Holomorphic functions of one variable:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY [u.a.]
Gordon and Breach Science Publishers
1983
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | IX, 274 Seiten graph. Darst. |
| ISBN: | 0677059507 |
Internformat
MARC
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| 100 | 1 | |a Colombo, Serge |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Holomorphic functions of one variable |c Serge Colombo |
| 264 | 1 | |a New York, NY [u.a.] |b Gordon and Breach Science Publishers |c 1983 | |
| 300 | |a IX, 274 Seiten |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Holomorphic functions | |
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Datensatz im Suchindex
| _version_ | 1804116761441730560 |
|---|---|
| adam_text | Contents
Preface ix
CHAPTER 1 THE FIELD OF COMPLEX NUMBERS AND THE
RIEMANN SPHERE
The Argand diagram 1
Curves and regions in the Gaussian plane 3
The Riemann sphere 7
Compactification of the Riemann sphere 10
CHAPTER 2 HOLOMORPHIC FUNCTIONS
Continuity and chordally continuous functions 13
Uniform continuity 15
Monogenic functions 16
Harmonic functions 19
Harmonic conjugate functions 20
Examples 21
Conformal mapping 22
The bilinear transformation 27
CHAPTER 3 POWER SERIES
Cauchy convergence principle 32
The O[f(z) and o[f(z)~ notation 33
Examples 34
Uniform convergence 35
Series of functions 37
Uniformly convergent integrals 39
Power series 42
The Cauchy Hadamard formula 43
The ratio formula 46
Product of power series 48
A power series defines a holomorphic function 50
v
Vi CONTENTS
Behaviour of the sum of a power series on the boundary of its
circle of convergence 53
Abel s theorem 55
Abel s transformation 58
Abel summability 60
Tauberian theorems 61
Miscelleneous problems 63
CHAPTER 4 SOME ELEMENTARY HOLOMORPHIC
FUNCTIONS
Polynomials 66
Rational functions 68
The exponential function 69
The circular functions 72
Periodicity of the exponential function 75
The hyperbolic functions 76
The logarithmic function 78
The general power ab 82
Examples of conformal mappings 85
CHAPTER 5 COMPLEX INTEGRATION AND CAUCHY
THEOREM
Jordan arcs and Jordan closed curves 91
Rectifiable arcs 93
Definition of a contour 97
The winding number 97
Definition of the complex line integral 98
Some basic complex line integrals 101
I f
A majorant for f(z) dz 104
I •/
(C)
Cauchy s theorem 105
Equivalent contours 110
Cauchy s integral formula 111
Liouville s theorem 115
The principle of the maximum modulus 116
Morera s theorem 116
Taylor s series 119
Zeros of a holomorphic function 121
CONTENTS Vii
Laurent s series 122
Additional remarks 127
CHAPTER 6 SINGULAR POINTS AND ANALYTIC
CONTINUATION
Regular points 130
Isolated singularities 133
Cluster points of zeros and poles 137
Singularities at z = oo 138
The behaviour of a holomorphic function near an essential
isolated singularity 139
Meromorphic functions 140
Other types of singularities 143
Analytic continuation 143
Analytic continuation along a curve 147
Natural boundaries 153
Pringsheim s theorem 153
Riemann surfaces 159
CHAPTER 7 THE CALCULUS OF RESIDUES
A fundamental theorem on residues 162
Residue at z= oo 166
Jordan s inequality 167
+ 00
Integrals of the form /(x) dx 168
— oo
Extension of the preceding method 175
Definite integrals of trigonometric functions 183
Miscellaneous exercises 185
+ 00
r
Evaluation of definite integrals of the form x /(x) dx with
J
0 a l ° 190
The logarithmic derivative and the principle of the argument 192
The fundamental theorem of algebra 196
The inverse function 198
The partial fraction expansion of cot nz 202
The representation of a transcendental meromorphic function by
a series of partial fractions 205
Viii CONTENTS
CHAPTER 8 POISSON JENSEN FORMULA
Jensen s formula 208
Second form of Jensen s formula 211
Poisson s integral formula 215
The Poisson Jensen formula 218
CHAPTER 9 INFINITE PRODUCTS. ENTIRE FUNCTIONS
AND CANONICAL PRODUCTS
Infinite products — definition 220
General properties of infinite products 222
Functions defined by infinite products 226
Entire functions 228
Weierstrass s primary factors 230
Rank on an infinite product — Canonical product 234
Entire functions of finite order 237
CHAPTER 10 THE GAMMA FUNCTION AND EULERIAN
INTEGRALS
The Gamma function 243
The Eulerian integral of second kind 248
Hankel s formula for [T(z)] 1 256
The logarithmic derivative of the gamma function 259
Stirling s asymptotic formula 262
Eulerian integrals of first kind 265
Appendix: The Builders of Analytic Function Theory 270
Index 273
|
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| id | DE-604.BV002300592 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T15:43:39Z |
| institution | BVB |
| isbn | 0677059507 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001511987 |
| oclc_num | 8280211 |
| open_access_boolean | |
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| owner_facet | DE-91G DE-BY-TUM DE-384 DE-703 DE-739 DE-706 DE-11 DE-188 |
| physical | IX, 274 Seiten graph. Darst. |
| publishDate | 1983 |
| publishDateSearch | 1983 |
| publishDateSort | 1983 |
| publisher | Gordon and Breach Science Publishers |
| record_format | marc |
| spelling | Colombo, Serge Verfasser aut Holomorphic functions of one variable Serge Colombo New York, NY [u.a.] Gordon and Breach Science Publishers 1983 IX, 274 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier Holomorphic functions Holomorphe Funktion (DE-588)4025645-5 gnd rswk-swf Holomorphe Funktion (DE-588)4025645-5 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001511987&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Colombo, Serge Holomorphic functions of one variable Holomorphic functions Holomorphe Funktion (DE-588)4025645-5 gnd |
| subject_GND | (DE-588)4025645-5 |
| title | Holomorphic functions of one variable |
| title_auth | Holomorphic functions of one variable |
| title_exact_search | Holomorphic functions of one variable |
| title_full | Holomorphic functions of one variable Serge Colombo |
| title_fullStr | Holomorphic functions of one variable Serge Colombo |
| title_full_unstemmed | Holomorphic functions of one variable Serge Colombo |
| title_short | Holomorphic functions of one variable |
| title_sort | holomorphic functions of one variable |
| topic | Holomorphic functions Holomorphe Funktion (DE-588)4025645-5 gnd |
| topic_facet | Holomorphic functions Holomorphe Funktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001511987&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT colomboserge holomorphicfunctionsofonevariable |