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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.

Язык:

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

-cuts      229 233 266
-cuts      229 233 266
$e_{1}$ , $e_{2}$ , $e_{3}$       150 157—162
$g_{2}$ , $g_{3}$       149 151—157 160
$j(\tau)$       151—157 159 160 163 308
      see Multiplier
      see Genus
$Q_{0}$       30
$Q_{0}$ , a group allied to      35 78 79
, the region, defined      44; see Fundamental region
$R_{0}$       see Fundamental region
$\lambda(\tau)$       157—163 308
$\mathfrak{B}(z)$       83 86 88 144—146 148—150 158—162 307
Abramowicz, C.      323
Accessible boundary points      189— 195 197 199
Alezais, R.      316
Algebraic functions      221 225 229— 249 252 266 278 308
Algebraic functions, of genus greater than zero      233—244 247—249 252 266 278
Algebraic functions, of genus one      239 240 278
Algebraic functions, of genus zero      229—233 245—247 252
Algebraic relations between automorphic functions      94—98 149 160 163 231
Analytic curve      201
Angles at vertices of ordinary cycle      62 89 93 112 130
Anharmonic ratio      see Cross ratio
Anharmonic ratios, group of      34 49 78 159
Area of $R_{0}$       75
Area of $R_{0}$ , theorems      167—169
Automorphic function, definition of      83
Automorphic functions, simple      86—88
Automorphic functions, simple, algebraic relations      94—98
Automorphic functions, simple, and differential equations      98—101
Automorphic functions, simple, behavior at vertices and parabolic points      88—91
Automorphic functions, simple, for finite group      102 131 136
Automorphic functions, simple, for modular group      155
Automorphic functions, simple, for modular sub-group      162
Automorphic functions, simple, for periodic and allied groups      144—146
Automorphic functions, simple, poles and zeros      91—94
Automorphic functions, simple, uniformization by means of      233—247 266—279
Axel, H.      322
Bagnera, G.      31 52
Baker, H. F.       $315^{2}$
Bianchi, L.      313 314 319
Bieberbach, L.      148 167 172 318 321 322 323
Biermann, O.      312 313
Birkhoff, G. D.      322
Blumenthal, O.       $316^{2}$ 317
Boundary, behavior of mapping function on      187—203
Boundary, circular arc      202
Boundary, continuation of mapping function across      201—203
Boundary, elements      195—198
Boundary, of R      47—49
Branch points      165 181 213 226 227 229 249 250 290 309
Brod $\acute{e}$ n, T.      317
Brouwer, L. E. J.      31 93
Burckhardt, H.      314
Burnside, W.      106 313
Cahen, E.      320
Carath $\acute{e}$ odory, C.      165 195
Caspar, M.      318
Cassel, G.       $313^{2}$ 314
Chain of points      176 179
Chazy, J.      319
circle      8—15 (see also Inversion Conformal Fixed Isometric Unit Etc)
Circular arc boundaries      202
Circular arc boundaries, triangle      305
Combination, method of      56—59 279
Combined regions, mapping of      203—205
Concentric slits      265 266
Conformal mapping      164—219
Conformal mapping of multiply connected region of planar character on region bounded by complete circles      279—283
Conformal mapping of neighborhood of parabolic point      90
Conformal mapping of neighborhood of regular singular point      298
Conformal mapping of plane on plane      3
Conformal mapping of plane on plane region      2
Conformal mapping of plane simply connected region on circle      179—187
Conformal mapping of region and subregion      166
Conformal mapping of region bounded by complete circles on second such region      282
Conformal mapping of semicircle on circle      188
Conformal mapping of simply connected finite-sheeted Conformal mapping regions      213—216
Conformal mapping of surface of genus zero on plane      229
Conformal mapping of two-sheeted circular region on circle      181
Conformal mapping, and groups of linear transformations      216—219
Conformal mapping, by $J(\tau)$       157
Conformal mapping, of circle on circle      32
Conformal mapping, of circle on plane finite region      169—175
Conformal mapping, of combined regions      203—205
Conformal mapping, of finite plane on finite plane      3
Conformal mapping, of half plane on circular arc triangle      305
Conformal mapping, of limit regions      205—213
Conformal mapping, of multiply connected region of planar character on slit region      262—265
Congruent configurations      37
Conjugate imaginary      8
Connectivity of algebraic surfaces      225—229
Connectivity of regions      221—229
Constant automorphic function      94
Continuation of mapping function across the boundary      201—203
Convergence in mapping limit region      207— 211
Convergence of iterative process      183 184
Convergence of series and products connected with the group      115 116
Convergence of series for $g_{2}$       151 152
Convergence of subsequence of a set of functions      267—271
Convergence of subsequence of mapping functions      271—273
Convergence of theta series      104—108
Cosine      85 144 220 233 307
Cotangent, series for      154
Courant, R.      148 319
Craig, C. F.      318
Cross (anharmonic) ratio      4 7 34 49 78 159
Cross-cut      222
cube      124—127 136—138
Curves defining boundary element      196—198
Cycles, ordinary      59—62 72 73 89 130
Cycles, parabolic      62—64 72
Cyclic groups      51—56 66 Parabolic Hyperbolic and
Dalaker, H. H.      323
de Brun, F.      313
Deformation of lengths by inversion in sphere      118
Deformation of lengths by linear transformation      24 25
Deformation theorem, for circle      171—175
Deformation theorem, general      175—177 274
Del Pezzo, P.      314
Delta$      150 151 155
Derivative of automorphic function      98 104 131
Derivative of automorphic function, Schwarzian      98 291 302
Determinant of linear transformation      2
Diameter of curve      198
Differential equations      98—101 284—309
Differential equations, connection with groups      284—286
Differential equations, hypergeometric equation      303—308
Differential equations, quotient of two solutions      287—293 296—298
Differential equations, regular singular points      293—298
Differential equations, triangle functions      305—308
Differential equations, with algebraic coefficients      308 309
Differential equations, with rational coefficients      299—308
Differential equations, with three singular points      303—308
Differential equations, with two singular points      303
Dihedral group      129
Discontinuous groups      35
Discrete set of points      277
Dixon, A. C.      315
Domain of existence of function      83
Doubly periodic functions      see Elliptic functions
Doubly periodic groups      35 38 139—146 148—151 238 243
Dyck, W.      312
D’Ovidio, E.      314
Edge separating square elements      251
Element, square      213 250
Elementary groups      66 117—147
Elementary groups, allied to periodic      140—143
Elementary groups, determination of all finite      129—136

