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Forsyth A.R. — Theory of differential equations. Part 3. Ordinary linear equations (Vol. 4)
Forsyth A.R. — Theory of differential equations. Part 3. Ordinary linear equations (Vol. 4)

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Название: Theory of differential equations. Part 3. Ordinary linear equations (Vol. 4)

Автор: Forsyth A.R.

Язык: en

Рубрика: Математика/Анализ/Дифференциальные уравнения/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1902

Количество страниц: 534

Добавлена в каталог: 09.04.2005

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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 $\infty$ 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 $\infty$ 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
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