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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.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1974

Количество страниц: 234

Добавлена в каталог: 06.12.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abel's integral equation      2 4 9 13 183—186
Abel, N.H.      2 4 183 184 219
Abramowitz, M.      219
Airy functions      180
Airy functions as hypergeometrics of complexity $\frac{0}{2}$      164
Algorithms for differintegration      136—148
Allegre, J.      133 202 219
Analog differintegration      148—154
Analytic functions      77 80 116
Analytic functions, differintegrals of      75
Applications of fractional calculus, to Abel's integral equation      183—186
Applications of fractional calculus, to biology      10
Applications of fractional calculus, to chemical physics      2
Applications of fractional calculus, to classical calculus      181—195
Applications of fractional calculus, to definition of function families      192—195
Applications of fractional calculus, to differential equations      2 8 10 13
Applications of fractional calculus, to diffusion of atmospheric pollutants      208
Applications of fractional calculus, to diffusion problems      197—218
Applications of fractional calculus, to electrical conduction in transmission lines      210
Applications of fractional calculus, to electrochemistry      2 14 15 204
Applications of fractional calculus, to functional equations      10
Applications of fractional calculus, to geometry      4
Applications of fractional calculus, to grain boundary grooving in metals      216—218
Applications of fractional calculus, to integral equations      2 10 12 13 15
Applications of fractional calculus, to location of candidate solutions for differential equations      189—192
Applications of fractional calculus, to mechanics      4
Applications of fractional calculus, to Navier — Stokes equation      12 14
Applications of fractional calculus, to prediction of peak pollutant concentrations      209
Applications of fractional calculus, to problems of elasticity      2
Applications of fractional calculus, to quantitative chemical analysis      205—207
Applications of fractional calculus, to rheology      2
Applications of fractional calculus, to solution of Bessel's equation      186—189
Applications of fractional calculus, to solution of Legendre's equation      191
Applications of fractional calculus, to tautochrone problem      2 4 5
Applications of fractional calculus, to theory of heat conduction      202—203
Applications of fractional calculus, to transmission line theory      2
Applications of fractional calculus, to transport problems      2
Applications of fractional calculus, to wave equation      11
Associated Legendre functions, as reducible transcendentals      161
Ayabe, Y.      160 221
Babbitt, J.D.      197 219
Barrer, R.M.      197 219
Basic hypergeometrics, definition of      168
Basic hypergeometrics, graphs of      168
Basic hypergeometrics, Laplace transformation of      171
Basic hypergeometrics, relationships among      171
Basis hypergeometrics      168—172
Basis hypergeometrics, complementary      168
Bassam, M.A.      12
Belavin, V.A.      2 13 219
Berg, E.J.      9
Bessel functions      13 97—98 124 125 177—178
Bessel functions as hypergeometrics of complexity $\frac{0}{2}$      164
Bessel functions as reducible transcendentals      161
Bessel functions, relationships among      98
Bessel's equation      186—189
Bessel's equation, solution via fractional calculus      186—189
beta function      see "Complete beta functions" "Incomplete
Bibliography, chronological      3—15
Binomial coefficients      20 28—29 178 180
Binomial coefficients of moiety argument, table of      118
Binomial functions      162 174
Binomial functions as hypergeometrics of complexity $\frac{1}{1}$      99
biology      10
Biorci, G.      154 219
Bishop, D.M.      2 14 222
Blumenthal, L.M.      10
Boole, G.      x 2 6 219
Boundary geometries for diffusion problems      198
Boundary geometries for diffusion problems, diagrams of      199
Bourlet, C.      x 2 219
Brenke, W.C.      9
Buschman, R.G.      12 13
Butzer, P.      14
Caffyn, J.E.      2 222
Carslaw, H.S.      197 219
Carson, J.R.      2 219
Cauchy's integral formula      1 7 8 14 54
Cayley, A.      8
Center, W.      6 7 9
Chain rule for differintegrals      80—81
Chain rule for multiple derivatives      36—37
Chronological bibliography on fractional calculus      3—15
Churchill, R.V.      134 136 150 219
Circuit for performing semiintegration      see "Semiintegrating circuits"
Civin, P.      53 219
Classical calculus      25—44
Classical derivatives      28
Classical integrals      29
Cole, K.S.      10
Complementary functions      4 5 8
Complete beta functions      21 65
Complex error functions      209
Composition rule      48 82—87
Composition rule for differentiable functions      85
Composition rule for differentiable functions, tabular summary      86
Composition rule for differintegrable units      82—84
Composition rule for differintegrable units, tabular summary      84
Composition rule for general differintegrable series      84—85
Composition rule for general differintegrable series, tabular summary      85
Composition rule for mixed integer orders      30—33
Composition rule for mixed integer orders, failure      32
