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Burkhardt H. — Theory of Functions of a Complex Variable
Burkhardt H. — Theory of Functions of a Complex Variable



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Название: Theory of Functions of a Complex Variable

Автор: Burkhardt H.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1913

Количество страниц: 432

Добавлена в каталог: 25.03.2006

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$A^{z}$, principal value of      309
$e^{2}$      216
$e^{2}$ is a symmetric automorphic function      225
$e^{2}$, fundamental region of      225
$Sin^{-1}w$      360
$Tan^{-1}z$, definition      313
$Tan^{-1}z$, mapping with      314
$Tan^{-1}z$, poles of the inverse of      315
$Tan^{-1}z$, 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 $tan^{-1}z$      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 $\sqrt{z}$      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 $\sqrt{z}$      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, $n^{z}$, the      365
Function, $s^{2}=1-z^{3}$, the      368
Function, $w=z^{2}$, the      82
Function, $w=z^{n}$, the      87
Function, $w=z^{n}$, the, determines a cyclic group      90
Function, $w=z^{n}$, the, fundamental region for      92
Function, $w=z^{n}$, the, is automorphic      88
Function, $z=w+i\sqrt{1-w^{2}}$, 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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