Elementary groups, finite      117—138
Elementary groups, of regular solids      123—129
Elementary groups, simply and doubly periodic      139 140
Elementary groups, with one limit point      139—146
Elementary groups, with two limit points      146 147
Elliptic cyclic groups      55 109 129 133 217
Elliptic functions      83 144—146 148—150 158 159 238 240 247 307 309
Elliptic modular functions      148—163
Elliptic modular functions, $J(\tau)$       151—157
Elliptic modular functions, $\lambda(\tau)$       157—159 160—162
Elliptic modular functions, algebraic relations      159 160 163
Elliptic modular functions, definition      148
Elliptic surface      227 257
Elliptic transformations      19 20 23 28 88 121 288 298
Emch, A.       $320^{3}$
Equation, indicial      294 298
Equations, differential      see Differential equations
Euler’s formula      128 228 246
Existence, domain of      83
Exponents at regular singular point      294 298—301 303
Extended groups      136—138
Falckenberg, H.      321
Families, normal      268
Fatou, P.      317
Finite groups      34 37 38 49 55 66 78 102 117—138 245 246 303 308
Finite groups, determination of all finite groups      129—136
Finite groups, fixed points of      132 133
Finite groups, of regular solids      123—129
Finite-sheeted simply connected regions      213—216
Fixed circles of linear transformation      19—22 28—32 66 67
Fixed points at infinity      75—82 105 139—147
Fixed points of linear transformation      6—8 24 67 88 109 130—135 141
Ford, L. R.      320
Forsyth, A. R.      314 321
Four group      129 133
Fricke, R.      148 319 320 323
Fubini, G.       $316^{4}$ 322
Fuchs, L.      31 12 317
Fuchsian functions      87
Fuchsian functions, and differential equations      302 303 306 309
Fuchsian functions, uniformization by means of      240 241—244 247 simple Elliptic
Fuchsian groups      66 67—82
Fuchsian groups, cycles      72 73
Fuchsian groups, fixed points at infinity      75—82
Fuchsian groups, fundamental region      69—71 73—75 77
Fuchsian groups, generating transformations      71 72
Fuchsian groups, limit points      67—69
Fuchsian groups, of first and second kinds      73—75
Fuchsian groups, of the first kind      68 73—75 108 237 308
Fuchsian groups, of the second kind      68 73—75 106—109 116 258
Fuchsian groups, the transformations      67
Fueter, R.      321 323
Fuhr, H.      322
Function      see Algebraic Automorphic Elliptic Elliptic Periodic Polyhedral Rational Triangle Etc. function
Function, groups      64—66 86 109
Fundamental region of a group      37—39 65 75 77 92 94 237 240 243
Fundamental region of a group, boundary of       47 48
Fundamental region of a group, definition of region       44
Fundamental region of a group, genus of      238
Fundamental region of a group, region $R_{0}$       69
Fundamental region of a group, regions congruent to       44—46
Fundamental region of a group, the cycles      59—64
Fundamental region of a group, the sides      47
Fundamental region of a group, the vertices      38 48 49 54 55 58 61 73 74 78 80 82 136 142 143 148 153 161 217 238 248 258 274 278 307
Garnier, R.      318 319
Generating transformations      34 39 50 51 65 71 72
Generating transformations, relations between      62
Genus, definition      227
Genus, of algebraic function      227
Genus, of fundamental region      238 (see also Algebraic functions)
