|
|
 |
| Авторизация |
|
|
|
| Поиск по указателям |
|
|
|
|
|
|
|
 |
|
|
|
| Forsyth A.R. — Theory of differential equations. Part 3. Ordinary linear equations (Vol. 4) |
|
|
|
| Предметный указатель |
Adjoint equation, and original equation are reciprocally adjoint 253
Adjoint equation, composition of 254
Adjoint equation, is reducible if original equation is reducible 254
Adjoint equation, Lagrange's 253
Adjoint equation, properties of, in relation to the number of regular integrals 257.
Algebraic coefficients, Appell's class of 484
Algebraic coefficients, associated with automorphic functions 488 see
Algebraic coefficients, character of integrals of, and in vicinity of a singularity 480
Algebraic coefficients, character of integrals of, in vicinity of branch-point 479
Algebraic coefficients, equations having Chapter x
Algebraic coefficients, mode of constructing integrals of 483
Algebraic equation, connected with differential resolvents 49.
Algebraic equation, roots of, satisfy a linear equation with rational coefficients 46 174
Algebraic integrals, and homogeneous forms 202.
Algebraic integrals, connected with theory of covariants 175
Algebraic integrals, connected with theory of finite groups 175
Algebraic integrals, construction of 184 198
Algebraic integrals, equations having 45 165 Chapter
Algebraic integrals, equations of fourth order having 201
Algebraic integrals, equations of second order having 176 et seq.
Algebraic integrals, equations of third order having 191 et seq.
Analytical form of group of integrals associated with multiple root of fundamental equation of a singularity 66
Analytical form of group of integrals associated with multiple root of fundamental equation of a singularity, likewise for multiple root of fundamental equation for a period or periods 416 454.
Annulus, integral converging in any see fundamental equation irregular Laurent
anormales 270.
Apparent singularity 117
Apparent singularity, conditions for 119.
Appell 209 484.
Asymptotic expansions, Poincare's theory of 338
Asymptotic expansions, represent normal integrals when functionally illusory 340.
ausserwesentlich 117.
Automorphic functions in general 510 et seq.
Automorphic functions, and conformal representation 491
Automorphic functions, and differential equations having algebraic coefficients 488
Automorphic functions, associated with linear equations of second order 495
Automorphic functions, constructed for a special case 500
Automorphic functions, when there are three 509 510
Automorphic functions, when there are two singularities 508
Automorphic functions, when there is one singularity 506
Barnes 448.
begleitender bilinearer Differentialausdruck 254.
Benoit 474.
Bessel's equation 1 13 34 100 101 126 164 330 333 393.
Bilinear concomitant of two reciprocally adjoint equations 254.
Bocher 161 169.
Bocher's theorem on equations of Fuchsian type with five singularities 161.
Boole 229.
Boulanger 195 197 198.
Brioschi 206 208 218.
Casorati 55 60 417.
Cauchy 11 20.
Cauchy's theorem used to establish existence of synectic integral of a linear equation 11.
Cayley 94 113 121 182 216 233 246 262 281 282 286.
Cayley's method for normal integrals 281
Cayley's method for normal integrals for subnormal integrals 284.
Cazzaniga 348.
Cels 254.
Characteristic equation belonging to a singularity 40.
Characteristic equation for determining factor of normal integrals 294
Characteristic equation for determining factor of normal integrals, effect of multiple root 298.
Characteristic equation for determining factor of normal integrals, effect of simple root of 294
Characteristic function of an equation 226.
Characteristic index and number of regular integrals of an equation 230 233
Characteristic index of reciprocally adjoint equations, the same 257.
Characteristic index, defined 221
Chrystal 7 83.
Circular cylinder, differential equation of 164 see
Class, equations of Fuchsian see Fuchsian type.
Cockle 49.
Coefficients, form of, near a singularity if all integrals there are regular 78.
Collet 20.
Conformal representation, and automorphic functions 491
Conformal representation, and fundamental polygon 493.
Constant coefficients, equation having 14—20.
Construction, of regular integrals, by method of Frobenius 78
Construction, of regular integrals, of normal integrals, periodic integrals see normal integrals simply-periodic doubly-periodic
Continuation process applied to synectic integral 20.
