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An introduction to complex analysis: classical and modern approaches

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Bibliographische Detailangaben
Beteiligte Personen:	Tutschke, Wolfgang 1934- (VerfasserIn), Vasudeva, Harkrishan L. (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Boca Raton [u.a.] Chapman & Hall/CRC 2005
Schriftenreihe:	Modern analysis series 7
Schlagwörter:	Analyse (wiskunde) Mathematical analysis Funktionentheorie
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013145128&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Umfang:	XVI, 460 S. graph. Darst.
ISBN:	1584884789

Internformat

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Datensatz im Suchindex

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adam_text	Contents 1 Preliminaries 1 1.1 The field of complex numbers ................. 1 1.1.1 Introduction ....................... 1 1.1.2 Complex numbers as pairs of real numbers ...... 1 1.1.3 Solvability of z* = -1 ................. 3 1.1.4 Definition of complex numbers using equivalence classes of polynomials ...................... 4 1.2 The complex plane ........................ 6 1.2.1 Geometrical interpretation of complex numbers .... 6 1.2.2 Absolute value. Conjugate complex numbers ..... 6 1.2.3 Interpretation of complex numbers as vectors ..... 7 1.2.4 Trigonometric form of complex numbers ........ 9 1.2.5 Geometrical interpretation of the product of complex numbers .......................... 9 1.2.6 Powers and roots of complex numbers ......... 10 1.2.7 Some special sets in the complex plane ......... 11 1.2.8 Examples ......................... 12 1.3 Metric spaces ........................... 14 1.3.1 The concept of a metric space .............. 14 1.3.2 Open sets ......................... 16 1.3.3 Convergence ....................... 18 1.3.4 Closed subsets ...................... 18 1.3.5 Definition of some more terms ............. 20 1.3.6 Completeness ....................... 22 1.3.7 Compact sets ....................... 24 1.3.8 Coverings ......................... 25 1.3.9 Examples ......................... 27 1.4 Mappings and functions. Continuity .............. 28 VIU 1.4.1 Basic definitions..................... 28 1.4.2 The extended complex plane. Spherical distance ... 31 1.4.3 Limits of functions .................... 34 1.4.4 Continuous functions ................... 35 1.4.5 Curves ........................... 38 1.4.6 Connectivity ....................... 39 1.4.7 Sequences of functions .................. 40 1.4.8 Infinite series ....................... 45 1.4.9 Power series ........................ 57 1.4.10 The complex exponential and trigonometric functions 67 1.4.11 Logarithm and its Riemann Surface .......... 75 1.4.12 Remark on the concept of a topological space ..... 80 1.4.13 Examples ......................... 81 1.5 Exercises to Chapter 1 ..................... 82 2 The classical approach 87 2.1 Ordinary complex differentiation ................ 87 2.1.1 Complex differentiability ................. 87 2.1.2 Rules of differentiation .................. 90 2.1.3 The Cauchy-Riemann system .............. 90 2.1.4 Holomorphic functions .................. 94 2.1.5 Differentiation of Power Series ............. 96 2.1.6 Differentiation of inverse functions ........... 100 2.2 Preliminaries of the Integral Calculus ............. 102 2.2.1 Line integrals of real valued functions ......... 102 2.2.2 The Green-Gauss Integral Theorem .......... 104 2.2.3 Line integrals of complex valued functions ....... 106 2.3 Complex Integral Theorems .................. 112 2.3.1 Cauchy s Integral Theorem ............... 113 2.3.2 Cauchy s Integral Formula ................ 115 2.4 Exercises to Chapter 2 ..................... 116 їх 3 An alternative approach 121 3.1 Partial complex differentiations ................. 121 3.1.1 Linearization of functions of one real variable ..... 121 3.1.2 Linearization of functions depending on several real variables .......................... 121 3.1.3 Linearization of functions depending on one complex variable .......................... 122 3.1.4 Definition of partial complex derivatives ........ 122 3.1.5 Differentiability rules for partial complex derivatives . 123 3.2 Complex Green-Gauss Integral Theorems .......... 123 3.3 Generalized Cauchy Integral Formula ............ 125 3.3.1 Application of complex version of the Green-Gauss For¬ mula to functions having isolated singularities ..... 125 3.3.2 The limit of the domain integral. Schmidt s Inequality 125 3.3.3 The limit of the line integral .............. 128 3.3.4 An integral representation formula for continuously dif- ferentiable functions ................... 128 3.4 The classical Cauchy Integral Formula * ........... 129 3.4.1 Another approach to Cauchy s Integral Formula . . . 