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| Авторизация |
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| Burkhardt H. — Theory of Functions of a Complex Variable |
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| Предметный указатель |
 , principal value of 309
 216
is a symmetric automorphic function 225
, fundamental region of 225
 360
 , definition 313
, mapping with 314
, poles of the inverse of 315
, principal value of 313
Abel’s theorem on multiplication of series 212
Abscissa 12
Absolute convergence of a power series 202
Absolute convergence of a series 170
Absolute value and amplitude of a complex number 218
Absolute value and amplitude of a product 19
Absolute value and amplitude of a quotient 21
Absolute value and amplitude of an integral 155
Absolute value of a sum 17
Absolute value, definition of 14
Absolutely convergent series, sum of an 170
Abteilungsweise monoton 140
Accumulation point 128
Addition and subtraction of number-pairs 3
Addition theorem for exponential functions 218
Addition theorem for trigonometric functions 218
Addition, geometrical representation of 16
Aggregate of points 127
Algebra, an, of number-pairs 3
Algebra, double 3
Algebra, geometrical representation of double algebra 3
Algebraic addition theorem 275
Amplitude and absolute value of a complex number 218
Amplitude of a product 19
Amplitude of a quotient 21
Amplitude of the double ratio 70
Amplitude of the logarithm 296
Amplitude, change in value of 284
Amplitude, definition of 15
Amplitude, many-valuedness of the 21
Amplitude, principal value of the 284
Amplitude, the, a continuous and a single-valued function of position 286 290 291 293
Analysis situs 328
Analytic continuation 382 384
Analytic continuation of an integral 385
Analytic continuation, across a line 402
Analytic function, an, definition of 384 386
Analytic functions of analytic functions 397
Analytic functions of analytic functions, single-valued 167
Angles in stereographic projection 48
Angles, opposite in sense 42 43
Angles, preserved 41 43
Approximation, degree of 153 169
Argand 12
Arithmetic, general, of real numbers 1 2
Assemblage of points 127
Associative law, the 5 10 16
Automorphic function, definition of 84
Automorphic function, example of 106 113
Automorphic function, symmetric 92
Axis of pure imaginary numbers 13
Axis of real numbers 13
Axis of reals, reflection on 39
Behavior of a function at infinity 229
Bilinear transformation, substitution 54
Binomial theorem, the 221
Bocher 232 282
Borel 235
Borel, theorem, the Heine — 162
Boundary, a natural, a line of singularities is 391
Boundary, natural, and singular points 389 393
Boundary, natural, not a cut 391
Boundary, point of a set of points 137
Bounds, least upper 128
Bounds, upper, lower 127 128
Branch-cut, cut, bridge 321
Branch-point of many-valued functions 393
Branch-point, definition of 292 321 395
Branch-point, examples of 324—326
Branch-point, when a pole, zero, essential singularity 335
Branch-point, when a singular point 395
Bridge, branch-cut, cut 321
Cahchy’s theorems on integration 190 195 199 247
Cahchy’s theorems on residues 236
Cahchy’s theorems, applied to Riemann’s surfaces 333
Cantor 131
Cantor, Dedekind axiom 3
Cartographic modulus 183
Cartography 47
Cassinian oval 373
Cauchy 168 178 179 199 203 213 267 271 401
Cauchy — Riemann differential equations 176 179 181
Cayley 43 421
Change of amplitude 284
Chapman 23
Circle of a power series 202
Circle of convergence 199 201
Circle transformation 54 56
Circle transformation, unit circle 15
Circuits about the origin 288
Circular points at infinity 70
Class of points 127
Classification of integrals by discontinuities 352
Closed, segment 138
Closed, set of points 138
Cluster point 128
Coefficients of a power series 206—210
Coefficients, method of undetermined 207
Coincidence of two functions 208 384
Coincidence of two power series 207
Collection of points 127
Collinear points, three 26
Collineations of space 77 79 80 410
Commutative law 5 7 16
Complex function 170
Complex function, continuity of a 171
Complex numbers on the helicoid 2gi
Complex numbers on the sphere 51
Complex numbers, conjugate 15 19
Complex numbers, geometrical representation of 1 12
Complex numbers, multiplication of 10
Complex numbers, real, pure imaginary parts of 10
Complex numbers, units of 8
Complex quantity 9
Condensation point 128
Confocal parabolas 85
Conformal representation 28 41 43 182
Conformal representation by  314
Conformal representation by singly periodic functions 224
Conformal representation by the logarithm 304
Conformal representation of a triangle on the half-plane 403 409
Conformal representation of the half-plane on a circle 421
Conformal representation, breaks down 184
Conformal representation, where dw/dz = 0 395
Congruent figures 30 31
Conjugate complex numbers 15 19
Connected, domain, simply 142
Connected, segment 138
Connected, set of points 138
Connectivity of the surface for  328
Constant, definition of a 28
Constant, when a function is 230
Construction of the Riemann’s surface, general 385
Continuation, analytic 382 384
Continuity 127
Continuity in a domain 145
Continuity of a rational function 133 170 172
