Conway J.B. — Functions of One Complex Variable :: Электронная библиотека попечительского совета мехмата МГУ

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Conway J.B. — Functions of One Complex Variable

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Название: Functions of One Complex Variable

Автор: Conway J.B.

Аннотация:

This book discusses a variety of problems which are usually treated in a second course on the theory of functions of one complex variable. It treats several topics in geometric function theory as well as potential theory in the plane. In particular it covers: conformal equivalence for simply connected regions, conformal equivalence for finitely connected regions, analytic covering maps, de Branges' proof of the Bieberbach conjecture, harmonic functions, Hardy spaces on the disk, potential theory in the plane. The level of the material is gauged for graduate students. Chapters XIII through XVII have the same prerequisites as the first volume of this text, GTM 11. For the remainder of the text it is assumed that the reader has a knowledge of integration theory and functional analysis. Definitions and theorems are stated clearly and precisely. Also contained in this book is an abundance of exercises of various degrees of difficulty.

Язык:

Рубрика: Математика/Анализ/Комплексный анализ/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1978

Количество страниц: 316

Добавлена в каталог: 02.04.2005

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Предметный указатель

Abel’s theorem      74
Absolute value      2
Absolutely convergent product      166
Absolutely convergent series      30 106
Analytic continuation along a chain of disks      216
Analytic continuation along a path      214
Analytic continuation of a germ      215
Analytic function      34 236
Analytic manifold      234
Analytic structure      234
Analytic surface      234
Angle between two paths      45
ARC      229
Arcwise connected      230
Argument      4
Argument principle      123
Arzela — Ascoli theorem      148
Barrier      269
Base space      228 232
Bernoulli numbers      76
Bessel function      122
beta function      186
Blaschke product      173
Bloch’s constant      295
Bloch’s theorem      293
Bohr — Mollerup theorem      179
Boundary      13
Bounded function      12
Branch of the logarithm      39 40
Cantor’s theorem      19
Caratheodory’s inequality      133
Casorati — Weierstrass theorem      109 111
Cauchy sequence      18
Cauchy — Riemann equations      41
Cauchy’s estimate      73
Cauchy’s integral formula      84
Cauchy’s theorem, first version      85
Cauchy’s theorem, fourth version      94
Cauchy’s theorem, second version      89
Cauchy’s theorem, third version      90
Cayley — Hamilton theorem      87
Chain of disks      216
Chain rule      34
Change of parameter      64
Closed ball      11
Closed curve      66
Closed map      99
Closed set      12 222
Closure      13
Compact      20 224
Complete analytic function      215 216
Complete analytic function of local inverses      239
Complete metric space      18
Component      16
Conformal mapping      46
Conformally equivalent      160
conjugate      2
Connected      14 224
Continuous function      24 224
Convergent sequence      18
Convergent Series      30 106
Convex function      134
Convex set      89 134
Coordinate patch      234
Cover      20
Covering space      245
Critical strip      193
Cross ratio      48
Curve      64
DeMoivre’s formula      5
Dense      13
Diameter of a set      19
Differentiable function      33
Dilation      47 56
Dini’s theorem      150
Dirichlet problem      252 269
Dirichlet region      269
Distance between a point and a set      25
Distance between two sets      27
Distance function      11
Elementary factor      168
Empty set      12
Entire function      77
Equicontinuous      148
Equivalence of paths      64
Essential singularity      105 111
Euler’s constant      75 177 185
Euler’s theorem      193
Exponent of convergence      286
Exponential function      32
Exponential series      32
Extended boundary      129
Extended plane      8
f.i.p.      21
FEP homotopic      93
Fiber      228
Final point      229
Finite genus      283
Finite intersection property      21
Finite order      285
Finite rank      282
Finitely generated ideal      208
Fixed ideal      209
Free ideal      209
Function element      214
Function of bounded variation      58
Functional equation of the gamma function      178
Functional equation of the zero function      192
Fundamental set      245
Fundamental theorem of algebra      77 100
Gamma function      176
Gaussian psi function      186
Gauss’s Formula      178
Genus      283
Geometric series      30
Germ      214
Goursat’s theorem      100
Great Picard theorem      300
Greatest common divisor      174
Green’s function      275
Hadamard’s factorization theorem      289
Hadamard’s three circles theorem      137
Hardy’s theorem      138
Harmonic conjugate      42 252
Harmonic function      41 252
Harnack’s inequality      260
Harnack’s theorem      261 268
Hausdorff space      224
Heine — Borel theorem      23
Holomorphic function      126 151
Homeomorphism      163 202
Homologous to zero      95
Homomorphism      6
Homotopic closed curves      88
Homotopic to zero      89
Hurwitz’s theorem      152
Ideal      174
Ideal generated by $\mathcal{J}$       208
Imaginary part      2
Independence of Path theorem      93
INDEX      81
Infinite order      285
Infinite product      164
Infinite rank      282
Initial point      229
Integration around a branch point      119

