1. Chapter 20 Complex variables

4. 20.2 Cauchy-Riemann relation A function f(z)=u(x,y)+iv(x,y) is differentiable and analytic, there must be particular connection between u(x,y) and v(x,y)

5. Chapter 20 Complex variables

7. 20.3 Power series in a complex variable

8. Chapter 20 Complex variables 20.4 Some elementary functions

9. Chapter 20 Complex variables

11. 20.5 Multivalued functions and branch cuts A logarithmic function, a complex power and a complex root are all multivalued. Is the properties of analytic function still applied? (A) (B) (A) (B)

12. Chapter 20 Complex variables Branch point: z remains unchanged while z traverse a closed contour C about some point. But a function f(z) changes after one complete circuit. Branch cut: It is a line (or curve) in the complex plane that we must cross, so the function remains single-valued.

13. Chapter 20 Complex variables

14. (A) (B)

15. Chapter 20 Complex variables 20.6 Singularities and zeros of complex function

16. Chapter 20 Complex variables

20. 20.10 Complex integral

21. Chapter 20 Complex variables

25. 20.11 Cauchy theorem

26. Chapter 20 Complex variables

28. 20.12 Cauchys integral formula

29. Chapter 20 Complex variables

31. 20.13 Taylor and Laurent series Taylors theorem:

32. Chapter 20 Complex variables

35. How to obtain the residue ?

36. Chapter 20 Complex variables

37. 20.14 Residue theorem

38. Chapter 20 Complex variables Residue theorem:

39. Chapter 20 Complex variables

40. 20.16 Integrals of sinusoidal functions

41. Chapter 20 Complex variables

42. 20.17 Some infinite integrals

43. Chapter 20 Complex variables

44. For poles on the real axis:

45. Chapter 20 Complex variables Jordans lemma

46. Chapter 20 Complex variables

47. 20.18 Integral of multivalued functions

48. Chapter 20 Complex variables