laurent 43 47 49 50 51 213
Laurent's theorem on the expansion of a function which converges within a ring 47
Leading term of a branch in vicinity of an ordinary point of the coefficients of the equation determined 167
legendre 194
Lerch 136
Lhuilier 325
Lhuilier's theorem on division of connected surface into polygons 325
Ligne de passage 339
Limit, natural 129
Limit, natural, of a power-series 129
Limitrophe 591
Lindemann 356 464
Linear differential equations of the second order, connected with automorphic functions 656
Linear substitution 512
Linear substitution, algebraical symbols to represent 583
Linear substitution, can be chosen so as to transform any circle into any other circle 514
Linear substitution, can be obtained by any number of pairs of inversions at circles 523
Linear substitution, changes a plane crescent into another of the same angle 514
Linear substitution, changes straight lines and circles into circles in general 513
Linear substitution, condition of periodicity 515
Linear substitution, elliptic 517
Linear substitution, elliptic, and real 519
Linear substitution, elliptic, is either periodic or infinitesimal 521
Linear substitution, equivalent to two translations, a reflexion and an inversion 512
Linear substitution, fixed points of 514
Linear substitution, group of 582 see
Linear substitution, hyperbolic 517
Linear substitution, hyperbolic, and real 519
Linear substitution, identical 583
Linear substitution, in homogeneous form 622
Linear substitution, index of composite 583
Linear substitution, infinitesimal 584
Linear substitution, interchangeable 586
Linear substitution, loxodromic 517 521
Linear substitution, multiplier of 515
Linear substitution, normal form of 582
Linear substitution, parabolic 517
Linear substitution, parabolic, and real 518
Liouville 161 210 218 230
Lippich 316 335
Logarithmic infinities, integral of third kind on a Riemann's surface must possess at least two 402
Loop 153
Loop-cut 315
Loop-cuts, changed into a cross-cut 320
Loop-cuts, defined 315
Loop-cuts, effect of, on connectivity 320
Loop-cuts, effect of, on number of boundaries 324
Loops, canonical form of complete system of simple 361
Loops, defined 153
Loops, effect of a loop, is unique 155
Loops, fundamental set of 360
Loops, simple cycle of 360
Loops, symbol to represent effect of 357
Loops, symbol to represent effect of, change of, when loop is deformed 358
Love 543
Loxodromic substitution 517
Loxodromic substitutions 517 521
Loxodromic substitutions, do not occur in connection with cycles of corners 613
Loxodromic substitutions, neither periodic nor infinitesimal 522
Lueroth 359 361
Magnification in conformal representation 11 492
Magnification in conformal representation, in star-maps 499
Maps 499
Mathews 617
Maximum and minimum values of potential function for a region lie on its boundary 422
Maxwell 408
Mehrdeutig 15
Mehrfach zusammenhaengend 314
Mercator's projection of sphere 498
Meromorphic 16
Meromorphic function, all singularities of uniform algebraical, are accidental 73
Meromorphic function, cannot, without change, be deformed across pole 34
Meromorphic function, defined 16
Meromorphic function, integral round a simple curve, containing one simple pole 27
Meromorphic function, integral round a simple curve, round a curve containing multiple pole 32
Meromorphic function, integral round a simple curve, round a curve containing several simple poles 28
Meromorphic function, integral unchanged by deformation of simple curve in part of plane where function is uniform 27
Meromorphic function, is form of uniform function with a limited number of accidental singularities 71
Meyer 576
Michell 541
Mittag-Leffler vii 57 112 147 275 278 279
Mittag-Leffler's theorems on functions having an unlimited number of singularities, distributed over a finite circle 117
Mittag-Leffler's theorems on functions having an unlimited number of singularities, distributed over the whole plane 112
Modular group 587
Modular group of substitutions 587
Modular group of substitutions, division of plane into polygons, associated with 588
Modular group of substitutions, for modulus of elliptic integral 635
Modular group of substitutions, for the absolute invariant of an elliptic function 637
Modular group of substitutions, is improperly discontinuous for real variables 585
Modular group of substitutions, relation between the fundamental substitutions of 590
Modular-function 633
Modular-function, as automorphic functions 657
Modular-function, connected with elliptic quarter-periods 633 see
Modular-function, defined 633
Moduli of periodicity, examples of 379
Moduli of periodicity, for cross-cuts 377
Moduli of periodicity, introduced in proof of existence-theorem 430
Moduli of periodicity, number of linearly independent on a surface 379
Moduli of periodicity, of function of first kind on a Riemann's surface 439
Moduli of periodicity, of normal elementary function of second kind are algebraic functions of its infinity 449
Moduli of periodicity, of normal elementary function of third kind are expressed as normal functions of first kind of its two infinities 451
Moduli of periodicity, properties of, for normal function of first kind 446
Moduli of periodicity, relation between, of a function of first kind and a function of second kind 442
