Fixed circle of elliptic Kleinian substitution, when the equation is generalised 613
Fixed points of linear substitution 514
Floquet 282
Form of argument for given value of function possessing an addition-theorem 300
Fortsetzung 55
Fractional factor for potential function 422
Fractional part of doubly-periodic function 220
Fredholm 46
Fricke vii 129 403 460 462 464 511 512 573
Frobenius 266 275 281
Frost 333
Fuchs 111 482 637
Fuchsian functions 619 see
Fuchsian group 591 606
Fuchsian group, class of 608
Fuchsian group, division of plane into two portions within and without the fundamental circle 603
Fuchsian group, families of 606
Fuchsian group, if real, conserves axis of real quantities 591
Fuchsian group, when real, it is transformed by one complex substitution and then conserves a circle 603
Fuchsian groups 606
Function possessing an algebraic addition-theorem, has only a finite number of values for one value of the argument 308
Function possessing an algebraic addition-theorem, if uniform, then either rational, or simply-periodic or doubly-periodic 308
Function possessing an algebraic addition-theorem, is either algebraic, or algebraic simply-periodic, or algebraic doubly-periodic 300
Function possessing an algebraic addition-theorem, satisfies a differential equation between itself and its first derivative 309
Function, branch, and branch-point, of a, defined 15
Function, constant along a line or area, if uniform, is constant everywhere 59
Function, continuation of a 55
Function, element of 56
Function, equations satisfied by real and imaginary parts of 11
Function, having a finite number of branches is a root of an algebraical equation 175
Function, holomorphic, defined 15
Function, integral algebraical 70
Function, integral transcendental 70
Function, meromorphic, defined 16
Function, monogenic analytic, definition of 56
Function, monogenic, defined 14
Function, potential 407 see
Function, properties of uniform, without essential singularities Chap. iv
Function, region of continuity of 55
Function, relations between real and imaginary parts of 9
Function, Riemann's general definition of 8
Function, uniform, multiform, defined 15
Functional dependence of complex variables, analytical conditions for 7
Functional dependence of complex variables, establishes conformal representation 11
Functional dependence of complex variables, form of, adopted 7
Functionality, monogenic, not coextensive with arithmetical expression 139
Functions, expression in series of see "Series of functions"
Fundamental circle for group 603
Fundamental circle of Fuchsian group 603
Fundamental circle of Fuchsian group, divides plane into two parts which are inverses of each other with regard to the circle 604
Fundamental circle of Fuchsian group, essential singularities of the group lie on 606
Fundamental equation for a Riemann's surface is determined by algebraical functions that exist on the surface 462
Fundamental loops 360
Fundamental parallelogram 200
Fundamental parallelogram for double periodicity 200 206
Fundamental parallelogram for double periodicity, is not unique 206
Fundamental polyhedron (of reference for space) 615
Fundamental region (of reference for plane) 591
Fundamental region (or polygon) for division of plane associated with a discontinuous group 591
Fundamental region (or polygon) for division of plane associated with a discontinuous group, can be taken so as to have edges of the first kind cutting the conserved line orthogonally 594 604
Fundamental region (or polygon) for division of plane associated with a discontinuous group, can be taken so as to have edges of the first kind cutting the conserved line orthogonally, angles of 597 see
Fundamental region (or polygon) for division of plane associated with a discontinuous group, can be taken so as to have edges of the first kind cutting the conserved line orthogonally, in this case, called a normal polygon 594
Fundamental region (or polygon) for division of plane associated with a discontinuous group, can be taken so as to have edges of the first kind cutting the conserved line orthogonally, in this case, which can be taken as convex 595
Fundamental region (or polygon) for division of plane associated with a discontinuous group, characteristics of 599
Fundamental set of loops 360
Fundamental substitutions 583
Fundamental substitutions of a group 583
