Oldham K., Spanier J. — The fractional Calculus: Theory and applications of differentiation and integration to arbitrary order :: Электронная библиотека попечительского совета мехмата МГУ

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Oldham K., Spanier J. — The fractional Calculus: Theory and applications of differentiation and integration to arbitrary order

$Oldham K., Spanier J. — The fractional Calculus: Theory and applications of differentiation and integration to arbitrary order$

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Название: The fractional Calculus: Theory and applications of differentiation and integration to arbitrary order

Авторы: Oldham K., Spanier J.

Аннотация:

The concept of differentiation and integration to noninteger order is by no means new. Interest in this subject was evident almost as soon as the ideas of the classical calculus were known—Leibniz (1859) mentions it in a letter to L'Hospital in 1695.1 The earliest more or less systematic studies seem to have been made in the beginning and middle of the 19th century by Liouville (1832a), Riemann (1953), and Holmgren (1864), although Euler (1730), Lagrange (1772), and others made contributions even earlier.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1974

Количество страниц: 234

Добавлена в каталог: 06.12.2009

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Предметный указатель

Gamma function, reciprocal, graph of      17
Gamma function, recurrence formula      16
Gamma function, reflection formula      18
Gamma function, relation to beta functions      21
Gamma function, relation to binomial coefficients      20
Gamma function, relation to psi function      23
Gauss functions      41 122 165 174
Gauss functions as reducible transcendentals      161
Gel'fand, I.M.      13
Gemant, A.      2 67 220
Generalized Abel equation      186
Generalized differentiation      see "Differintegration"
Generalized differentiation for operators      2
Generalized error function complement integrals      195
Generalized hypergeometric functions      see "Hypergeometrics"
Generalized integration      see "Differintegration"
Generalized logarithms      163(n) 175 192
Generalized logarithms, definition of      193
Generalized logarithms, representation of      193—194
Geometric factor to characterize boundary geometries in diffusion problems      198
Gradshteyn, I.S.      220
Graham, A.      2 220
Greatheed, S.S.      5
Greer, H.R.      7
Gregory, D.F.      5
Grenness, M.      2 14 144 204 220
Gruenwald, A.K.      x 1 7 48 220
Hadamard, J.      8
Hagstrom, K.G.      11
Hardy, G.H.      2 8 9 10 11 220
Hargreave, C.J.      6
Heat conduction in semiinfinite planar medium      202
Heat equation      5
Heat flux, relation to surface temperature through semidifferentiation      202
Heaviside function      61 105
Heaviside function, graph of differintegrals      106
Heaviside's operational calculus      see "Operational calculus"
Heaviside's unit function      see "Heaviside function"
Heaviside, O.      x xii 2 8 9 53 62 220
Herpe, G.      219
Higgins, T.P.      2 13 220
Hilbert transforms      15
Hilbert, D.      27 50 219
Hille, E.      x 220 221
Hirschmann, I.I.      12
Holmgren — Riesz transforms      12
Holmgren, H.      1 7 12 221
Holub, K.      133 221
Hyperbolic cosines as reducible transcendental      161
Hyperbolic sine integrals      128
Hyperbolic sine integrals as hypergeometrics of complexity $\frac{1}{3}$       165
Hyperbolic sines as hypergeometrics of complexity $\frac{0}{2}$       100
Hyperbolic sines as reducible transcendentals      161
Hypergeometric functions      11 13 14 15
Hypergeometric functions as reducible transcendentals      161
Hypergeometric functions of complexity $\frac{K}{L}$       15
Hypergeometric functions, Laplace transformation of      171
Hypergeometric functions, notation for      15
Hypergeometric functions, product with power of argument      see "Hypergeometrics"
Hypergeometrics      40—41 99—102 129—130
Hypergeometrics as differintegrable series      99
