Rectangle, area within, represented on a circle 502
Rectangle, area within, represented on a half-plane 544 545
Rectangle, area within, represented on an ellipse 504
Rectilinear polygon, convex, represented on half-plane 538
Rectilinear polygon, equation which secures the representation of 541
Rectilinear polygon, four sides 544 see "Square" "Quadrilateral"
Rectilinear polygon, function for representation of 540
Rectilinear polygon, limit in the form of a convex curve 548
Rectilinear polygon, three angular points (but not more) may be arbitrarily assigned in the representation 542
Rectilinear polygon, three angular points (but not more) may be arbitrarily assigned in the representation, determination of fourth for quadrilateral 547
Rectilinear polygon, three sides 543 see
Reducible (point) 199 200
Reducible circuit 327
Reducible circuits 327
Reducible points 199 200
Region of continuity 55
Region of continuity, bounded by the singularities 56
Region of continuity, of a power-series consists of one connected part 128
Region of continuity, of a power-series consists of one connected part, may have a natural limit 129
Region of continuity, of a series of uniform functions 132
Region of continuity, of a uniform function 55 126
Region of continuity, of multiform function 150
Regions in division of plane associated with discontinuous group, contiguous 591
Regions in division of plane associated with discontinuous group, corners of 591 see
Regions in division of plane associated with discontinuous group, edges of 591 see
Regions in division of plane associated with discontinuous group, fundamental 591
Regions in division of plane associated with discontinuous group, uniform correspondence between 591
Regular 16 52
Regular in vicinity of ordinary point, function is 52
Regular polygon, area of, conformally represented on a circle 548
Regular singularities 163
Regular singularities of algebraical functions 163
Regular solids, planes of symmetry of, dividing the surface of the circumscribed sphere 564
Representation conforme 11
Representation of complex variable on a plane 2
Representation of complex variable on a plane, and on Neumann's sphere 4
Representation, conformal 11 see
Residue 42
Residue of function, defined 42
Residue of function, when the function is doubly-periodic, the sum of its residues is zero 223
Resolution of Riemann's surface 351
Resolution of Riemann's surface, canonical 355
Resolution of Riemann's surface, how to choose cross-cuts for 352
Resolution of Riemann's surface, when in its canonical form 366
Resolution of surface, canonical 355
Retrosection 315
Revolution, surface of, conformally represented on a plane 496
Riemann v vii 8 10 14 21 133 180 186 312 336 368 408 409 447 456 459 464 500 526 656
Riemann — Roch's theorem on algebraic functions having assigned infinities 459
Riemann's definition of function 8
Riemann's fundamental lemma in integration 21
Riemann's surface 336
Riemann's surface, aggregate of plane sheets 336
Riemann's surface, can be taken in spherical form 346
Riemann's surface, canonical resolution of 355
Riemann's surface, class of 349
Riemann's surface, connectivity of, with one boundary 347
Riemann's surface, existence-theorem for functions on a given 405
Riemann's surface, form of, deformation to canonical form of 365
Riemann's surface, form of, when branch-points are simple 364
Riemann's surface, functions on see "First kind of functions" "Second "Third "Algebraic
Riemann's surface, integrals of algebraic functions on a 375
Riemann's surface, irreducible circuits on 350
Riemann's surface, ramification of 349
Riemann's surface, resolution of, by cross-cuts into a simply connected surface 351
Riemann's surface, resolution of, in canonical form 366
Riemann's surface, sheets of, joined along branch-lines 339
Riemann's surface, uniform functions of position on 369
Riemann's surface, uniform functions of position on, have as many zeros as infinities 372
Riemann's surface, uniform functions of position on, their expression and the equation satisfied by them 371
Riemann's surface, used to represent algebraic functions 338
Riemann's surface, with several boundaries 350
Riemann's theorem on conformal representation of any plane area, simply connected, on area of a circle 526
Riemann, J. 409
Ritter 620
Roch 459 464
Root 16
Roots of a function, defined 16
Rotations, connected with linear substitutions 621
Rotations, dihedral group of 623
Rotations, groups of for regular solids 623
Rotations, tetrahedral group of 625
Rouche 46
Rueckkehrschnitt 315
Salmon 356 367
Schlaefli 538
Schleife 153
Schlesinger 620
Schloemilch 2
Schoenflies 618
Schottky 525 619
Schroeder 137
Schwarz vii 13 57 129 136 297 308 405 490 505 506 526
Schwarz's symmetric continuation 57
Schwarzian derivative 529
Schwarzian derivative, used in conformal representation 529 550
Second kind of Abelian integrals 396
Second kind of pseudo-periodic function 274
Second kind of pseudo-periodic function, difference between sum of irreducible infinities and sum of irreducible zeros 281
Second kind of pseudo-periodic function, expressed in terms of the  -function 281
Second kind of pseudo-periodic function, Hermite's expression for 277 279
Second kind of pseudo-periodic function, limiting form of, when function is periodic of the first kind 278 280
Second kind of pseudo-periodic function, Mittag-Leffler's expression for, in intermediate case 279 280
Second kind of pseudo-periodic function, number of irreducible infinities same as the number of irreducible zeros 280
Second kind of pseudo-periodic function, used to solve Lame's differential equation 281