Geometric interpretation of linear transformation      13 14 26—28
Giraud, G.      320
Gordan, P.      315
Got, Th      322
Group of differential equation      286
Groups of linear transformations      33—66
Groups of linear transformations, cycles      59—64
Groups of linear transformations, cyclic groups      51—56
Groups of linear transformations, function groups      64—66
Groups of linear transformations, fundamental region      37—39
Groups of linear transformations, generating transformations      50 51
Groups of linear transformations, isometric circles      39—41
Groups of linear transformations, limit points      41—44
Groups of linear transformations, properly discontinuous      35 36
Groups of linear transformations, region       44—50
Groups of linear transformations, the method of combination      56—59
Groups of linear transformations, transforming a group      36 37 Kleinian Schottky Etc. groups)
Hecke, E.       $319^{2}$
Herglotz, L.      317
Hilb, E.       $318^{3}$ 319 321
Hilbert, D.      316 319
Homographic transformation      1
Humbert, G.       $312^{2}$ 321
Hurwitz, A.      148 185 311 313 317
Hurwitz’ theorem      185
Hutchinson, J. I.       $316^{2}$
Hyperbolic cyclic groups      52—54 147
Hyperbolic function      83
Hyperbolic transformations      18 19 23 28
Hyperelliptic functions      247—249
Hyperelliptic surface      227 229
Hypergeometric equation      303—308
Icosahedral group      129
Improperly discontinuous groups      36
Ince, E. L.      284
Indicial equation      294
Infinitesimal transformations      35
Infinity, point at      2 7 75—77 165 293 299 309
Integral, Poisson’s      177—179
Inverse of linear transformation      2 5 25 33
Inverse of quotient of solutions of differential equation      287—293
Inverse points with respect to circle      10—12 19 20 29
Inverse points with respect to sphere      117 119
Inversion in circle      10—15 26 28 29 45 104 137 202 280 282 305— 308
Inversion in sphere      117—119
Isometric circles      23—30
Isometric circles, and fixed circles      28—30
Isometric circles, and theta series      104 114
Isometric circles, deformation of lengths and areas      25
Isometric circles, geometric interpretation of linear transformation      26 27
Isometric circles, of group      39—42 67
Isometric circles, of product      40 53
Isometric circles, prescribed circles      57
Isometric circles, types of transformations      27 28
Isomorphic groups      36 218 236
Iterative process      179—186
Johansson, S.       $317^{2}$
Jordan curve      189
Jordan curves, regions bounded by      198—202
K $\ddot{o}$ nig, R.      319
Kapteyn, W.      314
Kempinski, S.      315
Klein, F.      124 148 311 314 315 317 318 319
Kleinian function      87
Kleinian groups      66
Kluyver, J. C.      315
Koebe, P.      134 140 186 188 191 205 233 262 274 279 288 323
Koebe’s lemma      188
Legendre’s differential equation      304
Levi, E. E.      317
Lewent, L.      323
Lewickyj, W.      321
Lichtenstein, L.      321
Liljestrom, A.      319 320
Limit points of group      41—44 46 47 315 58 62—64 67—69 70 85 108 277
Limit points of regions      205—213 216 218 268—271
Limit points of regions, application in uniformization      230 234 235 237 242 245 248 252 253 257 278 280 302
Lindemann, F.      316
Linear transformation      1—32
Linear transformation, and circle      8—15 32 282

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