Continued fractions used to obtain a fundamental equation 439.
Covariants associated with algebraic integrals 202
Covariants associated with algebraic integrals for equations of fourth order 204
Covariants associated with algebraic integrals for equations of second order 206.
Covariants associated with algebraic integrals for equations of third order 203 209
Craig, vi 411.
Crawford 474.
Curve, integral, denned 203 205.
Darboux 20 254 475.
Definite integrals see Laplace's definite integral double-loop
Determinant of a system of integrals 25
Determinant of a system of integrals, form of, near a singularity 77
Determinant of a system of integrals, its value 27
Determinant of a system of integrals, not vanishing, the system is fundamental 29
Determinant of a system of integrals, of a fundamental system does not vanish 30
Determinant of a system of integrals, special form of, for one particular system 34
Determinant of a system of integrals, when the coefficients are algebraic 481.
Determinant of a system of integrals, when the coefficients are periodic 406 445
Determinants, infinite see infinite determinants.
Determining factor, conditions for 265
Determining factor, for integrals of Hamburger's equations 288—292.
Determining factor, obtained by Thome's method 262 et seq.
Determining factor, of normal integral 262
Diagonal of infinite determinant 349.
Difference-relations 63 417.
Differential invariants see invariants differential.
Differential resolvents 49.
Dini 254 256.
Divisors, elementary see elementary divisors.
Double-loop integrals 333
Double-loop integrals, applied to integrate equations 334 et seq.
Doubly-periodic coefficients, equations having 441 et seq.
Doubly-periodic coefficients, fundamental equations for the periods 444 445.
Doubly-periodic coefficients, substitutions for the periods 443
Doubly-periodic integrals of second kind 447
Doubly-periodic integrals of second kind, belonging to Lame's equation 468
Doubly-periodic integrals of second kind, how constructed 471 475.
Doubly-periodic integrals of second kind, number of 448 450
Doubly-periodic integrals of second kind, Picard's theorem on 447
Eisenstein 525.
Element of fundamental system 30.
Elementary divisors, determine groups and sub-groups of integrals 62
Elementary divisors, effect of, upon number of periodic integrals when coefficients are periodic 416 450.
Elementary divisors, of certain determinants 41—43
Elementary divisors, of the fundamental equation 55
Elliott, M. 424 425 474.
Elliptic cylinder, differential equation of 164 399 431—441.
Expansion of converging infinite determinants 353.
Expansions, asymptotic see asymptotic expansions.
Exponent, of irregular integral as zero of an infinite determinant 368.
Exponent, properties of 75
Exponent, to which normal integral belongs 262
Exponent, to which regular integral belongs 74
Exponent, to which the determinant of a fundamental system belongs 77
Exponents, sum of, for Eiemann's P-function 139.
Exponents, sum of, for equations of Fuchsian type 126
Fabry 94 270.
Factor, determining see determining factor.
Fano 214 218.
Finite groups of lineo-linear substitutions, associated with equations of second order having algebraic integrals 182
Finite groups of lineo-linear substitutions, connected with polyhedral functions 181
Finite groups of lineo-linear substitutions, in one variable 176
Finite groups of lineo-linear substitutions, in three variables 200.
Finite groups of lineo-linear substitutions, in two variables 192
Finite groups of lineo-linear substitutions, the Laguerre invariant 196
Finite groups of lineo-linear substitutions, their differential invariants 195
Finite groups of lineo-linear substitutions, used for construction of algebraic integral 185
Finite groups of lineo-linear substitutions, used to construct equations of third order having algebraic integrals 197
First kind, periodic function of 410.
Floquet 231 234 259 411 448.
Fricke 489 515 525.
Frobenius 78 93 109 226 231 233 238 247 254 257 259.
Frobenius' method, applied to hypergeometric equation for special cases 147
| Frobenius' method, for the construction of integrals (all being regular) 78 et seq.
Frobenius' method, for the construction of integrals, when only some are regular 235 et seq.
Frobenius' method, used for construction of irregular integrals 379 et seq.
Frobenius' method, variation of, suggested by Cayley for some cases 114
Fuchs, L. 3 0 11 60 65 66 78 93 94 109 110 117 123 125 126 129 156 206 208 216 399 482.