Fundamental substitutions of a group, one for each pair of conjugate edges of region 593
Fundamental substitutions of a group, relations between 584 593 599
Fundamental systems of isothermal curves 579
Fundamental systems of isothermal curves, all families of algebraic isothermal curves are derived from, by algebraic equations 580
Fundamental systems of isothermal curves, given by a uniform algebraic function, or a uniform simply-periodic function, or a uniform doubly-periodic function 579
Galois 582
Gattung (kind of integral) 394
Gauss 2 11 84 408 491 496 500
General conditions for potential function 410 see
Generalised equations of Kleinian group 612 see
Generalised equations of Kleinian group, polyhedral division of space in connection with 614
Generalised equations of Kleinian group, polyhedral division of space in connection with, connected with polygonal division of plane by the group 615
Generalising factor of transcendental integral function 81
Genere 89
Geneve (class of connected surface) 324
Genre (applied to singularity) 148
Genre (applied to transcendental integral functions) 89
Genre (class of connected surface) 324
Geschlecht 324
Giramenta, punto di 15
Gleichmaessig unbedingt convergent 127
Gleichverzweigt 371
Gordan 174 209 359 361 367 403 455 456 586
Goursat 144 188 189 295 546 549 618
Graphical determination of, order of infinity of an algebraic function 164
Graphical determination of, the branches of an algebraic function in the vicinity of a branch-point 170
Graphical determination of, the leading term of a branch in the vicinity of an ordinary point of the coefficients of the equation 167
Graphical representation of periodicity of functions 198 199
Green 408
Greenhill 193
Grenze, natuerliche 129
Grenzkreis 111
Group of linear substitutions 582
Group of linear substitutions, conserved line in relation to the essential singularities 607
Group of linear substitutions, continuous, and discontinuous (or discrete) 584
Group of linear substitutions, dihedral 625
Group of linear substitutions, division of plane for any discontinuous group 591 see
Group of linear substitutions, division of plane for any discontinuous group, fundamental region for 591
Group of linear substitutions, finite 586 see
Group of linear substitutions, Fuchsian 591 606 see
Group of linear substitutions, fundamental substitutions of 583
Group of linear substitutions, fundamental substitutions of, connected with the pairs of conjugate edges of a region 593
Group of linear substitutions, fundamental substitutions of, relations between 584
Group of linear substitutions, Kleinian 610 see
Group of linear substitutions, modular, division of plane into polygons associated with 588
Group of linear substitutions, modular, division of plane into polygons associated with, relation between the fundamental substitutions 590
Group of linear substitutions, modular, with two fundamental substitutions 587
Group of linear substitutions, properly and improperly discontinuous 585
Group of linear substitutions, seven families of 606
Group of linear substitutions, tetrahedral 627
Group of linear substitutions, when real, conserves axis of real quantities 591
Group of substitutions 582
Grouping of branches of algebraical function at a branch-point 171
Grundzahl 317
Guenther 464
Guichard 104 147 217 218
Gutzmer 46
Gylden 126
Half-plane represented on a circle 506
Half-plane represented on a curvilinear polygon, bounded by circular arcs 549 see "Curvilinear
Half-plane represented on a rectilinear polygon 538 see
Half-plane represented on a sector 507
Half-plane represented on a semicircle 506
Half-plane represented on an infinitely long strip 508
Halphen 85 263 266 275 285 295 296
hankel 129 189
Harnack 6 10 409
Hauptkreis 603
Heine 189
Henrici 409
hermite vii 20 77 84 92 112 140 185 186 188 189 257 275 277 279 286 295 456 464 476 633 637
Hermite's sections for integrals of uniform functions 185
Herz 500
Hobson 6 83
Hoelder 54
Hofmann 366
Hole in surface, effect of making, on connectivity 320
Holomorphic 15
Holomorphic function, defined 15
Holomorphic function, expansion of, in the domain of an ordinary point 43 52
Holomorphic function, expansion of, within a ring of convergence by Laurent's theorem 47
Holomorphic function, has a derivative for points within, but not necessarily on the boundary of, its region 32
Holomorphic function, integral of, along a line 24