Composition rule for noninteger orders      63
Composition rule, examples illustrating failure      83 86—87
Composition rule, role in locating solutions of differential equations      190
Composition rule, use in inverting extraordinary differential equations      155
Composition rule, utility in finding differintegrals      96
Continued fractions      151—152
Convolution theorem for Laplace transformation      134
Cosine integrals as reducible transcendental      161
cosines      see "Sines and cosines"
Coulomb's law      211
Courant, R.      12 27 50 219
Crank, J.      197 219
Curvature correction for current semiintegral, graph of      206
Cyclodifferential functions      110—112 169(n)
Cyclodifferential functions, definition of      110
Cycloid as solution to tautochrone problem      185
Cycloid as solution to tautochrone problem, diagram of      185
Davis, H.T.      8 9 10 11
Davison, B.      197 219
Dawson's integral      163 175
Definite integrals, evaluation through fractional calculus      181—183
Delahay, P.      160 219
Delta function      see "Dirac delta function"
DeMorgan, A.      5
derivatives      see "Classical derivatives"
Difference quotients      28 48 56
Difference quotients for classical derivatives      28
Differential equations      2 8 10 13
Differential equations, conversion to lower order      190
Differentiation of hypergeometrics      40—44
Differentiation of powers      39—40 192
Differentiation of powers, diagram showing sign of resultant derivative      39
Differentiation to fractional order      see "Differintegration"
Differentiation, term by term      38
Differintegrable and nondifferintegrable functions, figure showing examples of each      47
Differintegrable functions      46—47
Differintegrable functions, definition of      47
Differintegrable series      69 70 82 93 116
Differintegrable series units      82
Differintegrable series units, definition of      46
Differintegrable series, definition of      46
Differintegral operators of imaginary order      14
Differintegral operators of integer order      25—44
Differintegral operators with respect to arbitrary function      55
Differintegral operators, analytic continuation for      49
Differintegral operators, comparison of definitions      55—57
Differintegral operators, connection with Fourier analysis      56
Differintegral operators, connection with Laplace transformation      11
Differintegral operators, defined      x 16 27(n)
Differintegral operators, definition of      45—60
Differintegral operators, definition of, as integral transform      see "Differintegral operators" "Riemann
Differintegral operators, definition of, as sum      see "Differintegral operators" "Gruenwald
Differintegral operators, definition of, based on Cauchy's integral formula      54 56
Differintegral operators, definition of, based on Cauchy's integral formula, diagram of contour used      55
Differintegral operators, definition of, in terms of difference quotients      1
Differintegral operators, definition of, in terms of series      1 5 7 53
Differintegral operators, differentiation of      48 50
Differintegral operators, equivalence of definitions      51—52
Differintegral operators, general properties      10 11 69—92
Differintegral operators, general properties, behavior far from lower limit      91
Differintegral operators, general properties, behavior near lower limit      90
Differintegral operators, general properties, chain rule      14 80—81
Differintegral operators, general properties, commutativity      87
Differintegral operators, general properties, composition rule      82—87
Differintegral operators, general properties, dependence on lower limit      87—89
Differintegral operators, general properties, differintegration term by term      69—75
Differintegral operators, general properties, homogeneity      75
Differintegral operators, general properties, inversion      86
Differintegral operators, general properties, Leibniz's rule      76—79
Differintegral operators, general properties, linearity      69
Differintegral operators, general properties, scale Differintegral operators, general properties, change      75—76
Differintegral operators, general properties, translation      89—90
Differintegral operators, Gruenwald definition      48 55
Differintegral operators, identity of definitions      51—52
Differintegral operators, modified Gruenwald definition      57
Differintegral operators, representation for analytic functions      57—59
Differintegral operators, Riemann — Liouville definition      1 6 9 56 see
Differintegral operators, Riemann — Liouville definition, use in evaluating integrals      181
Differintegral operators, some general definitions      52—57
Differintegral operators, summary of definitions      59—60
Differintegral operators, symbolism for      45
Differintegration      61—68 93—114
Differintegration of Bessel functions      97—98
Differintegration of binomial functions $[C-cx]^{p}$      93—94
Differintegration of constants      63
Differintegration of cosine function cos(x)      112
Differintegration of cyclodifferential functions      110—112
Differintegration of Dirac delta function      106
Differintegration of exponential functions exp(C-cx)      94—95
Differintegration of function $x^{q-1}exp(-1/x)$      112—113
Differintegration of function x-a      63—65
