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| Burkhardt H. — Theory of Functions of a Complex Variable |
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| Предметный указатель |
Path, improper 141
Path-curves 187
Perfect set of points 138
Period of trigonometric and exponential functions 222 223
Period, primitive 222
Period, strip 224—226
Periodic function 84 222 282
Periodic function, singly 224 268 273
Periodic function, values of 233
Permanently convergent power series 201
Picard 235 267 278
Pierpont 3 128
Pitch of the helicoid 292
Plane Riemann’s surface 292
Poincare 98
Point, at infinity 45 48
Point, boundary 137
Point, inner, of a set 137
Point, isolated 137
Point, limit 127 128
Points of the plane 12
Points, aggregate, assemblage, class, collection of 127
Points, diametral 53 80
Points, equianharmonic, harmonic 72
Points, sets of, in the plane 135
Points, sets of, on a straight line 127
Polar coordinates 13
Pole or nonessential singular point 227 247 254 389 392
Pole, at a branch-point 335
Pole, at infinity 105
Pole, continuity at a 172
Pole, many-fold 102
Pole, residue at a 237
Poles and zeros 242 243
Poles and zeros of the inverse of  315
Poles and zeros, number of 240—242
Positive half-plane 67
Potential, electrical 187
Potential, velocity 186 187
Power series 168
Power series, a continuous function 203
Power series, a regular function 203
Power series, absolute convergence of 202
Power series, circle of convergence of 202
Power series, coefficients of a 206—210
Power series, coincidence of 207
Power series, convergence of 201
Power series, derivative of 149 203
Power series, development in 199 207
Power series, permanently convergent 201
Power series, properties of 201
Power series, successive derivatives of 204
Power series, Taylor’s, Maclaurin’s 206
Power series, uniform convergence of 202 203
Primitive period 222
Principal value of the amplitude 284
Principal value of the logarithm 295 297 304
Principal value of the square root 319
Principle of reflection, the 399
Pringsheim 165
Product of two complex numbers, unique 10
Product of two series, convergence Of 212
Product, absolute value of 19
Product, amplitude of 19
Product, form for a transcendental integral function 375
Product, form for the sine 376
Product, Weierstrassian 373
Product, when zero 11
Progressive, regressive derivatives 163 i6S
Projection, angles preserved in stereographic 48
Projection, Mercator’s 307
Projection, stereographic 47 53 307
Pure imaginary numbers 10
Quadruple numbers 22
Quantity, complex (a complex number) 9
Quotient 20
Quotient of two conjugate complex numbers 21
Radius vector 13
Rational function, behavior of a, at infinity 104
Rational function, continuity of a 133 170 172
Rational function, definition of a 28 29 167 231 392
Rational function, degree of 102
Rational function, derivative of a 174
Rational function, indeterminate 45 103
Rational function, many-fold zero of a 102
Rational function, manyfold pole of a 102
Rational function, map of a 173
Rational function, order of a 102 103
Rational function, pole of a 242
Rational function, reciprocal of a 172 173 228
Rational function, sum of all the orders of a 106 242
Rational function, value of a, at infinity 105
Rational function, when infinity is a pole, a zero of a 105 106
Rational function, zero of a 242
Rational functions of z and  331
Rational functions of z and  344
Rational integral function 29 98 231 233 292 392
Rational integral function, has n zeros 242
Rational integral function, has n-fold pole at  242 392
Real axis 13
Real logarithm 296
Real numbers, combinations of 1
Real numbers, general arithmetic of 1 2
Real part of complex numbers 10
Real variable, functions of a 127
Reciprocal of a complex number 21
Reciprocal of a rational function 172 173 228
Reciprocal radii on the sphere 51
Reciprocal radii, transformation by 38 41
Rectangular contour 158
Reflected images 408
Reflection on a circle 407
Reflection on a straight line 402
Reflection on axis of reals 39
Reflection on equatorial plane 52
Reflection on unit circle 40
Reflection, principle of 399
Regressive, progressive derivatives 163 165
Regular function on the Riemann’s surface 334
Regular function, definition 178
Regular function, expanded in a power series 199
Regular function, mapping with a 182
Regular function, near a singular point 227 247 254
Regular function, successive derivatives of 204
Regular function, value in a domain 196
Regular functions, sums of 260
Removable discontinuities 103 228 252
Representation, conformal 28 41 43 182 184
Representation, geometrical, of addition and subtraction 16
Representation, geometrical, of complex numbers 1 12
Representation, geometrical, of double algebra 3
Representation, geometrical, of multiplication 18—20
Representation, isogonal 43
Representation, isogonal, of z-plane 29
Representation, isogonal, parametric 139
Representation, isogonal, similar in infinitesimal parts 43
Residue, at a logarithmic discontinuity 354
Residue, at a pole 237 241
Residues 237—239
Residues, Cauchy’s theorem on 236
Rhumb line or loxodrome 307
Riemann 29 47 168 178 179 214
Riemann’s surface of the amplitude 289
Riemann’s surface of the logarithm 297
Riemann’s surface of the square root 319
Riemann’s surface, general construction of the 385
Riemann’s surface, plane 292 293