Hypergeometrics as sum of hypergeometrics      44
Hypergeometrics as sum of hypergeometrics, use in finding differintegrals      101
Hypergeometrics of argument $x^{1/n}$       43 100—101
Hypergeometrics of complexity $\frac{0}{0}$       162
Hypergeometrics of complexity $\frac{0}{1}$       162
Hypergeometrics of complexity $\frac{1}{1}$       163
Hypergeometrics of complexity $\frac{1}{2}$       164
Hypergeometrics of complexity $\frac{1}{3}$       165
Hypergeometrics of complexity $\frac{2}{2}$       164—165
Hypergeometrics with K>L      165—166
Hypergeometrics with K>L, utility for asymptotic representations      166
Hypergeometrics, cancellation property of      42
Hypergeometrics, convergence properties of      165
Hypergeometrics, definition of      162
Hypergeometrics, differentiation and integration of      40—44
Hypergeometrics, differintegrals of      99—102
Hypergeometrics, examples of      162—165
Hypergeometrics, Laplace transformation      171—172
Hypergeometrics, recurrence relations for      42
Hypergeometrics, reduction of complexity      166—167
Hypergeometrics, reduction to complexity $\frac{0}{L-M}$       167
Hypergeometrics, regeneration from basis hypergeometrics      169
Hypergeometrics, symbolism for      40
I'a Bromwich, T.J.      9
Ichise, M      133 154 221
Incomplete beta functions      21 41 94 122 174
Incomplete beta functions as reducible transcendentals      161
Incomplete gamma functions      21 41 94 95 109 158 175
Incomplete gamma functions as reducible transcendental s      161
Incomplete gamma functions, recursion formula for      22
Infinite series, differentiation and integration of      38
Infinite series, evaluation through fractional calculus      183
Infinite transmission line, approximation by finite transmission line      212
Integral equations      2 10 12 13 15
Integrals      see "Classical integrals"
Integrating circuit      149
Integration of hypergeometrics      40—44
Integration of powers      39—40 192—193
Integration of powers, diagram showing sign of resultant integral      39
Integration to fractional order      see "Differintegration"
Integration, term by term      38
Intergranular groove      216—218
Intergranular groove, cross sectional diagram      217
Inverse hyperbolic functions      163
Inverse hyperbolic functions as reducible transcendentals      161
Inverse hyperbolic sines as hypergeometrics of complexity $\frac{2}{2}$       165
Inverse trigonometric functions      163
Inverse trigonometric functions as reducible transcendentals      161
Iterated integrals      37—38
Iterated integrals, Cauchy's formula for      38
Jaeger, J.C.      197 219
Johnson, W.C.      197 210 221
Jost, W.      197 221
Juberg, R.K.      15
Kalisch, G.K.      13
Kaufman, H.      115 222
Kelland, P.      4 5 6 7
Kesarwini, R.N.      13
Knopp, K.      49 221
Kober, H.      2 11 14 221
Kojima, T.      221
Koutecky function of polarography      160
Koutecky, J.      160 221
Krug, A.      1 8 53 221
Kummer functions      41 175
Kummer functions as hypergeometrics of complexity $\frac{1}{2}$       164
Kummer functions as reducible transcendentals      161
Kuttner, B.      2 12 221
L'Hospital, G.A.      1 3
Lacroix, S.F.      4
Lagrange, J.L.      1 3 221
Laplace transforms      11 115
Laplace transforms for performing circuit analysis      150
Laplace transforms of derivatives and integrals      133
Laplace transforms of differintegrals      133—136
Laplace transforms of differintegrals, formula for      134
Laplace transforms, role played in solving diffusion problems      201
Laplace, P.S.      3
Laurent, H.      8
Law of exponents      see "Composition rule"
Law of exponents, for operators of integral order      3
Lebesgue class      2 10
Legendre functions      122
Legendre functions as hypergeometrics of complexity $\frac{2}{2}$       164
Legendre functions as reducible transcendentals      161
Legendre's equation      191
Leibniz's rule      6 8 10 14 15 53 62
Leibniz's rule for derivatives      36
Leibniz's rule for differintegral operators, convergence difficulties      78—79