Differintegration of functions $x^{q}/[1-x]$ and $x^{p}/[1-x]$ and $[1-x]^{q-1}$      1 95—96
Differintegration of Heaviside function      105
Differintegration of Heaviside function, diagram illustrating      106
Differintegration of hyperbolic function sinh($\sqrt{x}$)      96—97
Differintegration of hypergeometric functions      99—102
Differintegration of hypergeometric functions, examples of $\frac{1}{3}$ complexity      112
Differintegration of logarithms      102—104
Differintegration of periodic functions      108—110
Differintegration of piecewise-defined functions      107
Differintegration of powers      63—68
Differintegration of powers, breakdown for p<= -1      67
Differintegration of powers, figure showing contour used in      66
Differintegration of powers, signs of resulting coefficients      68
Differintegration of sawtooth function      107—108
Differintegration of sawtooth function, diagram of      108
Differintegration of sine function sin($\sqrt{x}$)      96—97
Differintegration of sine function sin(x)      112
Differintegration of unit function      61—63
Differintegration of unit function, graphs of differintegrals      62
Differintegration of zero function      63
Differintegration to order one-quarter      217
Diffusion equation for semiinfinite geometries      199
Diffusion equation in presence of sources and sinks      207
Diffusion equation, general form      198
Diffusion in finite media      209—215
Diffusion in planar geometry      201—203
Diffusion in presence of sources and sinks      207—209
Diffusion in spherical geometry      204—207
Diffusion on curved surface      216—218
Dirac delta function      105
Dirac delta function, differintegrals of      106
Duff, G.F.D.      27 49 220
Eigenfunction of differintegral operator      122(n)
Eigenfunction of semidifferential operator      122(n)
Eigenfunction of semidifferential operator, graph of      123
Elasticity      2
Electrochemistry      2 14 15 204
Elliptic integrals as hypergeometrics of complexity $\frac{2}{2}$      165
Elliptic integrals as reducible transcendentals      161
Elliptic integrals of first kind      122
Elliptic integrals of second kind      122
Erdelyi, A.      2 11 12 13 54 55 76 115 220
Error function complement      194
Error function complement integrals      175 194—195
Error functions      175
Error functions as reducible transcendentals      161
Errors in analog semidifferentiation      215—216
Errors in analog semidifferentiation, graph of, for various terminations      215
Euler's constant      24
Euler's integral of second kind      21
Euler, L.      1 3 220
Exponential functions, differintegration of      94—95
Exponential integrals      166
Exponential integrals as reducible transcendentals      161
Exponential-like functions, as hypergeometrics of complexity $\frac{0}{1}$      162
Extraordinary differential equations      154—157
Extraordinary differential equations for describing groove depth in metals      217
Extraordinary differential equations, definition of      154
Extraordinary differential equations, inversion of      155
Extraordinary differential equations, series solutions of      159—160
Faa di Bruno's formula for differentiating composite function      37 80
Fabian, W.      11
Faddeeva, V.N.      209 220
Faradaic current, diagram illustrating curvature correction      206
Faradaic current, graph versus time      206
Faradaic current, semiintegral of      205—207
Faraday's constant      204
Faraday's electrochemical law      204
Feller W.      x 220
Festinger, J.C.      219
Fick's First Law      204
Fourier analysis      14
Fourier, J.B.J.      4 5
Fractional calculus applications of      see "Applications of fractional calculus"
Fractional calculus, historical survey      1—15
Fractional calculus, symbolic methods in      2
Fractional difference operators      12
Fractional differential operators      see "Differintegral operators"
Fractional differentiation      see "Differintegration"
Fractional integral operators      see "Differintegral operators"
Fractional integration      see "Differintegration"
Fractional integration by parts      11
Fractional integration for functions of more than one variable      2
Fresnel integrals      125 180
Fresnel integrals as reducible transcendentals      161
Friedman, B.      x 220
Function families      192—195
Function synthesis      169
Function synthesis diagrams      see "Synthesis diagrams"
Function synthesis of K=L transcendentals      172—175
Function synthesis of K=L-1 transcendentals      175—176
Function synthesis of K=L-2 transcendentals      177—180
Functional equations      10
Fundamental theorem of calculus      30
Gaer, M.C.      13 53 220
Gamma function      14 16—24 see
Gamma function for integer and half-integer arguments      17—18
Gamma function for integer and half-integer arguments, table of values      18
Gamma function, asymptotic expansion      19
Gamma function, duplication formula      18
Gamma function, Gauss multiplication formula      18
Gamma function, general properties      16—24
Gamma function, ratios      17 19—20
Gamma function, ratios, polynomials expressible as      19
Gamma function, ratios, polynomials expressible as, graphs of      19
Gamma function, reciprocal      17
Gamma function, reciprocal, asymptotic representation of      17
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