《Topological Methods in The Theory of Functions of A Complex Variable》

Foreword1

CHAPTER Ⅰ．PSEUDO-HARMONIC FUNCTIONS1

1．Introduction1

2．Pseudo-harmonic functions5

3．Critical points of U on G10

4．Critical points of U on (B)11

5．Level arcs leading to a boundary point zo not an extremum point of U14

6．Canonical neighborhoods of a boundary point zo not an extremum point of U17

7．Multiple points21

8．The sets Uc and their maximal boundary arcs w at the level c25

9．The Euler characteristic Ec26

10．Obtaining K＊c from Kc-e28

11．The variation of Ec with increasing c31

12．The principal theorem under boundary conditions A34

13．Th case of constant boundary values38

CHAPTER Ⅱ．DIFFERENTIABLE BOUNDARY VALUES43

14．Boundary conditions A，B，C43

15．Boundary conditions B48

16．The vector index J of the boundary values54

17．The vector index J as the degree of a map on a circle58

CHAPTER Ⅲ．INTERIOR TRANSFORMATIONS62

18．Locally simple curves62

19．Interior transformations65

20．First applications and extensions69

CHAPTER Ⅳ．THE GENERAL ORDER THEOREM79

21．An example79

22．Locally simple boundary images81

23．The existence of partial branch elements84

24．The order，angular order theorem90

25．Radó＇s theorem generalized93

CHAPTER Ⅴ．DEFORMATIONS OF LOCALLY SIMPLE CURVES AND AND OF INTERIOR TRANSFORMATIONS97

26．Objectives97

27．The μ-length of curves99

28．Admissible deformations of locally simple curves107

29．Deformation classes of locally simple curves116

30．The product of locally simple curves120

31．The product of deformation classes123

32．O-Deformations．Curves of order zero127

33．O-Deformations．Curves of order q？0130

34．Deformation classes of meromorphic functions and of interior transformations136