Handbook of Integral Equations (original) (raw)

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| | | | A. D. Polyanin and A. V. Manzhirov Handbook of Integral EquationsSecond 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 nnnth-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
— _n_th-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, nnnth-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

_L_2-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
_n_th-order differential equations, boundary value problems, 882
_n_th-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
— _n_th-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 _n_th-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 _n_th-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

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