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| Forsyth A.R. — Theory of differential equations. Part 3. Ordinary linear equations (Vol. 4) |
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| Предметный указатель |
Infinite determinants in general, may be capable of differentiation 359
Infinite determinants in general, properties of converging, in general 352 et seq.
Infinite determinants in general, uniform convergence of, when functions of a parameter 358
Infinite determinants in general, used to solve an unlimited number of linear equations 360
Initial conditions defined 4
Initial conditions defined, effect of, upon form of synectic integral 9.
Initial conditions defined, values 4
Integral curve 203 205.
Integrals rendered uniform functions of a variable, by means of Zetafuchsian functions 518 520.
Integrals rendered uniform functions of a variable, in general 510 et seq.
Integrals rendered uniform functions of a variable, when there are three singularities 509 510
Integrals rendered uniform functions of a variable, when there are two singularities 508
Integrals rendered uniform functions of a variable, when there is one singularity 506
Integrals, doubly-periodic, irregular, normal, regular, simply-periodic, subnormal, synectic see under these titles respectively.
Integrals, irregular see irregular integrals
Invariants of fundamental equation, connected with a period or periods 405 445
Invariants of fundamental equation, connected with singularity 38 40
Invariants of fundamental equation, for irregular integrals 398
Invariants of fundamental equation, when coefficients are algebraic 482.
Invariants, differential, for equations of the fourth order 201 213
Invariants, differential, for equations of the third order 195
Invariants, differential, Laguerre's 196.
Invariants, differential, Sehwarzian derivatives as, for equations of the second order 182
Irreconcileable paths, defined 23.
Irreducible equations, exist 347
Irreducible equations, Frobenius' method of constructing 248.
Irregular integrals, converge within an annulus 366
Irregular integrals, formal expression for, obtained by infinite determinants 376
Irregular integrals, groups and sub-groups of, obtained by generalisation of Frobenius' method 379
Irregular integrals, in the form of Laurent series 364
Irregular integrals, made uniform functions of a new variable by means of automorphic functions see automorphic functions.
Irregular integrals, these constitute a fundamental system 387
JORDAN 197 200 333 334 338 341.
Juergens 65 113.
Klein 150 153 155 158 161 176 185 187 190 197 206 489 515 525.
Klein's normal form of equation of second order and Fuchsian type 158
Klein's normal form of equation of second order and Fuchsian type, method for equations of second order having algebraic integrals 176.
Kneser 341.
Kummer 146.
Kummer's group of integrals of the hypergeometric equation 144.
Lagrange 251.
Laguerre 196.
Lame's generalised equation 160.
Lamp's equation 1 126 159 160 165 168 338 448 464—473.
Laplace's definite integral, contour of 323
Laplace's definite integral, developed into normal integrals, where these exist 324 et seq.
Laplace's definite integral, satisfying equation with rational coefficients 318
Laurent series expressing an irregular integral 364
Laurent series expressing an irregular integral, proof of convergence within an annulus 366.
Legendre's equation 1 13 34 103 126 160 163.
Liapounoff 319 425—431.
Liapounoff's theorem, applied to evaluate Laplace's definite integral 324
Liapounoff's theorem, method of discussing uniform periodic integrals 425.
Lindemann 431 434 437.
Lindstedt 439.
Linear algebraic equations, infinite system of, solved by means of infinite determinants 360.
Linear differential equation, definition of 2.
Lineo-linear substitutions see finite groups.
Logarithms, condition that some regular integral shall be free from 110.
Logarithms, identical relations, polynomial in powers of, cannot exist 69
Logarithms, quantity affected by, can satisfy a uniform linear differential equation and determine its fundamental system 66
Logarithms, regular integrals free from 106
Lommel 331.
Macdonald 333.
Markoff 169.
Member of a fundamental system 30.
Minors of infinite determinants 354.
Mittag — Leffler 399 463.
Modular function, Eisenstein's function similar to 525.
Modular function, used to render integrals of differential equations uniform in special case 510
Multiple root, group of integrals associated with see multiple root.
Multiplier of periodic integral of second kind 410
Multiplier of periodic integral of second kind is a root of the fundamental equation of the period 406.
Muth 42.
Normal form of infinite determinant 350.
Normal form, (after Frobenius) of equation having some integrals regular 227
Normal form, (after Klein) of equation of Fuchsian type 158
Normal form, of component factors of such an equation, and of a composite equation 228
Normal integrals and the number of 280
Normal integrals of Hamburger's equation of order n 288 et seq.
Normal integrals, aggregate of, satisfy another differential equation 271
Normal integrals, are asymptotic representation of Laplace's integral 340.