129 3.4.2 A second proof of Cauchy s Integral Theorem ..... 129 3.5 Comparison * .......................... 130 3.5.1 Partial complex derivatives of functions having an ordi¬ nary complex derivative ................. 130 3.5.2 Ordinary complex differentiability of solutions of the Cauchy-Riemann system ................. 130 3.6 Exercises to Chapter 3 ..................... 131 4 Local properties 135 4.1 Existence of higher order derivatives .............. 135 4.1.1 A method for proving local properties of holomorphic functions ......................... 135 4.1.2 The holomorphy of Une integrals with respect to com¬ plex parameters ...................... 135 4.1.3 Cauchy s Integral Formula for the derivatives of a holo¬ morphic function ..................... 137 4.2 Local power series representation ................ 138 4.2.1 Power series representation of the Cauchy kernel . . . 138 χ 4.2.2 Local power series for holomorphic functions ала an in¬ tegral representation of the coefficients ......... 139 4.2.3 Cauchy s estimate of the coefficients .......... 142 4.2.4 The power series of the product of two holomorphic functions ......................... 142 4.2.5 Division of power series ................. 143 4.3 Distribution of zeros ....................... 144 4.4 The Weierstrass Convergence Theorem ............ 145 4.4.1 Statement of the problem ................ 145 4.4.2 Formulation and proof .................. 145 4.4.3 Termwise differentiability ................ 146 4.5 Connexion with plane Potential Theory ............ 147 4.5.1 Holomorphic functions as solutions of the Laplace equa¬ tion ............................ 147 4.5.2 Representation of the Laplace operator by partial com¬ plex differentiations ................... 148 4.6 Complex Integral Theorems revisited * ............ 148 4.6.1 Goursat s Theorem .................... 149 4.6.2 G. Fichera s proof of the Goursat Theorem ...... 152 4.6.3 A measure-theoretic approach to Cauchy s Integral The¬ orem ............................ 155 4.6.4 Consequences of Complex Integral Theorems under weaker assumptions ................... 163 4.7 Exercises to Chapter 4 ..................... 163 5 Global properties 169 5.1 Analytic continuation ...................... 169 5.1.1 Definition of analytic continuation ........... 169 5.1.2 The Unique Continuation Theorem ........... 170 5.1.3 The uniqueness of analytic continuation ........ 171 5.1.4 Analytic continuation of the limit fonction of a power series ........................... 172 5.1.5 Analytic continuation across a curve .......... 180 5.1.6 Global behaviour of holomorphic functions with non- isolated wo-points .................... 181 5.2 Maximum Modulus Principle (Maximum Principle) ..... 182 Xl 5.2.1 The basic statement ................... 182 5.2.2 Holomorphic functions with constant modulus .... 182 5.2.3 Mean Value Property of holomorphic functions .... 183 5.2.4 Proof of the Maximum Modulus Principle ....... 183 5.2.5 A Maximum Modulus Principle for bounded domains 184 5.2.6 The Minimum Modulus Principle (Minimum Principle) 184 5.3 Entire functions ......................... 186 5.3.1 Definition and basic properties ............. 186 5.3.2 Liouville s Theorem ................... 187 5.3.3 Functions of polynomial growth ............. 187 5.4 Fundamental Theorem of Algebra ............... 188 5.4.1 Statement of the problem ................ 188 5.4.2 Proofs of the Fundamental Theorem of Algebra using complex analysis ..................... 189 5.4.3 Special case: Existence of the roots of complex numbers 192 5.4.4 Argand s proof of the Fundamental Theorem of Algebra 193 5.4.5 Additional proofs of the Fundamental Theorem of Al¬ gebra ........................... 195 5.4.6 Factorization of polynomials ............... 195 5.5 Exercises to Chapter 5 ..................... 195 β Isolated singularities 199 6.1 Classification ........................... 199 6.1.1 Definition of isolated singularities ............ 199 6.1.2 Removable singularities ................. 199 6.1.3 Poles ............................ 201 6.1.4 Essential singularities .................. 202 6.2 Laurent series .......................... 202 6.2.1 Holomorphic functions in an annulus .......... 202 6.2.2 Holomorphic functions in a punctured disk ...... 205 6.3 Characterization by the principal part ............. 206 6.4 Meromorphic functions ..................... 208 6.5 Behaviour at essential singularities ............... 210 6.6 Behaviour at infinity ...................... 211 xii 6.7 Partial fractions of rational functions ............. 213 6.7.1 An application of the Division Algorithm ....... 213 6.7.2 Representation of rational functions by partial fractions 214 6.8 Meromorphic functions on the Sphere ............. 216 6.9 Exercises to Chapter 6 ..................... 217 7 Homotopy 223 7.1 Statement of the problem .................... 