Leibniz's rule for differintegral operators, integral form      79 182
Leibniz's rule for differintegral operators, symmetric form      79
Leibniz's rule for differintegral operators, when factors are analytic      77
Leibniz's rule for differintegral operators, when one factor is polynomial      77—78
Leibniz's rule for multiple integrals      34—35
Leibniz, G.W.      x 1 3 221
Letnikov, A.V.      7 8
Levy, P.      9
Lions, J.L.      12 14
Liouville, J.      xi 1 4 5 6 7 8 49 53 95 191 221
Lipschitz classes      2
Littlewood, J.E.      2 10 11 220
Liverman, T.P.G.      13
Logarithms      163
Logarithms as hypergeometrics of complexity $\frac{1}{1}$       102
Logarithms as reducible transcendentals      161
Logarithms, differintegrals of      102—104
Logarithms, differintegrals of, graphs of      104
Logarithms, generalization of      192—194
Logarithms, generalization of, location in function synthesis diagram      194
Love, E.R.      11 14
Lower limit      87—89
Lutskaya, N.K.      13 219
Magnus, W.      220
Matsuda, H.      160 221
Mechanics      4
Mellin transforms      11
Meyer, R.F.      133 203 221
Mikusinski, J.G.      x 221
Miroshnikov, A.I.      13 219
Modified Bessel functions      97—98 125
Modified Bessel functions as hypergeometrics of complexity $\frac{0}{2}$       100
Modified Struve functions      125
Modified Struve functions, notation for      15
Moore, F.K.      197 221
Moritz, R.E.      8
Mullins, W.W.      217 218 221
Multiple integrals, dependence on lower limit      33—34
Multiple integrals, symbolism for      26 27
Murphy, G.M.      156 158 159 202 221
Muskat, M.      197 221
Nagayanagi, Y.      221
Naraniengar, M.T.      9
Navier — Stokes equation      12 14
Nekrassov, P.A.      8 54 221
Nemec, L.      133 221
Nerve impulse propagation      10 210(n)
Nessel, R.      14
Nigmatullin, R.Sh.      13 219
Numerical differintegration, algorithms for      136—148
Numerical differintegration, algorithms for, based on Gruenwald definition      136
Numerical differintegration, algorithms for, based on linear interpolation      140
Numerical differintegration, algorithms for, based on modified Gruenwald definition      137—138
Numerical differintegration, algorithms for, based on Riemann — Liouville definition      138—139
Numerical differintegration, algorithms for, relative error in      141—144
Numerical differintegration, algorithms for, weighting factors in      144
Numerical differintegration, algorithms for, weighting factors in, table of      146—147
Numerical differintegration, approximations used in, diagram illustrating      139
Numerical differintegration, coincidence of algorithms for      144
Numerical differintegration, comparison of rival algorithms      144—145
Numerical differintegration, comparison of rival algorithms, in differintegrating $1-x^{\frac{3}{2}}$       145 148
Numerical differintegration, comparison of rival algorithms, in differintegrating $\sqrt{x}$       145
Numerical differintegration, errors in      141—144
Numerical differintegration, errors in, table of      144
Numerical differintegration, nomenclature for      136
Numerical differintegration, nomenclature for, diagram illustrating      137
O'Shaughnessy, L.      8
Oberhettinger, F.      220
Ohm's law      150 211 214
Oldham, K.B.      2 14 15 133 143 144 154 160 200 204 220 221 222
Oltramare, G.      8
Operational calculus      9 10 11 12 13
Ordinary derivatives      see "Classical derivatives"
Ordinary differential equations, solution via fractional calculus      186—189
Ordinary integrals      see "Classical integrals"
Osler, T.J.      2 14 15 54 55 56 79 81 182 183 192(n) 222
Parseval's integral formula      79
Peacock, G.      4 6 7 8
Pennell, W.O.      10
Periodic functions      108—110
Periodic functions as hypergeometrics of complexity $\frac{0}{2}$       163
Peters, A.S.      12
Piecewise-defined functions      105 107—108
Pincherle, S.      8
Poritsky, H.      11
Post, E.L.      2 6 8 9 10 48 222
Prabhakar, T.R.      15
Product rule      see "Leibniz's rule"
Psi function      23 103
Psi function, recursion formula for      23
Quasi-semiinfinite medium, condition for      200(n)
Radius of convergence of differintegrable series      70—71
Rayleigh's formulas, generalizations of      97
Reducible transcendentals      161
Resistive-capacitative transmission line      see "Transmission line"
Rheology      2 67
Ridella, S.      154 219
Riemann Sums      48 56
Riemann sums for classical integrals      29
Riemann zeta function      142—143
Riemann zeta function, table of values      143
Riemann — Liouville integral      49 55
Riemann — Liouville transforms      115
Riemann, B.      xi 1 6 7 49 53 222
Riesz, M.      2 11 12 27 49 222
Ritt, J.F.      x 2 222
Roberts, G.E.      115 222
Robertson, W.M.      217 222
Rodrigues' formula, generalization of      192
Ross, B.      xiii 3 15
Rubel, L.A.      14 53 220
Ryzhik, I.M.      220
Samko, S.G.      13
Sawtooth function      107—108
Sawtooth function, graphs of differintegrals      108
Scale-change property, use in finding differintegrals      100 109
Schuyler, E.      8
Scott Blair, G.W.      2 53 67 220 222
Semi derivatives      7 50
Semicalculus      14
Semiderivatives and semiintegrals      115—131
Semiderivatives and semiintegrals, definition of      115(n)
Semiderivatives and semiintegrals, graphs for cos(x)      126
Semiderivatives and semiintegrals, graphs for sin(x)      126
Semiderivatives and semiintegrals, role played in solving diffusion problems      197
Semiderivatives and semiintegrals, table of definitions      115—116
Semiderivatives and semiintegrals, table of general properties      116—117
Semiderivatives and semiintegrals, tables of, for Bessel and Struve functions      127—128
Semiderivatives and semiintegrals, tables of, for binomials      120—121
Semiderivatives and semiintegrals, tables of, for complete elliptic integrals      131
Semiderivatives and semiintegrals, tables of, for constants and powers      118—119
Semiderivatives and semiintegrals, tables of, for exponentials and related functions      122—124
Semiderivatives and semiintegrals, tables of, for generalized hypergeometric functions      129—130
Semiderivatives and semiintegrals, tables of, for Heaviside function      131
Semiderivatives and semiintegrals, tables of, for logarithms      130
Semiderivatives and semiintegrals, tables of, for trigonometric and hyperbolic functions      124—125
Semidifferential equations      10 120 157—159
Semidifferential equations for diffusion in planar geometry      201
Semidifferential equations for diffusion in semiinfinite media      197
Semidifferential equations for diffusion in spherical geometry      204
Semidifferential equations for open-circuited finite transmission line      213
Semidifferential equations for short-circuited finite transmission line      213
Semidifferential equations, definition of      157
Semidifferential equations, examples of      157 159
Semidifferential equations, occurrence in electrochemistry      159—160
Semidifferential equations, techniques for solving      157
Semidifferential equations, techniques for solving, Laplace transformation      159
Semidifferential equations, techniques for solving, transformation to ordinary differential equation      157
Semidifferential equations, used to relate current to applied voltage signal in infinite transmission line      212
Semidifferential equations, used to relate electrochemical current to surface concentration      205

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