Fuchsian equations 123 495
Fuchsian equations, independent variable a uniform function of quotient of integrals of 496—499
Fuchsian equations, mode of determining coefficients in 501
Fuchsian equations, used as subsidiary to linear equations of any order 515.
Fuchsian functions in the expression of integrals as uniform functions 520.
Fuchsian functions, associated with linear equations of general order 515 517
Fuchsian functions, associated with linear equations of the second order 500 502 515
Fuchsian group see Fuchsian function Zetafuchsian
Fuchsian type of second order with any number of singularities 150
Fuchsian type, Bocher's theorem on 161
Fuchsian type, equations of Chapter iv pp.
Fuchsian type, equations of, having five singularities 161
Fuchsian type, form of 123
Fuchsian type, forms of, when  is an ordinary point 152
Fuchsian type, having polynomial integrals 166
Fuchsian type, having rational integrals 169.
Fuchsian type, Klein's normal 158
Fuchsian type, Lame's equation transformed so as to be of 160
Fuchsian type, properties of exponents 126
Fuchsian type, when is a singularity 155 158
Fuchsian type, when fully determined by singularities and exponents 128
Fundamental equation for simple period 405
Fundamental equation for simple period is invariantive 406
Fundamental equation for simple period, analytical expression of 419.
Fundamental equation for simple period, form of 407
Fundamental equation for simple period, integral associated with a simple root 408
Fundamental equation for simple period, integrals associated with a multiple root 408
Fundamental equation when coefficients are algebraic 482
Fundamental equation when coefficients are algebraic, relation to indicial equation 482.
Fundamental equation when integrals are irregular, expressed as an infinite determinant 389
Fundamental equation when integrals are irregular, expressed in finite terms 392
Fundamental equation when integrals are irregular, various methods of obtaining 399.
Fundamental equation, belonging to a singularity, is same for all fundamental systems 38—40
Fundamental equation, fundamental system of integrals associated with 50
Fundamental equation, invariants of 40
Fundamental equation, Poincare's theorem on 40
Fundamental equation, properties of, connected with elementary divisors 41—43
Fundamental equation, roots of, how related to roots of indicial equation 94.
Fundamental equation, when a root is multiple 53
Fundamental equation, when roots are simple 52
Fundamental equations for double periods 444 445
Fundamental equations for double periods, effect of multiple roots of 451.
Fundamental equations for double periods, number of these integrals 450
Fundamental equations for double periods, roots of, determine doubly-periodic integrals of the second kind 448
Fundamental equations for double periods, their form 447
Fundamental polygon for automorphic functions 490 493 500.
Fundamental system of integrals, affected by elementary divisors of fundamental equation 57
Fundamental system of integrals, aggregate of groups associated with roots of indicial equation Fundamental system of integrals, make fundamental system 95.
Fundamental system of integrals, defined 30
Fundamental system of integrals, form of, near singularity 50
Fundamental system of integrals, if root is multiple 53
Fundamental system of integrals, if root of fundamental equation is simple 52
Fundamental system of integrals, its determinant is not evanescent 30
Fundamental system of integrals, properties of 30 31
Fundamental system of integrals, tests for 31 32
Fundamental system, constituted by group of integrals belonging to a multiple root of fundamental equation see group of integrals.
Fundamental system, of integrals when coefficients are simply-periodic 408 419
Fundamental system, of irregular integrals 387
Fundamental system, when coefficients are algebraic 480.
Fundamental system, when coefficients are doubly-periodic 449—457
Giinther 11 299 399.
Gordan 182.
Grade of normal integral 269.
Graf 333.
Greenhill 466.
Group of integrals associated with multiple root of fundamental equation for period when coefficients are simplyperiodic 415
Group of integrals associated with multiple root of fundamental equation for period when coefficients are simplyperiodic, analytical expression of 419
Group of integrals associated with multiple root of fundamental equation for period when coefficients are simplyperiodic, arranged in sub-groups, according to elementary divisors 416
Group of integrals associated with multiple root of fundamental equation for period when coefficients are simplyperiodic, further expression of, when uniform 421.