223 7.2 Homotopic curves ........................ 225 7.3 Path independent line integrals ................. 227 7.4 Simply connected domains ................... 229 7.4.1 The concept of simple connectedness .......... 229 7.4.2 Cauchy s Integral Theorem in homotopy formulation . 229 7.4.3 Monodromy Theorem .................. 230 7.5 Solution of first order systems ................. 232 7.5.1 A property of path independent line integrals ..... 232 7.5.2 Local solution of first order systems .......... 233 7.5.3 Global solutions of first order systems ......... 234 7.6 Conjugate solutions ....................... 234 7.7 Inversion of complex differentiation .............. 235 7.8 Morera s Theorem ........................ 237 7.9 Potentials of vector fields .................... 238 7.9.1 The concept of a potential ................ 238 7.9.2 Some physical interpretations of vector fields ..... 239 7.9.3 Curl-free and source-free vector fields ......... 240 7.10 Exercises to Chapter 7 ..................... 241 8 Residue theory 245 8.1 Statement of the problem .................... 245 8.2 Winding numbers ........................ 245 8.3 The integration of principal parts ............... 246 8.3.1 Tennwise integration ................... 246 8.3.2 A complex version of the Fundamental Theorem of Dif¬ ferential and Integral Calculus ............. 247 XIU 8.3.3 Integration of meromorphic functions with first order poles ............................ 249 8.4 Residue Theorem ........................ 249 8.5 Calculation of residues ..................... 251 8.5.1 The case of first order poles ............... 252 8.5.2 The case of poles of order к > 2............ 253 8.5.3 Determination of residues using Laurent series .... 253 8.6 Exercises to Chapter 8 ..................... 254 9 Applications of residue calculus 257 9.1 Total number of zeros and poles ................ 257 9.1.1 Representation by a boundary integral ......... 257 9.1.2 A proof of Fundamental Theorem of Algebra based on a boundary integral representation ........... 258 9.1.3 Rouché s Theorem .................... 259 9.1.4 Another proof of Fundamental Theorem of Algebra us¬ ing Rouché s Theorem .................. 259 9.2 Evaluation of definite integrals ................. 260 9.2.1 Evaluation of integrals involving certain periodic func¬ tions between the limits 0 and 2π ........... 260 9.2.2 Evaluation of improper real integrals .......... 263 9.2.3 Integrals involving many-valued functions ....... 273 9.3 Sum of certain series ...................... 277 9.4 Exercises to Chapter 9 ..................... 280 10 Mapping properties 287 10.1 Continuously differentiable mappings ............. 287 10.1.1 Invertible linear mappings. The general case ..... 287 10.1.2 The exceptional case ................... 290 10.1.3 Calculation of the angle of rotation ........... 291 10.1.4 Orientation-preserving mappings ............ 292 10.2 Conformai mappings ...................... 293 10.2.1 Conformai mappings by holomorphic functions .... 293 10.2.2 Behaviour at zeros of the derivative .......... 294 10.2.3 Inversion of multivalent functions * .......... 295 10.2.4 Another proof of the local existence of the inverse func¬ tion ........................... 296 10.2.5 Domain invariance .................. 296 10.2.6 Behaviour of the Cauchy-Riemann system under con- formal mappings * ................... 297 10.2.7 Conformai equivalence * ................ 297 10.2.8 Quasiconformal mappings * .............. 297 10.3 Examples of conformai mappings ................ 299 10.3.1 Some elementary conformai mappings ......... 299 10.3.2 The Möbius fractional linear transformations ..... 302 10.3.3 Mappings of the unit disk onto itself .......... 307 10.3.4 Complex plane onto itself ................ 313 10.3.5 Schwarz Reflection Principle .............. 314 10.4 Univalent functions * ...................... 316 10.4.1 Definition and basic properties ............. 316 10.4.2 Bieberbach s conjecture ................. 316 10.4.3 Univalent functions outside the unit disk ........ 317 10.4.4 Proof of Bieberbach s conjecture for аг ........ 318 10.4.5 Koebe s Covering Theorem ............... 320 10.4.6 Limits of univalent functions .............. 320 10.5 Riemann s Mapping Theorem * ................ 321 10.5.1 Statement of the problem ................ 321 10.5.2 Outline of the proof; extremal problems in classes of holomorphic functions .................. 322 10.5.3 Proof of Riemann s Mapping Theorem ......... 322 10.5.4 Summary of the solution of the main problems of Con- formal Mappings ..................... 326 10.6 Construction of flow Unes ................... 328 10.6.1 The level curves of real and imaginary parts of holomor¬ phic functions ....................... 328 10.6.2 Construction of curl-free and source-free vector fields in the plane ......................... 