復函數論導論(英文)

preface

i the complex number system
1 the algebra and geometry of complex numbers
2 exponentials and logarithms of complex numbers
3 functions of a complex variable
4 exercises for chapter i

ii the rudiments of plane topology
1 basic notation and terminology
2 continuity and limits of functions
3 connected sets
4 compact sets
5 exercises for chapter ii

iii analytic functions
1 complex derivatives
2 the cauchy-riemann equations
3 exponential and trigonometric functions
4 branches of inverse functions
5 differentiability in the real sense
6 exercises for chapter iii

iv complex integration
1 paths in the complex plane
2 integrals along paths
3 rectifiable paths
4 exercises for chapter iv

v cauchy’’s theorem and its consequences
1 the local cauchy theorem
2 winding numbers and the local cauchy integral formula
3 consequences of the local cauchy integral formula
4 more about logarithm and power functions
5 the global cauchy theorems
6 simply connected domains
7 homotopy and winding numbers
8 exercises for chapter v

vi harmonic functions
1 harmonic functions
2 the mean value property
3 the dirichlet problem for a disk
4 exercises for chapter vi

vii sequences and series of analytic functions
1 sequences of functions
2 infinite series
3 sequences and series of analytic functions
4 normal families
5 exercises for chapter vii

viii isolated singularities of analytic functions
1 zeros of analytic functions
2 isolated singularities
3 the residue theorem and its consequences
4 function theory on the extended plane
5 exercises for chapter viii

ix conformed mapping
1 conformal mappings
2 msbius transformations
3 paemann’’s mapping theorem
4 the caratheodory-osgood theorem
5 conformal mappings onto polygons
6 exercises for chapter ix

x constructing analytic functions
1 the theorem of mittag-leffier
2 the theorem of weierstrass
3 analytic continuation
4 exercises for chapter x

appendix a background on fields
1 fields
2 order in fields
appendix b winding numbers revisited
1 technical facts about winding numbers
index