Continuity of complex functions 171
Continuity of functions of two variables 143 144
Continuity on an interval 131
Continuity, at a point 131
| Continuity, at a pole 172
Continuity, at infinity 173
Continuity, uniform 132
Continuous function, a = a power series 203
Continuous function, continuous mapping 148 151
Contour, rectangular 158
Convergence of a power series 201
Convergence of a product of two series 212
Convergence of an infinite series 169 170
Convergence, absolute, of a power series 202
Convergence, absolute, of a series 170
Convergence, circle of 199 201
Convergence, permanent, of a power series 201
Convergence, unconditional 170
Convergence, uniform 155 156 200 202 203
Convergent sequence, limit of a 168
Coordinates of a point 12
Coordinates of a point, polar 13
Correspondence, one-to-one 12 13 29 127
Cosine 216—221
Cosine, a symmetric automorphic function 226
Cosine, period of the 222
Cube roots of unity 25 72
Curve 136
Curve, = a set of points 136 138 139
Curve, general definition of a 139 140
Curve, parametric representation of a 139
Curve, simple 140
Curvilinear integral 156 158 i8g
Cut 321
Cut, Dedekind, partition 128
Cut, not a natural boundary 391
Darboux 152 402
De Moivre’s theorem, generalization of 309
Decomposition into partial fractions 268
Dedekind cut, partition 128
Dedekind — Cantor axiom 3
Definitely infinite 255 256
Deformation of the surface for 329
Degenerate transformation 55
Degree of a function 99 102
Degree of a rational function 102
Degree of approximation 153 169
Dense in itself, a set of points 138
Derivatives of a determinant 166
Derivatives of a power series 149 203
Derivatives of a rational function 174
Derivatives of an integral 206
Derivatives, definition 148
Derivatives, higher, successive 179
Derivatives, independent of approach 175 176 178
Derivatives, logarithmic 240 374
Derivatives, partial 150
Derivatives, progressive, regressive 163 165
Derivatives, successive, of a power series 204
Derivatives, successive, of a regular function 204
Derivatives, term by term 149 156
Derived, function 149
Derived, sets of points 131
Descartes 12
Determinant of a transformation 55 65
Determinant, derivative of a 166
Development in a circular ring 249 253
Development in a power series 199 206
Development in a power series, unique 207
Development in a strip 257
Development, near a pole 229
Diametral, two complex numbers 53 80
Differential equations, Cauchy — Riemann 176 179 181
Differential equations, partial 280
Differentiation, the inverse of integration 155 192
Direct similarity transformation 35
Direct similarity transformation, direct circle transformation 56
Discontinuity, removable 103 228 252
Discontinuous, finite group 113
Displacement (a translation) 30 33
Distributive law 7
Division by zero 45
Division of a line segment, harmonic 61
Division of complex numbers 20 21
Division of number-pairs 6
Domain 140
Domain, improper 141
Domain, mapped continuously 148 151
Domain, simply connected 142
Double algebra and its geometrical representation 3
Double ratio of four points 69 70
Double ratio, amplitude of 70
Double ratio, invariant 65 72
Electrical potential 187
Elliptic linear substitution 94 95
Equation of nth degree has n roots 242
Equation, a complete 299
Equation, has no more than n roots 99
Equator on sphere 51
Equatorial plane, reflection on 51
Equianharmonic points 72
Equipotential, lines of 187
Essential singularity, at a branch-point 335
Essential singularity, definition of 389 390 392
Euler 81 217
Euler’s relations 217 316
Even, odd functions 272
expansion see “Development”
Expansion in a circular ring 249 253
Expansion in a power series 199
Expansion in a strip 257
Exponential function 216—221 279
Exponential function, addition theorem for 218
Exponential function, periodicity of 222
Finite discontinuous group 113
Fixed (invariant) points of a transformation 33
Fixed (invariant) points of the linear fractional transformation 57 64
Flaechenstuecke (region) 136
Flow of a fluid 185
Flow, lines of 187
Forsyth 142 261
Fourier’s series 257
fractional numbers 2
Fresnel integrals 215
Function,  , the 365
Function,  , the 368
Function,  , the 82
Function,  , the 87
Function, , the, determines a cyclic group 90
Function, , the, fundamental region for 92
Function, , the, is automorphic 88
Function,  , the 355
Function, automorphic 84
Function, behavior at infinity 104 229
Function, complex 170
Function, constant, when 230
Function, continuity of a rational 133 170 172
Function, continuous = a power series 203
Function, continuous where finite 255
Function, definition and history of 28 29 178
Function, definition of a rational 28 29 167 231 392
Function, definition of a regular 178
Function, definition of an analytic 384 386
Function, degree of a rational 103
Function, derivative of a rational 174
Function, even, odd 272
Function, example of automorphic 106 113
Function, exponential 216—221 279
Function, generalization of the transcendental 216
Function, holomorphic 252
Function, infinite value of a 173
Function, linear fractional 54
Function, linear integral 32—38
Function, map of a rational 173
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