Moduli of periodicity, values of, for canonical cross-cuts 377
Modulus 3
Modulus for cross-cut, constant 377
Modulus of periodicity (cross-cut) 377
Modulus of variable 3
Moebius 325 512 573
Monadelphic 313
Monodromic 15
Monogenic 14
Monogenic analytic function 56
Monogenic functionality not coextensive with arithmetical expression 139
Monogenic, analytic function 56
Monogenic, defined 14
Monogenic, function has any number of derivatives 14
Monotropic 15
Multiform 15
Multiform function, defined 15
Multiform function, defined by algebraic equation 161 see
Multiform function, elements of, in continuation 56
Multiform function, expression of, in vicinity of a branchpoint 158
Multiform function, is uniform on Riemann's surface 337 343
Multiple circuit 327
Multiple circuits 327
Multiple connection 314
Multiple periodicity 208
Multiple periodicity of uniform function of several variables 209
Multiplicateurs, fonctions a 464
Multiplication-theorem 297
Multiplicity of pole 65
Multiplicity of zero 61
Multiplier of linear substitution 515
Multiplier of substitution 515
Multipliers of factorial functions at cross-cuts 465
Multipliers of factorial functions at cross-cuts, forms of, when cross-cuts are canonical 466
Multiply connected surface 314
Multiply connected surface, boundaries of, affected by cross-cuts 323
Multiply connected surface, connectivity modified by cross-cuts 319
Multiply connected surface, connectivity modified by loop-cuts 320
Multiply connected surface, connectivity modified by slit 321
Multiply connected surface, defined 315
Multiply connected surface, divided into polygons, Lhuilier's theorem on 325
Multiply connected surface, number of circuits in complete system of circuits on 330
Multiply connected surface, relation between boundaries of, and connectivity 324
Multiply-periodic uniform functions of n variables, cannot have more than 2n periods 209
Multiply-periodic uniform functions of n variables, obtained by inversion of functions of first kind 453
Natuerliche Grenze 129
Natural limit 129
| Natural limit, for pseudo-automorphic function with certain families of groups 643
Natural limit, of a power-series 129
Natural limit, of part of plane 558
Negative edge of cross-cut 374 375 438
Nekrassoff 111
Netto 583
Neumann vii 5 6 37 153 161 316 347 354 408 409 456 464 468
Neumann's sphere used to represent the variable 4
Neumann's sphere used to represent the variable, used for multiform functions 153
Niveaupunkte, points where a function acquires any, the same, value 227
Noether 356 464
Non-essential singularity 53
Normal (connected) surface 334
Normal elementary function of second kind 448 see
Normal elementary function of third kind 450 see
Normal form of linear substitution 582
Normal function of first kind, second kind, third kind 446 448 450
Normal functions of first kind 446 see
Normal polygon for division of plane 594
Normal polygon for division of plane, can be taken convex 595 see
Normal polygon for substitutions 594
Normal surface, Klein's, as a surface of reference of given connectivity and number of boundaries 334 365
Number of zeros of periodic functions see "Doubly-periodic functions second "Doubly-periodic third
Number of zeros of pseudo-automorphic functions see "Pseudo-automorphic functions"
Number of zeros of uniform function in any area 61 63 68 72
Octahedral (and cubic) division of surface of circumscribed sphere 565
Octahedral (and cubic) 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 570
Open cycles of corners in normal polygon for division of plane by Fuchsian group 596 see
Open cycles of corners in normal polygon for division of plane by Fuchsian group, do not occur in division of plane by Kleinian group 613
Order of a doubly-periodic function 220
Order of doubly-periodic function 220
Order of infinity of a multiform function determined 164
Order, of connection, adelphic 317
Ordinary point 52
Ordinary point of a function 52
Ordinary point of a function, domain of 52
Orthomorphosis 11
Painleve 140
Parabola, area within, represented on a circle 506
Parabola, area without, represented on a circle 505
Parabolic substitution 517
Parabolic substitutions 517 518
Parabolic substitutions, neither periodic nor infinitesimal 522
Parabolic substitutions, occur in connection with cycles of corners 607 613
Parallelogram for double periodicity, edges and corners in relation to zeros and to accidental singularities of functions 218
Parallelogram for double periodicity, fundamental 200 206
Parallelogram for double periodicity, is fundamental for a function of the second order within it 224
Parallelogram, fundamental or primitive 200 206
Parametric integer of thetafuchsian functions 650
Path of integration 18
Path of integration, can be deformed in region of holomorphic function without affecting the value of the integral 26
Path of integration, on a Riemann's surface, and not over a branch-point 350
Path of integration, on a Riemann's surface, can be deformed except over a discontinuity 373
Period 198
Periodic linear substitutions 515
Periodic linear substitutions, are elliptic 519
Periodicity for cross-cut, modulus of 377
Periodicity of uniform functions, of one variable 198
Periodicity of uniform functions, of several variables 209
Periodicity, modulus of 377 see
Periods of a function of one variable 198
