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Ford L. — Automorphic Functions

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Название: Automorphic Functions

Автор: Ford L.

Аннотация:

In 1915, the author published in the series of Edinburgh Mathematical Tracts a brief introduction to "Automorphic Functions." This booklet has been long out of print. Except for this little volume, no book on the subject has ever appeared in English. This is regrettable, in view of the importance of the subject to those whose interests lie in the field of Functions of a Complex Variable and of its numerous contacts with other domains of mathematical thought.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1-st edition

Год издания: 1929

Количество страниц: 341

Добавлена в каталог: 04.03.2013

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Предметный указатель

Linear transformation, carries circle into circle      9
Linear transformation, carries inverse points into inverse points      11
Linear transformation, carrying three points into three points      7
Linear transformation, carrying unit circle into self      31
Linear transformation, corresponding to rotation of the sphere      122
Linear transformation, elliptic      19 23 28
Linear transformation, equivalent to even number of inversions      14
Linear transformation, fixed circles of      19—22 28—32
Linear transformation, fixed points of      6—8
Linear transformation, geometric interpretations of      13 14 26 27
Linear transformation, hyperbolic      18 23 28
Linear transformation, inverse of      2 5
Linear transformation, isometric circle of      23—30
Linear transformation, loxodromic      20 23 28
Linear transformation, multiplier of      15—18
Linear transformation, parabolic      21—23 28
Linear transformation, sufficient conditions for      2 3 32 273—277 282
Linearly independent solutions      284
Loop-cut      222 225 256 266
Loxodromic cyclic groups      52—54 147
Loxodromic transformations      20 21 23 28
Mapping      see Conformal mapping
Maurer, L.      322
Method of combination      56—59 279
Modular functions      see Elliptic modular functions
Modular group      35 79—81 151—157
Modular group, subgroups of      81 82 159
Montel, P.      268
Morris, R.      317
Multiplier of linear transformation      15—18 24 37 140 141
Myrberg, P. J.      320
Nevanlinna, F.      323
Normal families of functions      268
Octahedral group      127 129
One limit point, groups with      139—146
Order of fixed point      130
Ordinary cycles      59—62 72 73 89 130
Ordinary point of differential equation      293
Ordinary point of group      42
Oscillation of function on curve      275
Osgood, W. F.      154 186 191 241 266 319
Painlev $\acute{e}$ , P.      313
Parabolic cycles      62—64 72
Parabolic cyclic groups      55
Parabolic point      63 64
Parabolic point, behavior at $J (\tau)$       153—155
Parabolic point, behavior at $\lambda(\tau)$       160—162
Parabolic point, behavior at Schwarzian derivative      99
Parabolic point, behavior at simple automorphic function      87
Parabolic point, behavior at theta function      110—114
Parabolic transformations      21—23 28 130 139 288 298
Parallel slits      266
Parametric equations      220 254
Perfect set      43 68
Period of linear transformation      20
Period of linear transformation, parallelogram      38 139—146 148—151 240
Period of linear transformation, strip      37 38 139—144
Periodic functions      see Simply periodic functions Elliptic
Periodic groups      34 35 37 38 139—146 148—151 238 306
Periodic groups, allied      140—146 306
Picard, E.      35 311 313 314
Picard, group of      35 36
Pick, G.      314 317
Planar character, regions of      256
Plemelj. J.      320
Poincar $\acute{e}$ theta series      see Theta series
Poincar $\acute{e}$ , H.      103 115 286 31 15 316 320
Point at infinity      see Infinity
Poisson’s integral      177—179
Poles of $J(\tau)$       155
Poles of $\lambda(\tau)$       162
Poles of simple automorphic function      91—94
Poles of theta functions      110 112—115
Polyhedral functions      136 247 303 306 308 309
Price, H. F.      321
Prime end      195
Primitive periods      150 151
Principal circle      67
Products connected with group      115 116 258—261
Projection, stereographic      119 120 307
Properly discontinuous groups      35
Prym, F.      319
Quotient of solutions of differential equation      284 287—293
Quotient of solutions of differential equation, at regular singular point      296—298
Radial slits      266
Rational functions      3 96 97 100 102 131 144—146 229—233 240 247 299 305
Rausenberger, O.      311
Reality of $J(\tau)$       156 157
Reality of $\lambda(\tau)$       162
Reality of triangle function      306
Reflection in line      11 13 26 28 194 306
Reflection in plane      119 136—138 308
Regions bounded by complete circles      279—283
Regions bounded by complete circles, and $R_{0}$       see Fundamental region
Regions bounded by complete circles, congruent to       44—46
Regions bounded by complete circles, connectivity of      221—229
Regions bounded by complete circles, of planar character      256
Regions bounded by complete circles, simply connected      see Simply connected regions
Regions bounded by Jordan curves      198—202
Regular singular points of differential equations      293—298
Regular solids, groups of      123 124 127—129 133 138 308
Relations, algebraic      see Algebraic relations
Removable singularities      274—276
Rey — Pastor, J.      320
Richmond, H. W.      318
Riemann surface      96 97 165 221 225—229 249 266 287