328 10.6.3 Flow lines ......................... 329 10.6.4 Examples of the construction of flow Unes ....... 330 10.7 Exercises to Chapter 10..................... 332 11 Special functions 339 11.1 Prescribed principal parts .................... 339 11.2 Prescribed zeros ......................... 346 11.3 Infinite products ........................ 348 11.3.1 Statement of the the problem .............. 348 11.3.2 Infinite products of complex numbers ......... 349 11.3.3 Infinite products of complex-valued functions ..... 355 11.3.4 Derivatives of infinite products ............. 359 11.4 Weierstrass products * ..................... 362 11.4.1 Statement of the problem ................ 362 11.4.2 Entire functions without zeros ............. 363 11.4.3 Weierstrass Primary Factors ............... 364 11.4.4 Preliminaries for the application of Weierstrass Pri¬ mary Factors ....................... 366 11.4.5 The Weierstrass Factorization Theorem ........ 367 11.4.6 Some examples ...................... 368 11.5 Gamma function ........................ 369 11.5.1 Definition of the gamma function ............ 369 11.5.2 Functional equation of the gamma function ...... 372 11.5.3 Some elementary properties of the gamma function . . 372 11.5.4 An integral representation of the gamma function in the right half-plane ...................... 375 11.5.5 A partial fraction representation of the gamma function 379 11.6 The Riemann zeta function ................. 381 11.6.1 Definition of Riemann s zeta function ......... 381 11.6.2 Connexion between zeta function and prime numbers 382 11.6.3 Analytic continuation of the Riemann zeta function . 383 11.6.4 Relationship between gamma function and zeta function 386 11.7 EDiptic functions * ....................... 388 11.7.1 Weierstrass zeta function ................ 388 11.7.2 Weierstrass p-function ................. 390 11.7.3 Periods of meromorphic functions ............ 393 11.7.4 Properties of elliptic functions ............. 400 XVI 11.7.5 Construction of elliptic functions with prescribed prin¬ cipal parts ......................... 406 11.7.6 Related topics ...................... 408 11.8 Exercises to Chapter 11..................... 416 12 Boundary value problems 425 12.1 Preliminaries ........................... 425 12.1.1 Harmonic functions and the Dirichlet problem .... 425 12.1.2 Maximum Principle ................... 426 12.2 The Poisson Integral Formula ................. 428 12.2.1 Derivation of Poisson s Integral Formula ........ 428 12.2.2 Construction of solutions of the Laplace equation by Poisson s Integral Formula ................ 430 12.2.3 Preliminaries for the proof of Theorem 101...... 431 12.2.4 Proof of Theorem 101.................. 432 12.2.5 Examples ......................... 433 12.2.6 Poisson kernel and its conjugate ............ 435 12.3 Cauchy Type Integrals * .................... 436 12.3.1 Statement of the problem ................ 436 12.3.2 Cauchy Type Integrals with constant density ..... 437 12.3.3 Cauchy Type Integrals with Holder-continuous density 440 12.3.4 The Plemelj Formulae .................. 444 12.4 Desired holomorphic functions *................ 447 12.4.1 The Schwarz problem .................. 447 12.4.2 Solution of the Schwarz problem in a disk ....... 448 12.4.3 The Riemann boundary value problem ......... 449 12.5 Exercises to Chapter 12..................... 450 References 455 Index 457
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spellingShingle	Tutschke, Wolfgang 1934- Vasudeva, Harkrishan L. An introduction to complex analysis classical and modern approaches Modern analysis series Analyse (wiskunde) gtt Mathematical analysis Funktionentheorie (DE-588)4018935-1 gnd
subject_GND	(DE-588)4018935-1
title	An introduction to complex analysis classical and modern approaches
title_auth	An introduction to complex analysis classical and modern approaches
title_exact_search	An introduction to complex analysis classical and modern approaches
title_full	An introduction to complex analysis classical and modern approaches Wolfgang Tutschke ; Harkrishan L. Vasudeva
title_fullStr	An introduction to complex analysis classical and modern approaches Wolfgang Tutschke ; Harkrishan L. Vasudeva
title_full_unstemmed	An introduction to complex analysis classical and modern approaches Wolfgang Tutschke ; Harkrishan L. Vasudeva
title_short	An introduction to complex analysis
title_sort	an introduction to complex analysis classical and modern approaches
title_sub	classical and modern approaches
topic	Analyse (wiskunde) gtt Mathematical analysis Funktionentheorie (DE-588)4018935-1 gnd
topic_facet	Analyse (wiskunde) Mathematical analysis Funktionentheorie
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013145128&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
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