Periods of a function of one variable, cannot exceed two in number independent of one another if function be uniform 205
Periods of a function of one variable, cannot have a real ratio when the function is uniform 200
Phragmen 297 407 538
Picard 54 141 282 296 464 617
Pincherle 146
Plane used to represent variation of complex variable 2
Pochhammer 188
Poincare viii 92 141 144 295 297 512 518—520 523 583 619
poisson 408
Polar discontinuity 16
Pole 16 53
Poles of a function defined 16 53
Polyadelphic 314
Polyhedral division of space in connection with generalised equations of group of Kleinian substitutions 614
Polyhedral functions 575
Polyhedral functions, connected with conformal representation 566
Polyhedral functions, for double-pyramid 567 632
Polyhedral functions, for icosahedron and dodecahedron 573
Polyhedral functions, for octahedron and cube 570
Polyhedral functions, for tetrahedron 568 630
Polytropic 15
Position on Riemann's surface, has as many zeros as infinities 372
Position on Riemann's surface, most general uniform function of 369
Position on Riemann's surface, their algebraical expression 371
Positive edge of cross-cut 374 375 438
Potential function 407
Potential function, real, boundary conditions assigned to 410
Potential function, real, classes of complex functions derived from, with the respective conditions 436
Potential function, real, classes of, determined 436
Potential function, real, conditions satisfied by, when derived from a function of position on a Riemann's surface 407
Potential function, real, defined 407
Potential function, real, general conditions assigned to 410
Potential function, real, Green's integral-theorems connected with 411
Potential function, real, introduction of assigned infinities 435
Potential function, real, introduction of cross-cut moduli for, on a doubly connected surface 430
Potential function, real, introduction of cross-cut moduli for, on a triply connected surface 433
Potential function, real, introduction of cross-cut moduli for, on any multiply connected surface 434
Potential function, real, is determined by general conditions and boundary values, for any simply connected surface 429
Potential function, real, is determined by general conditions and boundary values, for area conformally representable on area of a circle 423
Potential function, real, is determined by general conditions and boundary values, for area containing a winding-point 428
Potential function, real, is determined by general conditions and boundary values, for combination of areas when it can be obtained for each separately 425
Potential function, real, is uniquely determined for a circle by general conditions and continuous finite boundary values 414
Potential function, real, is uniquely determined for a circle by general conditions and continuous finite boundary values, integral expression obtained for, satisfies the conditions 417
Potential function, real, is uniquely determined for a circle by general conditions and continuous finite boundary values, properties of, for a circle 421
Potential function, real, is uniquely determined for a circle by general conditions and continuous finite boundary values, the boundary values for circle may have finite discontinuities at a limited number of isolated points 418
Potential function, real, maximum and minimum values of, in a region, lie on the boundary 422
Potential function, real, number of linearly independent, everywhere finite 434 445
Power-series, as elements of an analytical function 56 128
Power-series, region of continuity of, consists of one connected part 128
Power-series, region of continuity of, may have a natural limit 129
Primary factor 82
Primfunction 82
Primitive parallelogram 206
Primitive parallelogram of periods 206
Pringsheim 137 201
Product-form of transcendental integral function with infinite number of zeros over whole plane 80
Properly discontinuous groups 585
Prym 353 354 369 409 456 464
Pseudo-automorphic functions 643 see
Pseudo-periodic functions Chap. xii
Pseudo-periodic functions, of the first kind 273
Pseudo-periodic functions, of the second kind 274
Pseudo-periodic functions, of the second kind, properties of see "Second kind"
Pseudo-periodic functions, of the third kind 274
Pseudo-periodic functions, of the third kind, properties of see "Third kind"
Pseudo-periodic functions, on a Riemann's surface see "Factorial functions"
Pseudo-periodicity 256 259 273 274
Pseudo-periodicity of the  -function 260
Pseudo-periodicity of the  -function 255
Puiseux 168
Quadrilateral, area of, represented on half-plane 546
Quadrilateral, determination of fourth angular point, three being arbitrarily assigned 547
Querschnitt 314
Raffy 482
Ramification (of Riemann's surface) 349
Ramification of a Riemann's surface 349
Ramification, point de 15
Ratio of periods of uniform periodic function cannot be real 200
Rational algebraical function 70
Rational points 141
Rational points in an area 141
Rational transcendental function 70
Rausenberger 295 586
Real and imaginary parts of functions, each can be deduced from the other 12
Real and imaginary parts of functions, equations satisfied by 11
Real and imaginary parts of functions, how related 9
Real potential function 407 see
Real substitutions 517 591 see
Reconcileable circuits 327
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