| Holomorphic function, integral of, round a simple curve 23
Holomorphic function, integral of, when line is deformed 26
Holomorphic function, integral of, when simple curve is deformed 27
Holomorphic function, superior limit for modulus of derivatives of 33
Holzmiiller 2 345 512
Homen 144
Homogeneous form of linear substitutions 622
Homogeneous substitutions 622
Homogeneous substitutions, dihedral group of 624
Homogeneous substitutions, two derived from each linear substitution 622
Homographic substitution connected with spherical rotation 620
Homographic transformation, or substitution 512 see
Homologous (points) 200
Homologous points 200 591
Homoperiodic 224
Homoperiodic functions 224
Homoperiodic functions are connected by an algebraical equation 236
Homoperiodic functions, when equivalent 225
Homoperiodic functions, when in a constant ratio 224
Houeel 2
Humbert 456 464
Hurwitz 406 585 588 637 657
Hyperbolic substitution 517
Hyperbolic substitutions 517 519
Hyperbolic substitutions, do not occur in connection with cycles of corners 607 614
Hyperbolic substitutions, neither periodic nor infinitesimal 522
Hypergeometric series, cases of algebraical solution 567
Hypergeometric series, solution of differential equation for, connected with conformal representation of curvilinear triangle 555
Icosahedral (and dodecahedral) division of surface of circumscribed sphere 565
Icosahedral (and dodecahedral) division of surface of circumscribed sphere, equation giving the conformal representation on a half-plane of each triangle in the stereographic projection of the divided surface 573
Identical substitution 583
Imaginary parts of functions, equations satisfied by real and 11
Imaginary parts of functions, how related to real parts 9
Improperly discontinuous groups 585
Improperly discontinuous groups, example of 615
Index of a composite substitution 583
Index of a composite substitution, not entirely determinate 584
Infinite circle, integral of any function round 36
Infinitesimal curve, integral of any function round 35
Infinitesimal substitution 522 584
Infinities of doubly-periodic functions, and, in sum, are congruent with their sums 228
Infinities of doubly-periodic functions, irreducible, are in number equal to the irreducible zeros 227
Infinities of doubly-periodic functions, of pseudo-periodic functions see "Second kind" "Third
Infinities of potential function on a Riemann's surface 435
Infinities, of a function defined 16
Infinities, of algebraic function 163
infinity 16
Integral function, algebraical 70
Integral function, transcendental 70
Integral of multiform function, also when the curve is otherwise deformed under conditions 183
Integral of multiform function, between two points is unaltered for deformation of path not crossing a branch-point or an infinity 181
Integral of multiform function, deformed path adopted for 190
Integral of multiform function, round a curve containing branch-points and infinities is unaltered when the curve is deformed to loops 182
Integral of multiform function, round a loop 189
Integral of multiform function, round a small curve enclosing a branchpoint 183
Integral of multiform function, with more periods than two, can be made to assume any value by modifying the path of integration between the limits 208
Integral of uniform function round parallelogram of periods, general expression for 222
Integral of uniform function round parallelogram of periods, is zero when function is doubly-periodic 221
Integral with complex variables, defined 18
Integral with complex variables, elementary properties of 19 20
Integral with complex variables, of any function round infinitesimal circle 35
Integral with complex variables, of any function, round any curve enclosing all the branch-points 37
Integral with complex variables, of any function, round infinitely great circle 36
Integral with complex variables, of holomorphic function along a line is holomorphic 24
Integral with complex variables, of holomorphic function round simple curve is zero 23
Integral with complex variables, of meromorphic function round simple curve containing one simple pole 27
Integral with complex variables, of meromorphic function, round curve containing multiple pole 32
Integral with complex variables, of meromorphic function, round simple curve, containing several simple poles 28
Integral with complex variables, of uniform function along any line 184
