1. Chapter 3 Integral of Complex Function §3.1 Definition and Properties §3.2 Cauchy Integral Theorem §3.3 Cauchy’s Integral Formula §3.4 Analytic and Harmonic Functions

2. Review

3. §3.1 Definition and Properties 1. Def. C smooth (or piecewise smooth) f : C→ C.

4. C —— path of integral f —— integrand Z —— integration variable the limit is called the integral of f along C, denoted by. If C is closed, we can write.

5. 2. Evaluation

6. TH 3.1.1 (3.1.2) Corollary: (3.1.4)

7. Ex.1 center of a circle radius

8. 3. Properties ①②③ ④ ⑤ the length of C(Pf. P38) ⑥

9. ① ② Ex.

10. ③ Note: I independent of integration path.

11. Ex.3.1.2

12. Ex.3.1.3

13. Note: integration of f(z) dependent on integration path.

14. §3.2 Cauchy Integral Theorem —— continuous C-R equation TH.3.2.1(Cauchy TH)

15. TH 3.2.2 P42 upper limit lower limit

16. TH 3.2.3 P45

17. Def.3.2.1 Properties ① G anti derivative of f on D G analytic on D ② G 1 and G 2 anti derivative of f on D G 1 =G 2 +constant on D. TH 3.2.5 (Fundamental Theorem of Contour Integral)

18. Ex.3.2.1

19. Ex.3.2.2

20. Generalized Cauchy theorem in multi-connected domains TH 3.2.5 D multi connected with multi closed contours Γ, f(z) analytic in D and on Γ.

21. - Deformation Theorem

22. Closed Deformation Theorem Ex.3.2.3 Solutions:

24. Homework: P59-60: A1-A7

25. §3.3 Cauchy’s Integral Formula ＆ High Order Derivative Analysis:

26. 1. Cauchy’s Integral Formula: (TH 3.3.1) Pf. ∵ f(z) continuous at z 0,

28. Note 1. f(z) on D depend on f(z) on C D: domain 2. f =g analytic on C f =g on D 3. f: → C analytic.

29. Ex:

30. Solution:

33. 2. Existence of higher derivative TH 3.3.2. f analytic on C & on D, Pf. n=1

35. Note. f(z) analytic on D f (n) (z) exist on D & analytic on D. n=1,2,… -the difference with real function

36. Ex:

38. §3.4 Analytic and Harmonic Function Def. real harmonic on D, if is called harmonic function on D. Def.u, v harmonic on D, v is harmonic conjugate of u if

39. Note. v harmonic conjugate of u u harmonic conjugate of v i.e. u+iv analytic on Dv+iu analytic on D Properties: TH.3.4.2

40. (3). v harmonic conjugate of u -u harmonic conjugate of v i.e. u+iv analytic on D v-iu analytic on D (4). v harmonic conjugate of u on D u harmonic conjugate of v on D u, v constants on D. (5). v 1,v 2 harmonic conjugate of u on D v 1 =v 2 + constant on D. Pf. u+iv 1 analytic on D, u+iv 2 analytic on D i(v 1 -v 2 ) analytic on Dv 1 -v 2 analytic on D (real) v 1 -v 2 =constant.

41. Question: Does u have a harmonic conjugate (u x =v y, u y =-v x ) on D? Does there exist an analytic f :D →C, u=Re f ? (v=Im f ) Ans. No in general. yes if D is simply connected.

42. D simply connected domain, u harmonic on D, Similarly,

43. Ex.3.4.2 Solution:

44. Ex.3.4.3 method2 method1

45. Homework: P60-61: A8-A17