Second kind, doubly-periodic function of the 274
Second kind, of functions on a Riemann's surface 436
Second kind, of functions on a Riemann's surface, elementary function of, is determined by its infinity and moduli 448
Second kind, of functions on a Riemann's surface, normal elementary function of 448
Second kind, of functions on a Riemann's surface, normal elementary function of, moduli of periodicity of 449
Second kind, of functions on a Riemann's surface, normal elementary function of, used to construct algebraic functions on a Riemann's surface 457
Second kind, of functions on a Riemann's surface, relation between moduli of periodicity of functions of, and those of a function of first kind 442
Second kind, of integrals on a Riemann's surface 396 see
Second kind, of integrals on a Riemann's surface, compared with elliptic functions 272
Second kind, of integrals on a Riemann's surface, elementary integrals of 396
Second kind, of integrals on a Riemann's surface, elementary integrals of, determined by its infinity except as to integral of first kind 398
Second kind, of integrals on a Riemann's surface, elementary integrals of, number of 399
Second kind, of integrals on a Riemann's surface, general value of 398
Second kind, of integrals on a Riemann's surface, two distinct forms of characteristic equation 271
Second order of doubly-periodic functions see also "Doubly-periodic functions"
Second order of doubly-periodic functions, of first class and even, illustrated by Weierstrassian elliptic functions 249
Second order of doubly-periodic functions, of second class and even 267
Second order of doubly-periodic functions, of second class and odd 243
Second order of doubly-periodic functions, of second class and odd, addition-theorem for 247
Second order of doubly-periodic functions, of second class and odd, connected with Jacobian elliptic functions 246
Second order of doubly-periodic functions, properties of Chap. xi
Secondary periodic functions 275 see
Secondary-periodic functions 275
Section 140 186
Section (cross-cut) 314
Sections for integrals of uniform functions, Hermite's 185
Sector on a half-plane 507
Seidel 137
Semicircle represented on a circle 507
Semicircle represented on a half-plane 506
Sequence of interchange along branch-lines determined 341
Series of functions, expansion in 115
Series of functions, may represent different functions in distinct parts of its region of continuity 137
Series of functions, region of continuity of 132
Series of functions, represents the same function throughout any connected part of its region of continuity 132
Series of powers, expansion in 43
Series of powers, function determined by, is the same throughout its region of continuity 128
Series of powers, natural limit of 129
Serret 583
Sheet 336
Sheets of a Riemann's surface 336
Sheets of a Riemann's surface, joined along branch-lines 339
Sheets of a Riemann's surface, relation between variable and 338
Siebeck 579
Simple branch-points 174
| Simple branch-points for algebraic function 174
Simple branch-points for algebraic function, canonical arrangement of 364
Simple branch-points for algebraic function, in connection with loops 357
Simple branch-points for algebraic function, number of 175 356
Simple circuit 327
Simple connection 313
Simple curve 21
Simple curve, defined 21
Simple curve, used as boundary 322
Simple cycle of loops 360
Simple cycles of loops 360
Simple cycles of loops, number of independent 361
Simple element for tertiary periodic functions, of negative class 293
Simple element for tertiary periodic functions, of positive class 291
Simple element of positive class for tertiary-periodic function 291
Simply connected surface 313
Simply connected surface, circuits drawn on, are reducible 329
Simply connected surface, defined 315
Simply connected surface, effect of cross-cut on 316
Simply connected surface, effect of cross-cut on, and of loop-cut on 320
Simply connected surface, winding surface containing one winding-point is a 348
Simply infinite system of zeros, function having 83
Simply periodic functions 198
Simply periodic functions, graphical representation 198 211
Simply periodic functions, limited class of, considered 217
Simply periodic functions, of most elementary form 215
Simply periodic functions, possess algebraical addition-theorem 298
Simply periodic functions, properties of, with an essential singularity at infinity 212
Simply periodic functions, when uniform, can be expressed as series of powers of an exponential 213
Singular line 140
Singular point 16
Singular points 16
Singularities, accidental 16 see
Singularities, bound the region of continuity of the function 57
Singularities, discrimination between 53 66
Singularities, essential 17 see
Singularities, must be possessed by uniform functions 64
Singularities, of algebraical functions, regular 163
Singularity of a coefficient of an algebraic equation is an infinity of a branch of the function 164
Singularity, accidental 16 53
Singularity, essential 17 53
Slit, effect of, on connectivity of surface 321
Species of essential singularity 148
Species of singularity 148
Sphere conformally represented on a plane 497
Sphere conformally represented on a plane, Mercator's projection 498
Sphere conformally represented on a plane, stereographic projection 498
Spherical form of Riemann's surface 346
Spherical form of Riemann's surface, is bounded 347
Spherical form of Riemann's surface, related to plane form 347
Spherical surface with holes, connectivity of 321
Spheroid, oblate, conformally represented on plane 500
Square, area within, represented on a circle 502 545
Square, area within, represented on a half-plane 544 546
