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| Burkhardt H. — Theory of Functions of a Complex Variable |
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| Предметный указатель |
Function, mapping with a regular 182
Function, monogenic 178
Function, n-valued at a point 242
Function, order of the rational 102 103
Function, periodic 84 222 284
Function, rational integral 29 98 231 233 392
Function, rational, of z and  331
Function, rational, the, indeterminate 45 103 174 228
Function, reciprocal of a rational 172 173
Function, regular = a power series 203
Function, regular, expanded in a power series 199
Function, regular, near a singular point 227 247 254
Function, regular, on the Riemann’s surface 335
Function, single-valued analytic 167
Function, single-valued on the Riemann’s surface 333
Function, single-valued on the sphere 389
Function, singly periodic 224 268 273
Function, successive derivatives of a regular 204
Function, sum of all orders of a rational 106 242
Function, symmetric automorphic 92
Function, transcendental 124 167 282 392
Function, transcendental integral 201 392
Function, transcendental integral, at infinity 233
Function, trigonometric 216—221
Function, value of a periodic 233
Function, value of a rational at infinity 105 173
Function, value of a regular in a domain 196
Function, when  is a pole of a rational 105
Function, when is a zero of a rational 106
Function, with removable discontinuity 228
Function, “element” of the 384
Functions of  and  342 368
Functions of a real variable 127
Functions, analytic, of analytic functions 397
Functions, coincidence of 208 384
Functions, hyperbolic 217 219—221
Functions, many-valued 284
Functions, rational, of z and  344
Functions, zeros and poles of a rational 242
Fundamental laws, the 1 5 7 10
Fundamental operations 22 23
Fundamental region for  225
Fundamental region for  91
Fundamental region, definition of 84
Fundamental theorem of algebra 229 232 240
Gauss 9 12 43 308
Gauss’s series 214
Generalization of De Moivre’s theorem 309
Generalization of transcendental functions 216
Geometrical representation of addition 16
Geometrical representation of complex numbers on the sphere 51
Geometrical representation of multiplication 18—20
Geometrical representation of subtraction 18
Gmeiner, A. and O. Stolz 22
Gordan 232
Goursat 190
Goursat — Hedrick 182 232
Green’s Theorem 280
Group, cyclic 90
Group, definition of 57
Group, finite discontinuous 93 113
Group, invariant of a 72
Hankel, H. 23
Harkness and Morley 186 188 280 384
Harmonic, division of a line segment 61
Harmonic, points 72
Heine 132
Heine — Borel theorem 162
Helicoidal surface 290
Helicoidal surface, complex numbers on, a 291
Helicoidal surface, pitch of the 292
Herz 308
Higher derivatives 179 (see also “Successive derivatives”)
Hoffman, Fr. 329
Holomorphic function 252
Hyperbolic functions 217 219—221
Hyperbolic linear substitution 94—98
Hypergeometric series 214
Images, reflected 408
Imaginary axis 13
Imaginary axis, imaginary numbers as number-pairs 3
Imaginary axis, imaginary part of complex numbers 10
Improper domain 141
Improper path 141
Independent of the path, the derivative is 175 176 178
Independent of the path, the integral is 191
Indeterminate value of a function 45 103 174 228
Inferior limit 130
Infinite series of partial fractions 263
Infinite series, convergence of 169 170
Infinite series, sum of 169
Infinite value of a complex variable 45 47
Infinite value of a function 173
Infinite value on the sphere 51
Infinite value, fundamental laws for 46
Infinite, definitely 255 256
Infinitesimal, definition of 182
Infinitesimal, triangle 183
Infinity as an inner point 238
Infinity, behavior of a function at 229
Infinity, circular points at 70
Infinity, corresponds to zero 45 104
Infinity, point at 45 48
Infinity, regarded as 1/0 45
Infinity, two views of 173
Infinity, V-fold of a rational function 102
Infinity, value of a function at 105 173
Inner point of a set 137
Integers of arithmetic 1 2
Integral function, linear 32—38
Integral function, rational 29 98 231 233
Integral function, transcendental, as a product 375
Integral function, transcendental, at infinity 233
Integral function, transcendental, definition 201 392
Integral of a rational function 351
Integral of a regular function 188
Integral, absolute value of 155
Integral, along a closed curve 190 195 196
Integral, analytic continuation of an 385
Integral, classified by discontinuities 352
Integral, curvilinear 156 158 189
Integral, definition of a definite 151
Integral, derivative of an 206
Integral, equal to zero 190 191 193 195
Integral, independent of the path 191
Integral, transformation, linear 32—38
Integral, upper limit of an 189
Integral, upper, lower 152
Integral, with logarithmic discontinuities 353
Integrals, Fresnel 215
integrand 154
Integration of a series term by term 155 200
Integration, arithmetic definition of 151
Integration, between complex limits 188
Integration, Cauchy’s theorem on 190 199
Integration, inverse of differentiation 155 192
Integration, path of 189
Interval, partition of an 151
Interval, segment 127
Invariant of a group 72
Invariant, (fixed) points 33 57 64
Invariant, double ratio is 65
Inverse circular and the logarithmic functions 365