Normal integrals, arising out of Laplace's definite integral 329
Normal integrals, belonging to equation of third order 304 308 309
Normal integrals, Cayley's method of obtaining 281
Normal integrals, conditions that Hamburger's equation of second order may have 279
Normal integrals, constructed by Thome's method 262 et seq.
Normal integrals, defined 262
Normal integrals, number of, belonging to equation of order n 295 298
Normal integrals, of equations with rational coefficients 313
Number of regular integrals of an equation and its characteristic index 230
Number of regular integrals of an equation and its characteristic index, and the number for the adjoint equation 257.
Number of regular integrals of an equation and its characteristic index, can be less than maximum value 233
Ordinary point, synectic integral in domain of 4.
Origin of infinite determinant 349
Origin of infinite determinant can be moved in the diagonal without changing the value of the determinant 350.
Osgood 85 122.
P-function, discussion of see Riemami's P-function.
Painleve 195 198 199.
Papperitz 142.
Parabolic cylinder, equation of 165.
Paths, deformation of, without crossing singularity 22
Paths, effect of, round a singularity Chapter n passim.
Paths, if reversed in continuation process, restore initial values 21
Paths, reconcileable, and irreconcileable 23
Pepin 206.
Period, fundamental equation for simple 405
Period, fundamental equations for double 445 see
Periodic coefficients, doubly see doubly-periodic coefficients.
Periodic coefficients, equations having uniform Chapter ix 403
Periodic coefficients, simply see simply-periodic coefficients
Physics, equations of mathematical, and equations of Fuchsian type having five singularities 161.
Picard vi 317 319 341 443 447 448 460 471.
Pochhammer 105 159 333 338.
Poincare 40 61 105 246 270 271 315 317 330 338 347 348 353 399 441 482 489
Poincare's theorem on aggregate of normal integrals of a given equation 271
Poincare's theorem on aggregate of normal integrals of a given equation, applications of automorphic functions to equations having algebraic coefficients 488 et seq.
Poincare's theorem on aggregate of normal integrals of a given equation, asymptotic expansions 338 et seq.
Poincare's theorem on aggregate of normal integrals of a given equation, development of Laplace's definite integral that satisfies equation with rational coefficients 318 et seq.
Poincare's theorem on aggregate of normal integrals of a given equation, theorem on the integration of linear equations by means of zetafuchsian functions 517 523.
Polygon, fundamental see fundamental polygon.
Polyhedral functions, and finite groups 181
Polyhedral functions, associated with equations of second order having algebraic integrals 182
Polyhedral functions, used for construction of algebraic integral 185.
Polynomial integrals, equations having 166
| Polynomial integrals, how far determinate 167.
Potential, equation for the, solved by means of Lame's equation 465.
Principal diagonal of infinite determinant 349.
Puiseux diagram used 267 269 274 285 300 310 311.
Quarter-period in elliptic functions, equation of 1 129 337 510.
Rank, equations of, greater than unity replaced by equations of rank unity 342 et seq.
Rank, of differential equation, defined 271
Rational coefficients, equations having 313 et seq.
Rational coefficients, Laplace's definite integral solution of 318.
Rational coefficients, normal integrals of 314
Rational integrals, equations having 169.
Real singularity 117
Real singularity, conditions for 119.
Reconcileable paths 23.
Reducibility of equations, defined 223
Reducibility of equations, extent of, when some integrals are regular 226 248
Reducibility of equations, if they possess normal or subnormal integrals 273.
Reducible, adjoint of a reducible equation is 253
Reducible, equation, having a reducible adjoint, is 254
Reducible, equations having regular integrals, are 224 226 248
Reducible, equations, having normal or subnormal integrals, are 273.
Regular integrals, conditions that all may be free from logarithms 106
Regular integrals, conditions that some may be free from logarithms 110
Regular integrals, construction of, by method of Frobenius 78
Regular integrals, denned 4 74
Regular integrals, equations having all integrals everywhere regular Chapter iv see
Regular integrals, form of coefficients near a singularity if all integrals are 78
Regular, conditions that they exist 237
Regular, equations having no integrals 231 233
Regular, equations having some integrals, are reducible 224 226
Regular, equations when only some integrals are Chapter vi
Regular, form of coefficients 221
Regular, how many integrals of adjoint equation are 257.
Regular, integrals possessed by an equation, number of 230
Regular, integrals, when they exist, constructed by method of Frobenius 235 et seq.
Resolvents, differential 49.
Riemann's P-function, definition of 136
Riemann's P-function, determines a differential equation 141 163 165
Riemann's P-function, forms of differential equation thus determined 143
Riemann's P-function, group of integrals deduced for hypergeometric equation 144.