Group of integrals associated with multiple root of fundamental equation for period when coefficients are simplyperiodic, they constitute a fundamental system for equation of lower order 420
Group of integrals associated with multiple roots of fundamental equations for periods when coefficients are doublyperiodic 451
Group of integrals associated with multiple roots of fundamental equations for periods when coefficients are doublyperiodic, analytical expression of 452 457
Group of integrals associated with multiple roots of fundamental equations for periods when coefficients are doublyperiodic, further development of, when uniform 459.
Group of integrals for hypergeometric equation 144.
Group of integrals, aggregate of, make a fundamental system 96
Group of integrals, associated with multiple root of fundamental equation 53
Group of integrals, associated with multiple root of indicial equation in method of Frobenius 80
Group of integrals, can be fundamental system of equation of lower order 72.
Group of integrals, compared with Hamburger's groups 113.
Group of integrals, general analytical form of 65
Group of integrals, general theorem on 93
Group of integrals, Hamburger's sub-groups of 62
Group of integrals, resolved into sub-groups, by elementary divisors 57
Group of irregular integrals associated with multiple root of characteristic infinite determinant 381 et seq.
Group of irregular integrals associated with multiple root of characteristic infinite determinant, resolved into sub-groups 382.
Groups of substitutions, finite see finite groups
Groups of substitutions, infinite see automorphic functions.
Gruenfeld 259.
Gubler 333.
Gylden 462.
Halphen 254 255 281 315 316 448 464 465 473.
Hamburger 38 60 62 63 64 113 277 280 283 286 399 482.
Hamburger's equations 276 et seq.
Hamburger's equations of general order n with normal or subnormal integrals 288 et seq.
Hamburger's equations of second order with normal integrals 279
Hamburger's equations of third order with normal or subnormal integrals 301 et seq.
Hamburger's equations, the number of normal integrals 280
Hamburger's sub-groups of integrals see sub-groups of integrals.
hankel 103 333.
Harley 49.
Heffter 55 156.
Heine 164 166 431 441.
hermite 15 20 448 463 465 468 473.
Hermite, on equation with constant coefficients 15—20
Hermite, on equation with doubly-periodic coefficients 465.
Heun 159.
Heymann 50.
Hill, G.W. 348 396 398 399 402 482.
Hobson 334 337.
Homogeneous forms see covariants.
Homogeneous linear equations, defined 2
Homogeneous linear equations, discussion limited to 3.
Homogeneous relations between integrals when they are algebraic 203 217
Homogeneous relations between integrals when they are algebraic and of higher degree 214.
Homogeneous relations between integrals when they are algebraic of second degree for equations of third order 230
Horn 333 341 342 346 347.
Humbert 167.
Hypergeometric function, used to render integrals of differential equations uniform in special case 509.
Hypergeometric series, equation of 1 13 34 103 126 144—150 173 338 501 509.
Identical relations, polynomial in powers of a logarithm, cannot exist 69.
Index, characteristic see characteristic index
Index, properties of 75.
Index, to which regular integral belongs 74
Indicial equation for equation with not all integrals regular 222 227
Indicial equation, group of integrals associated with a multiple root of 86
Indicial equation, integral associated with a simple root of 86
Indicial equation, relation of, to the fundamental equation 482.
Indicial equation, roots of, how connected with roots of fundamental equation 94
Indicial equation, significance of, in the method of Frobenius 85
Indicial equation, when all integrals are regular 85 94
Indicial equation, when coefficients are algebraic 482
Indicial function of adjoint equation, as affecting the number of regular integrals 259.
Indicial function, degree of, as affecting the number of regular integrals 230 233
Indicial function, when all integrals are regular 94
Indicial function, when not all integrals are regular 227
Infinite determinant is a periodic function of its parameter 375
Infinite determinant, effect of a multiple root of 381 et seq.
Infinite determinant, effect of simple root of 380
Infinite determinant, expressed in finite terms 392.
Infinite determinant, giving exponent of irregular integral 368
Infinite determinant, leads to the fundamental equation of the singularity 389
Infinite determinant, modified to another determinant 369
Infinite determinants in general 349
Infinite determinants in general, applied to construct irregular integrals of differential equations 363 et seq.
Infinite determinants in general, convergence of 350
|
|
|
| Реклама |
|
|
|
|
|
|
|
|
О проекте