Interior      13
Intermediate Value Theorem      27
inversion      47
Isolated singularity      103
Isomorphism      238
Jensen’s formula      280
Jordan curve theorem      213
Landau’s Constant      295
Laplace’s equation      252
Laurent Series Development      107
Lebesgue’s Covering Lemma      21
Left side of a circle      53
Leibniz’s Rule      68 73
Lifting      246
Limit inferior of a function      129
Limit inferior of a sequence      30
Limit point      18 222
Limit superior of a function      129
Limit superior of a sequence      30
Line integral      63
Linear fractional transformation      47
Liouville’s theorem      77 123 238
Lipschitz function      25
Little Picard theorem      297
Locally arcwise connected      230
Locally bounded      153
Locally Lipschitz      154
Logarithmically convex function      135
Maximal ideal      174
Maximum Modulus theorem, first version      79 128 238
Maximum Modulus theorem, second version      128
Maximum Modulus theorem, third version      129
Maximum principle for harmonic functions, first version      253
Maximum principle for harmonic functions, fourth version      264
Maximum principle for harmonic functions, second version      254
Maximum principle for harmonic functions, third version      264
Mean value property      253 260
Mean value theorem      253
Meromorphic function      123
Metric space      11
Metrisable space      223
Minimum principle      129 255
Mittag — Leffler’s theorem      205
Mobius transformation      47
Monodromy theorem      219 221 247
Montel — Carathedory theorem      300
Montel’s theorem      153
Morera’s theorem      86
Multiplicity of a zero      76
MVP      253 260
Neighborhood system      226
Normal set of functions      146
Normal subgroup      57
Open ball      11
Open cover      20
Open map      99
Open mapping theorem      99 238
Open set      12 222
Order      285
Orientation      52
Orientation principle      53
Parallelogram law      3
Partial fractions      106
Path      45 229
Pathwisc connected      230
Periodic function      38
Perron family      266
Perron function      268
Phragmen — Lindelof theorem      138
Picard’s Great theorem      300
Picard’s Little theorem      297
Poisson kernel      256
Poisson — Jensen formula      281
Pole      105 111
Polygon      15
Polynomially convex hull      200
Power series      30
Prime ideal      174
primitive      65 94
Principal branch of the loagrithm      40
Principal ideal      208
Projection map      228
Proper ideal      174
Properly covered      245
Punctured disk      103
Radius of convergence      31
Rank      282
Real part      2
Rectifiable path      62
Region      40
Regular function      151
Relative topology      225
Relatively prime      208
Removable singularity      103 111
Residue      112
Residue theorem      112
Riemann hypothesis      193
Riemann mapping theorem      160 277
Riemann surface      232
Riemann zeta function      187
Riemann’s functional equation      192
Right side of a circle      53
Roots of unity      5
Rotation      47
Rouche’s Theorem      125
Runge’s theorem      198
Schottky’s theorem      298
Schwarz reflection principle      211 256
Schwarz’s lemma      130
Sequentially compact      21
Sheaf of germs      228
Simple group      57
Simple root      98
Simply connected      93 202
Singular part      105
Singularity at infinity      111
Stalk      228
Standard form      283
Star shaped set      89
Stereographic projection      9
Subharmonic function      264
Superharmonic function      264
Symmetric points      50
Symmetry principle      52
Terminal point      229
Topological space      221
Topology induced by a neighborhood system      227
Total boundedness      22
Total variation      58
Trace of a path      62 229
Translation      47
Triangle inequality      3 11
Triangular path      86
Trivial zeros      193
Uniform convergence      28 29
Uniformly continuous      25
Uniformly convergent integral      154
Unrestricted analytic continuation      219
Vitali’s theorem      154
Wallis’s formula      176
Weierstrass approximation theorem      263
Weierstrass factorization theorem      170 207
Weierstrass M-test      29
Weierstrass Pe function      207
Weierstrass zeta function      207
Winding number      81 250

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