| CHAPTER Ⅰ. ELEMENTS OF THE THEORY Ⅰ. GENERAL PRINCIPLES. ANALYTIC FUNCTIONS 1. Definitions 2. Continuous functions of a complex variable 3. Analytic functions , 4. Functions analytic throughout a region 5. Rational functions 6. Certain irrational functions 7. Single-valued and multiple-valued functions Ⅱ. POWER SERIES WITH COMPLEX TERMS. ELEMENTARY TRANSCENDENTAL FUNCTIONS 8. Circle of convergence 9. Double series 10. Development of an infinite product in power series 11. The exponential function 12. Trigonometric functions 13. Logarithms 14. Inverse functions: arc sin z, arc tan z 15. Application to the integral calculus 16. Decomposition of a rational function of sin z and cos z into simple elements 17. Expansion of Log (1+z) 18. Extension of the binomial formula Ⅲ. CONFORMAL REPRESENTATION 19. Geometric interpretation of the derivative 20. Conformal transformations in general 21. Conformal representation of one plane on another plane 22. Riemann's theorem 23. Geographic maps 24. Isothermal curves EXERCISES CHAPTER Ⅱ. THE GENERAL THEORY OF ANALYTIC FUNC-TIONS ACCORDING TO CAUCHY Ⅰ. DEFINITE INTEGRALS TAKEN BETWEEN IMAGINARY LIMITS 25. Definitions and general principles 26. Change of variables 27. The formulae of Weierstrass and Darboux 28. Integrals taken along a closed curve 31. Generalization of the formulae of the integral calculus 32. Another proof of the preceding resultS Ⅱ. CAUCHY'S INTEGRAL. TAYLOR'S AND LAURENT'S SERIES. SINGULAR POINTS. RESIDUES 33. The fundamental formula 34. Morera's theorem 35. Taylor's series 36. Liouville's theorem 37. Laurent's series 38. Other series 39. Series of analytic functions 40. Poles 41. Functions analytic except for poles 42. Essentially singular points 43. Residues Ⅲ. APPLICATIONS OF THE GENERAL THEOREMS 44. Introductory remarks 45. Evaluation of elementary definite integrals 46. Various definite integrals 47. Evaluation of r(p) r(1-p) 48. Application to functions analytic except for poles 49. Application to the theory of equations 50. Jensen's formula 51. Lagrange's formula 52. Study of functions for infinite values of the variable Ⅳ. PERIODS OF DEFINITE INTEGRALS 53. Polar periods 54. A study of the integral fozdz//1-z2 55. Periods of hyperelliptic integrals 56. Periods of elliptic integrals of the first kind EXERCISES CHAPTER Ⅲ SINGLE-VALUED ANALYTIC FUNCTIONS CHAPTER Ⅳ ANALYTIC EXTENSION CHAPTER Ⅴ ANALYTIC FUNCTIONS OF SEVERAL VARIABLES EXERCISES INDEX |
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