Riemann’s surface, “sheets” of the 293
Root of a number-pair, square 9
Roots of unity 25 72
| Roots, limits of the 100 101
Roots, manyfold 99
Rotating and stretching 32 33
Rotating the plane into itself 31
Scale of similarity 183
Schwarz, H.A. 402 407 420
Screw surface 290
Segment 127
Segment, closed, connected 138
Sequence, convergent 129
Sequence, increasing 130
Sequence, limit of convergent 168
Series of partial fractions, infinite 263
Series, absolute convergence of 170
Series, convergence of 169 170 201
Series, convergence of a product of two 212
Series, Fourier’s 257
Series, Gauss’s 214
Series, hypergeometric 214
Series, Laurent’s 247—251 254
Series, power see “Power series”
Series, sum of an infinite 169
Set of points as lines, surfaces 138
Set of points, closed, connected, perfect 138
Sets of points on a straight line 127
Sets of points, dense 138
Sets of points, derived 131
Sets of points, inner point of a 137
Similar figures transformed into each other 36
Similar triangles, condition for 36
Similarity, scale of 183
Similarity, transformation 32 35
Simple curve 140
Simply connected domain 142
Sine 216—221
Sine is a symmetric automorphic function 226
Sine, inverse 360
Sine, period of 222
Sine, product for the 376
Single-valued analytic functions 167
Single-valued function on the Riemann’s surface 333
Singly periodic function 224
Singly periodic function, conformal representation by 224
Singly periodic function, general theorems on 273
Singly periodic function, partial fractions of 268
Singular point, essential 389 390 392
Singular point, isolated 227 247 254 389
Singular point, nonessential or pole 227 228 389 392
Singular point, regular function near a 227 247 254
Singularities and natural boundaries 389 393
Singularities, branch-points are, when 393 395
Singularities, line of 391
Singularities, line of, as a natural boundary 391
Singularities, removable 103 228 252
Situs, analysis 328
Sphere 47
Sphere, Neumann’s 47 77
Square root 316—319
Square root of a number-pair 9
Square root, a single-valued function of position 322
Square root, branch-points of 322
Square root, connectivity of its surface 328
Square root, definition 317
Square root, deformation of its surface 329
Square root, expansion in a power series 337
Square root, principal value of 319
Square root, Riemann’s surface of 319
Squares, indefinitely small 185 187
Stereographic projection 47 53 307
Stolz, O. and A. Gmeiner 22
Stretching and rotating 32 33
Stretching the plane into itself 31
Strip, period 224—226
Subgroup 57
Substitution, elliptic, hyperbolic, loxodromic, parabolic 94—98
Substitution, linear (= bilinear) 54
Subtraction, geometrical representation of 18
Successive derivatives of a power series of a regular function 204
Sum of an absolutely convergent series 170
Sum of an infinite series 169
Sum of regular functions 260
Sum, absolute value of a 17
Superior limit 130
surface 136
Surface as a set of points 138
Surface, general definition of 139 140
Surface, helicoid, screw 290
Surface, plane, Riemann’s 292 293
Surface, Riemann’s see “Riemann’s surface”
Taylor’s series 206 209 213 383
Taylor’s series for the logarithm 297 298
Term by term differentiation 149 156
Term by term integration 155 200
Theorem on integration, Cauchy’s 190 195 199 247
Theorem on residues, Cauchy’s 236
Theorem, addition, for trigonometric and exponential functions 218
Theorem, binomial 221
Theorem, De Moivre’s, a generalization of 309
Theorem, Fundamental, of Algebra 229 232 240
Theorem, Green’s 280
Theorem, Laurent’s 247 254
Theorem, Liouville’s 230
Theorem, Mittag-Leffler’s 262
Theorems, general, on singly periodic functions 273
Thomae 152 R68
Thomson images 40
topology 328
Total variation of a quantity 175
Transcendental functions 124 167 282 392
Transcendental functions, generalization of 216
Transcendental integral function as a product 375
Transcendental integral function, at infinity 233 255
Transcendental integral function, definition 201
Transformation by rotating the sphere 52
Transformation of the plane 29
Transformation of the plane by a translation 30
Transformation of the plane by reciprocal radii 38
Transformation of the plane by rotation 31
Transformation of the plane by stretching 31
Transformation of the sphere into itself 51
Transformation on the sphere, linear 77
Transformation, bilinear 54
Transformation, circle 54 56
Transformation, degenerate 55
Transformation, determinant of the 55
Transformation, identical 57
Transformation, into itself, linear 84
Transformation, inverse 57 114
Transformation, involutoric 38 40 75
Transformation, linear see “Substitution linear”
Transformation, linear fractional 54
Transformation, linear integral 32—38
Transformation, orthomorphic 43
Transformation, periodic 93 95
Transformation, reciprocal radii 38 41
Transformation, reciprocal radii, on the sphere 51
Transformation, similarity 32 35
Transformation, “direct” similarity 35
Transformations, group of 90 113
Translation, parallel 28 30 33
Transmission of electricity 187
Transmission of heat 187
Triangles, infinitesimal 183
Triangles, similar 136
Trigonometric functions 216—221
Trigonometric functions, addition theorem for 218
Trigonometric functions, periodicity of 222
Triple numbers 22
Unconditional convergence 170
Undetermined coefficients, method of 207
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