Integral with complex variables, over area changed into integral round boundary, by Riemann's fundamental lemma 21
Integrals of the first kind, second kind, third kind, Abelian 394 396 400
Integrals, at opposite edges of cross-cut, values of, differ by a constant 375
Integrals, discontinuities of, excluded on a Riemann's surface 378
Integrals, first kind of 394
Integrals, first kind of, are not uniform functions of position 395
Integrals, first kind of, general value of 396
Integrals, first kind of, number of independent, of first kind 395
Integrals, general value of, on a Riemann's surface 379
Integrals, infinities of, of algebraic functions 390
Integrals, of algebraic functions 387
Integrals, of algebraic functions, when branch-points are simple 389
Integrals, second kind of 396 see
Integrals, second kind of, elementary, of second kind 396 see
Integrals, third kind of 400 see
Integrals, third kind of, connected with integrals of second kind 403
Integrals, third kind of, elementary, of third kind 402 see
Integrals, when cross-cuts are canonical 377
Integration, Riemann's fundamental lemma in 21
Interchange of argument and parameter 451
Interchange of argument and parameter in normal elementary function of the third kind 453
Interchange, cyclical, of algebraical function 171
Interchange, cyclical, of branches of a function affected by a branch-point 156
Interchange, sequence of, along branch-lines determined 341
Interchangeable substitutions 586
Invariants of elliptic functions 250
Invariants, derivatives of elliptic functions with regard to the 265
Inversion problem 455
Inversion problem of functions of the first kind with several variables leading to multiply periodic functions 453
Inversion-problem 455
Inversions at circles, even number of, lead to lineo-linear relation between initial and final points 523
Irreducible (point) 199 200
Irreducible circuit 327
Irreducible circuits 327
Irreducible circuits, cannot be drawn on a simply connected surface 329
Irreducible circuits, complete system contains same number of 328
Irreducible circuits, round two branch-points 350
Irreducible, also of automorphic functions 651
Irreducible, likewise the number of level-points 227
Irreducible, points 199 200 591 638
Irreducible, sum of irreducible points is independent of the value of the doubly-periodic function 228
Irreducible, zeros of doubly-periodic function are the same in number as irreducible infinities 226
Isothermal 576
Isothermal curves, all, are conformal representations of fundamental systems by algebraical equations 580
Isothermal curves, families of plane algebraical 576
Isothermal curves, form of equation that gives such families as the conformal representation of parallel straight lines 579
Isothermal curves, isolated may be algebraical by other relations 581
Isothermal curves, three fundamental systems of 579
jacobi 88 189 194 200 238 456 500 501
JORDAN 35 129 188 193 195 583
Kinds of edges in region for Fuchsian group 592 see
Kinds of pseudo-periodic functions, examples of other 295
Kinds of pseudo-periodic functions, three principal 273 274
Kirchhoff 514
Klein vii viii 129 334 335 369 403 408 456 460 462 464 501 511 512 517 549 573 583 619
Kleinian functions 619 see
Kleinian group 610
Kleinian group, conserves no fundamental line 610
Kleinian group, generalised equations of, applied to space 612
Kleinian group, generalised equations of, conserve the plane of the complex variable 612
Kleinian group, generalised equations of, double (or fixed) circle of elliptic substitution of 613
Kleinian group, polygonal division of plane by 613
Kleinian group, polyhedral division of space in connection with generalised equations of 614
Kleinian group, relation between polygonal division of plane and polyhedral division of space associated with 615
Kleinian groups 610
Koenigsberger 230 456
Koepcke 136
Korkine 497 500
Krause 295
Krazer 456
Lacet 153
Lachlan 557 562
Lacunary functions 141
Lagrange 497 499 500
Laguerre 89 91 92
Laguerre's criterion of class of transcendental integral function 91
lame 281 576
Lame's differential equation 281
Lame's differential equation, can be integrated by secondary periodic functions 283
Lame's differential equation, general solution for integer value of n 284
Lame's differential equation, general solution for integer value of n, special cases of n = l and n = 2 285
|
|
|
О проекте