Riemann — Schwarz triangle functions      see Triangle functions
Riemann, B.      186 305
Ritter, E.      313 315
Rost, G.      319
Rotation      13 28 31
Rotations, and translations, groups of      140— 146
Rotations, groups of      34 37 38
Rotations, of sphere      120—123
Rothe, H.      318
Sansonc, G.      322
Schlesinger, L.       $313^{3}$ 323
Schoenflies, A.      31 42
Schottky groups      59
Schottky type, groups of      59 277
Schottky, F.      59 312 318
Schwarz, H. A.      98 165 304 305 311 313
Schwarzian derivative      98 291 302
Schwarz’s lemma      165 182 206
Series connected with group      115 116
Series, theta      see Theta series
Severing a surface      228 229
Side, pole or zero on      91
Sides of R      47 48 51 65 71 72 238 239
Sigma cross-cut      222
Simple automorphic functions      see Automorphic functions
Simply connected regions      70 179 222
Simply connected regions, finite-sheeted      213—216 (see also Algebraic functions of genus zero)
Simply periodic functions      34 144 247
Simply periodic groups      34 37 38 139—146
Sine      34 83 88 220 233
Singular point of differential equation      293
Slit regions      262—265
Smirnoff, V.       $322^{2}$
Solids, regular      see Regular solids
Sphere, inversion in      117—119
Sphere, rotations of      120—123
Spiesz, O.      323
Square element      213 250
St $\ddot{a}$ ckel, P.      314
Stahl, H.      312 317
Stereographic projection      119 120 307
Stouff, X.      312 313 314
Stretchings      13 24
Stretchings, group of      38
Subgroups, $\Gamma_{u}$       76
Subgroups, of modular group      81 82 159
Subregion      166
Symbolic notation      4—6
Taylor, E. H.      191
Tetrahedral group      129
Theta functions, properties of      108— 115
Theta series      102—116
Theta series, behavior at parabolic point      111
Theta series, convergence      104—108
Theta series, convergence, for Fuchsian group of second kind      106—108
Theta-fuchsian series and functions      104
Theta-kleinian series and functions      104
Tinirnov, T.      321
Transcendental functions      249—255
Transformations, linear      see Linear transformations
Transformations, linear, of differential equations      290 300 301
Transforming a group      36 107
Transforms of       44—46
Transforms of $R_{0}$       70
Translations      13 139
Triangle functions      138 305—308
Two limit points, groups with      146 147
Uhler, A.      322 323
Uniformization      220—283
Uniformization by automorphic functions      233—249 266—279
Uniformization by automorphic functions belonging to Schottky groups      277 279
Uniformization by elementary and Fuchsian functions      229—255
Uniformization by elliptic functions      240 247
Uniformization by Fuchsian functions of first kind      240 244 247—249
Uniformization by rational functions      229—233 240 247
Uniformization by simply periodic functions      247
Uniformization of algebraic functions of genus greater than zero      233—244 247—249 266—279
Uniformization of algebraic functions of genus one      237—244 278
Uniformization of algebraic functions of genus zero      229—233 245—247
Uniformization of transcendental functions      249—255
Uniformization, the concept      220
Unit circle      30—32
Van Vleck, E. B.      321
Vertex, behavior at $J(\tau)$       155 156
Vertex, behavior at automorphic function      88 89
Vertex, behavior at polyhedral function      131
Vertex, behavior at theta function      109 110
Vertex, pole or zero at      91
Vertices of       48 59—62 72 73
Vertices of cycle, angles at      62 89 93 112 130
Viterbi, A.      315
Vivanti, G.      148
von Mangoldt, H.      312
Watson, G. N.      284 294
Wcllstein, J.      316
Weber, H.      312
Weierstrassian function      see $\mathfrak{B}(z)$
Whittaker, E. T.      116 247 284 294 315 316
Whittaker’s groups      247—249
Whittaker’s product      116
Wirtinger, W.      316
Young, J. W.      316
Zero, genus      see Algebraic functions
Zeros of $J(\tau)$       156
Zeros of $\lambda(\tau)$       162
Zeros of simple automorphic functions      91—94
Zeros of theta functions      110 112—115

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