic cylinder function, 1034
• Weber transform, 265, 518
• Weierstrass elliptic function, 1041
• Weierstrass function, 1041
• Weierstrass transform, 237, 518
• weight function, Jacobi, 793
• well-posed problem, 623
• Whittaker confluent hypergeometric function, 274, 1027
• Whittaker equation, 1027
• Wiener–Hopf equation, 574, 626, 679
• — first kind, 285, 538, 574, 606
• — Krein's method, 679
• — second kind, 373, 547, 571, 626, 660, 679
• — second kind, exceptional case, 678
• — second kind, homogeneous, 672
• — second kind, index, 661
• — second kind, nonhomogeneous, 677
• — second kind, solution, 681
• — Volterra equation, 528
• Wiener–Hopf method, 671
• — scheme, 676
• Wronskian, confluent hypergeometric function, 1026
• Wronskian, Legendre function, 1034

Y

• Y-transform, 264, 516, 518

Z

• Zakharov–Shabat method, 898
• zero measure, set, 1058
• zeros of Bessel functions, 1019