Methods of the Theory of Functions of a Complex Variable

M.A.Lavrentjev and B.V.Shabat

Publishing House "Nauka", Moscow 1951

I Basic concepts   II. CONFORMAL MAPPING    
1.1 Complex Numbers   2.1 General propositions. Examples    
1. Complex Numbers   27. Concept of conformal mapping    
2. Geometrical illustration   28. Fundamental problem    
1.2 Functions of a Complex Variable   29. Correspondence of boundaries    
3. Geometrical concepts   30. Examples    
4. Functions of a complex variable   2.2 The simplest conformal mappings    
5. Differentiability and analyticity   31. Bi-linear Mappings    
1.3 Elementary Functions   32. Particular cases    
6. The Functions w = zn and w = z1/n   33. Examples    
7. Joukovsky's function w = (z + 1/z)   34. Mapping of sickles    
8. Exponential function and logarithm   2.3 Symmetry principle and mapping of polygons    
9. Trigonometric and hyperbolic functions   35. Symmetry Principle    
10. The general power w = za   36. Examples    
1.4 Integration of functions of a complex variable   37. Mapping of polygons    
11. Integral of a function of a complex variable        
12. Cauchy's theorem   38. Supplementary remarks    
13 Extension to multiply connected domains   39. Examples    
14. Cauchy's formula and the theorem of the mean   40. Rounding of corners    
15. Maximum Principle and Schwarz's Lemma   REFERENCES OF CHAPTER II    
16. Uniform convergence   III. Boundary Value Problems of Function Theory. Their Basic Concepts    
17. Higher order derivatives   3.1 Harmonic functions    
1.5 Representation of analytic functions by series        
18. Taylor Series        
19. Power series        
20. Uniqueness Theorem        
21. Laurent series        
22. Singularities        
23. Residue Theorem. Argument principle        
24. Point at Infinity        
25. AnalyticContinuation. Generalization of the concept of analytic function        
26. Riemann Surfaces        
REFERENCES OF CHAPTER 1        

Index of Chapters I and II