Inverse of the logarithm 305 306
Inverse sine, defn. 360
Inverse tangent, defn. 313
Inverse tangent, poles of the 315
Inverse transformation, the 57 114
Involutoric transformation 38 40 75
Irrational numbers 2 127 128
| Isogonal representation 43
Isolated point of a set 137
Isolated point of a set, isolated singular point 227 247 254 389
Isometric, isothermal system of curves 185
JORDAN 152
Kirchhoff 96
Klein 94 119 413
Lagrange 308
Laplace 180
Laurent’s series 247—251 254
Laurent’s theorem applied to the surface for  339
Law, the associative 5 10
Law, the commutative 5 7
Law, the distributive 7
Laws, the fundamental 1 5 7 10
Least upper bound 128
Lennes, Veblen and 28 131 140 162
Level, lines of 187
Lie, S. 23
Limit of a convergent sequence 168
Limit of a convergent sequence of a complex fund Ion 167 171
Limit of a convergent sequence of a function of two real variables 144
Limit of a convergent sequence, superior, inferior 130
Limit of a convergent sequence, uniform approach to 132 141 155
Limit of a convergent sequence, upper, of an integral 189
Limit points 127 128
Limit points in the plane 137
Line of singularities 391
Line of singularities is a natural boundary 391
Linear fractional transformation 54
Linear integral function, transformation 32—38
Linear substitution, transformation 54
Linear transformation on the sphere 77
Linear transformation, into itself, a 84
Lines as sets of points 138
Lines of equipotential 187
Lines of flow 187
Lines of level 187
Lines, niveau 187
Logarithm of a real positive, negative number 297
Logarithm, a many-valued function 297
Logarithm, a regular function 297
Logarithm, amplitude of the 296
Logarithm, conformal representation by the 304
Logarithm, definition of the 294
Logarithm, expansion of the, in a Taylor’s series 297 298
Logarithm, inverse of the 305 306
Logarithm, natural 294
Logarithm, principal value of the 295 297 304
Logarithm, real 296 304
Logarithm, Riemann’s surface of the 297
Logarithmic, arid inverse circular functions 365
Logarithmic, derivative 240 374
Logarithmic, discontinuities of integrals 353
Logarithmic, residue 354
Lower bound =, lower integral 152
Lower bound =, lower limit 130
Lower bound =, lower, upper bounds 127 128
Lower bound =, upper integral 152
Loxodrome or rhumb line 307
Loxodromic linear substitution 94
Maclaurin’s development in a power series 206 207
Many-fold root 99
Many-valued functions 284
Many-valuedness of the amplitude 21
Map of a rational function 173
Map of a rectangle upon a circular ring 257
Map of a triangle on the half-plane 403 409
Map of surfaces, construction of, reference 308
Map of the z-plane 29
Mapping, general continuous 148 151 182
Mapping, with  83 85—87
Mapping, with 88
Mapping, with a regular function 182
Mapping, with preservation of angles 43
Meray, Ch. 168
Mercator’s projection 307
Meridians on the sphere 51
Method of undetermined coefficients 207
Mittag-Leffler’s partial fractions 373
Mittag-Leffler’s theorem 262
Modulus 14
Modulus of periodicity 352
Modulus, cartographic 183
Moebius 54 72
Monogenic function 178
Monotonic 134
Monotonic, partitively 140
Morley, Harkness and 186 188 280 384
Motions in space as a collineation 80
Multiplication of complex numbers 10
Multiplication of number-pairs 6
Multiplication of series 212
Multiplication, geometrical representation of 18—20
n-valued function at a point 242
Natural boundaries and singular points 389 393
Natural boundaries, a line of singularities 391
Natural logarithm 294
negative numbers 2
Neighborhood of a point 136
Neighborhood, circular, rectangular 137
Neumann’s sphere 47 77
Niveau lines 187
Non-collinear, three points 24
Non-essential singular point or pole 227 228 247 254 389 392
Norm 14 19
nth root of z, the 365
Number of zeros and poles 240—242
Number-pairs as complex numbers 7
Number-pairs as “imaginary” numbers 3
Number-pairs in the form (a + bi) 9
Number-pairs, addition and subtraction of 3—6
Number-pairs, division of 6
Number-pairs, equality of 4
Number-pairs, multiplication of 6 7
Number-pairs, opposite 5
Number-pairs, units for 7
Numbers, a notation for points 127
Numbers, complex, geometrical representation of 1 12
Numbers, complex, multiplication of 10
Numbers, complex, units of 8
Numbers, irrational 127 128
Numbers, negative, fractional, irrational 2
Numbers, opposite complex 15
Numbers, real, combinations of 1
Numbers, real, general arithmetic of 1 2
Numbers, “imaginary”, as number-pairs 3 7
Odd, even function 272
One-to-one correspondence 12 13 29 127
Opposite complex numbers 15
Order at a pole, at a zero 103
Order of a rational function 102 103
Order of a rational function, sum of all 106 242
Orthomorphic transformation 43
Osgood 25 142 186 232 384
Painleve, P. 400
Parabolic linear substitution 94—97
Parallel displacement 30
Parallel translation in the plane 28 30
Parametric representation of a curve 139
Partial derivative 150
Partial differential equations 280
Partial fractions, decomposition into 268
Partial fractions, infinite series of 263
Partial fractions, Mittag-Leffler’s 373
Partition of an integral 151
Partition, Dedekind 128
Partitively monotonic 140
Path 140
Path of integration 189
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