Square, area without, represented on a circle 545
Stahl 619 653
Stereographic projection of curvilinear triangle on the surface of a sphere 562
Stereographic projection of sphere on plane as a conformal representation 498
Stickelberger 266 456
Stieltjes 144
Stokes 408
Stolz vi
Straight line changed into a circle by a linear substitution 513
Strip of plane, infinitely long, represented on a cardioid 536
Strip of plane, infinitely long, represented on a circle 508
Strip of plane, infinitely long, represented on half-plane 508
Sub-categories of cycles 607
Subcategories of cycles of corners 607
Substitution, homogeneous 622
Substitution, linear or homographic 512 see
Sum of residues of doubly-periodic function, relative to a fundamental parallelogram, is zero 222
Surface, class of 324
Surface, connected 312
Surface, connectivity of 317
Surface, connectivity of, affected by cross-cuts 319
Surface, connectivity of, and by slit 321
Surface, connectivity of, by loop-cuts 320
Surface, effect of any number of cross-cuts on 316
Surface, has a boundary assigned 314 322 329
Surface, Lhuilier's theorem on division of, into polygons 325
Surface, Riemann's see "Riemann's surface"
Surface, supposed bifacial, not unifacial 325
Symbol for loop 357
Symbol for loop, change of, when loop is deformed 358
Symmetric continuation, Schwarz's 57
Synectic 15
System of branch-lines for a Riemann's surface 341
System of zeros for transcendental function, cannot be triply-infinite arithmetical series 88
System of zeros for transcendental function, doubly-infinite 84
System of zeros for transcendental function, simply-infinite 83
System of zeros for transcendental function, used to define its class 89
Taglio trasversale 314
Tannery vi 137
Tannery's series of functions representing different functions in distinct parts of its region of continuity 137
Teixeira 145
Tertiary periodic functions 275 see
Tertiary-periodic functions 275
Test, analytical, of a branch-point 157
Tetrahedral division of surface of circumscribed sphere 564
Tetrahedral 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 568
Tetrahedral function, automorphic for tetrahedral group 630 see
Tetrahedral group 625
Tetrahedral group, function automorphic for 632
Tetrahedral group, of rotations 625
Tetrahedral group, of substitutions 627
Tetrahedral group, of substitutions, in another form 628
Thetafuchsian function 642
Thetafuchsian functions 642
Thetafuchsian functions, exists either only within the fundamental circle, or over whole plane, according to family of group 643
Thetafuchsian functions, number of irreducible accidental singularities of 644
Thetafuchsian functions, number of irreducible zeros of 648
Thetafuchsian functions, parametric integer for 650
Thetafuchsian functions, pseudo-automorphic for infinite group 644
Thetafuchsian functions, quotient of two with same parametric integer is automorphic 651
Thetafuchsian functions, their essential singularities 642
Third kind of Abelian integral 400
Third kind of pseudo-periodic function 274
Third kind of pseudo-periodic function, canonical form of characteristic equations 275
Third kind of pseudo-periodic function, expression in terms of -function 288
Third kind of pseudo-periodic function, of negative class 291
Third kind of pseudo-periodic function, of negative class, expansion in trigonometrical series 293
Third kind of pseudo-periodic function, of negative class, expressed in terms of Appell's element 293
Third kind of pseudo-periodic function, of positive class 288
Third kind of pseudo-periodic function, of positive class, expressed in terms of simple elements 290
Third kind of pseudo-periodic function, relation between number of irreducible zeros and number of irreducible infinities 286
Third kind of pseudo-periodic function, relation between sum of irreducible zeros and sum of irreducible infinities 287
Third kind, doubly-periodic function of the 274
Third kind, of functions on a Riemann's surface 436
Third kind, of functions on a Riemann's surface, elementary functions of 449
Third kind, of functions on a Riemann's surface, normal elementary function of 450
Third kind, of functions on a Riemann's surface, normal elementary function of, interchange of argument and parameter in 453
Third kind, of functions on a Riemann's surface, normal elementary function of, moduli of periodicity of 451
Third kind, of functions on a Riemann's surface, normal elementary function of, used to construct Appell's factorial functions 466
Third kind, of integrals on a Riemann's surface 400
Third kind, of integrals on a Riemann's surface, elementary integrals of 402 see
Third kind, of integrals on a Riemann's surface, must have two logarithmic infinities at least 402
Third kind, of integrals on a Riemann's surface, sum of logarithmic periods of, is zero 401
Thomas 463
Thomson (Lord Kelvin) 408
Three principal classes of functions on a Riemann's surface 436 see of "Second of "Third of
Todhunter 20
Transcendental function, rational 70
Transcendental integral function 70
Transcendental integral function, cannot have triply-infinite arithmetical series of zeros 88
Transcendental integral function, class of, determined by zeros 89
Transcendental integral function, having doubly-infinite system of zeros 84
Transcendental integral function, having doubly-infinite system of zeros, Weierstrass's product form of 87
Transcendental integral function, having simply-infinite system of zeros 83
Transcendental integral function, it has  for an essential singularity 74
Transcendental integral function, most general form of 80
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