Riemann's P-function, transformations of 137
Riemaun 137 140.
Roots of fundamental equation and of indicia! equation, how related 94.
Salmon 43.
Sauvage 42.
Schlesinger vi 113 218.
Schwarz 492.
Scott, R.F. 41.
Second kind, equation with periodic coefficients has integrals which are 411 447
Second kind, number of such integrals 411 417 448 450 see doubly-periodic
Second kind, periodic functions of 410 447
Simply-periodic coefficients, analytical expression of these integrals 415.
Simply-periodic coefficients, equations having 403 et seq.
Simply-periodic coefficients, possess integrals which are periodic of second kind 411
Simply-periodic integrals of second kind 411
Simply-periodic integrals of second kind, their analytical expression 412.
Singularities of a differential equation 3
Singularities of a differential equation, how treated when coefficients are algebraic 490 et seq.
Singularities of a differential equation, real or apparent 117
Singularities of a differential equation, with conditions for discrimination 119
Singularity, effect of path round 36
Singularity, equation connected with, is invariantive 38.
Stieltjes 169 437.
Sub-groups of irregular integrals see group of irregular integrals irregular
Sub-groups of periodic integrals, analytical expression of 419.
Sub-groups of periodic integrals, are analogous to Hamburger's sub-groups of regular integrals 417
Sub-groups of periodic integrals, determined by elementary divisors of the fundamental equation of the period 416
Sub-groups, can be fundamental system of an equation of lower order 72.
Sub-groups, general analytical form of 65
Sub-groups, Hamburger's 62
Sub-groups, in a group of integrals associated with multiple root of fundamental equation 57
Sub-groups, number of, is equal to number of elementary divisors of fundamental equation 62
Subnormal integrals, aggregate of, satisfy another equation 271
Subnormal integrals, Cayley's method of obtaining 284
Subnormal integrals, denned 270
Subnormal integrals, how constructed 270
Subnormal integrals, of Hamburger's equation of order n 299 et seq.
Subnormal integrals, of Hamburger's equation of second order 286
Subnormal integrals, of Hamburger's equation of third order 309 313.
Subsidiary equation for integration of any linear equation in terms of uniform functions, Fuehsian equations used as 517.
Substitutions, finite groups of lineo-linear substitutions see finite groups.
Sylvester's eliminant used 46.
Synectic integral in domain of ordinary point 4
Synectic integral in domain of ordinary point is linear in initial values 9
Synectic integral in domain of ordinary point is unique as determined by initial conditions 8
Synectic integral in domain of ordinary point vanishes if all initial values vanish 9
Synectic integral in domain of ordinary point, continuation of 20.
Synectic integral in domain of ordinary point, modes of establishment of 10 11
System of functions, this property used to reduce an equation see reducible equations.
System of functions, when linearly independent, can satisfy a linear differential equation of which they are a fundamental system 44
System of functions, when the coefficients in the equation are rational 45 223
System of integrals, determinant of 25
System of integrals, fundamental see fundamental system.
Tannery 44 94 109 129 131 135.
Ternariants see covariants.
Thetafuchsian functions used 520 et seq.
Third kind, equation having integrals which are 411.
Third kind, periodic functions of 410
Thome 74 221 231 232 233 234 257 259 262 483.
Thome's method of obtaining the determining factor of a normal integral 262 et seq.
Tisserand 431 441.
Transformation of equations of rank greater than unity to equations of rank unity 342 et seq.
Type, equations of Fuchsian see Fuchsian type
Type, of equations, as associated with automorphio functions 516.
Uniform doubly-periodic integrals 459
Uniform doubly-periodic integrals, illustrated by Lame's equation 464 et seq.
Uniform doubly-periodic integrals, modes of constructing 460 468 471 475
Uniform functions, in general 510 et seq.
Uniform functions, integrals of equations expressible as, by means of automorphic functions see automorphic functions
Uniform functions, Poineare's theorem on 518.
Uniform functions, simple examples of 506 508 509 510
Uniform simply-periodic integrals 421
Uniform simply-periodic integrals, Liapounoff's method of dealing with 425.
Valentiner 197.
van Yleck 169.
Vogt 399.
von Koch 348 359 398 399 482.
Weber 165 333.
Weierstrass 42 85 117 277.
wesentlich 117.
Whittaker 515.
Williamson 320.
Zetafuchsian functions 520
Zetafuchsian functions, most general expression of 523.
Zetafuchsian functions, properties of 521
Zetafuchsian functions, used to express the integral of any linear equation 522—524
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