Chapter 2. Analysis Functions

1. §2.1 The concept of analytic function Chapter 2. Analysis Functions §2.2 Necessary and sufficient conditions of analytic functions. §2.3 Elementary functions

2. §2.1 The concept of analytic function 1. Derivative ＆differential Def . 2.1.1 have derivative at , if exists, the limit is called as derivative of at , denoted by , or Otherwise no derivative at .

3. Also Note: The forms are similar, however the derivative of the complex functionis more complicated, strict with the differentiability.

4. Ex 1. Ex 2.

5. ① let then (1) ② let then (2)

6. For Note 1. differentiable at , but not in any nbd of . Note 2. , i.e. u & v i.e. Ref & Imf have continuous partial derivative of any order. exists.

7. Note 3. is domain, , exists Pf. , ① let , then is real. ② let , then is pure imaginary.

8. · Ex.3 not differentiable on C. · Ex.4 not exist, Note 4. In general, f involve , not exist. Properties. 1. differentiable at continuous at .

9. 2.The operations are same as real functions.

10. differentiable at or linear main part of

11. Def. exists, is the differential of at , denoted as when Derivative Differentiable

12. 2. The concept of an analytic function Def . ① analytic at , if , differentiable at any . Also, is holomorphic at is regular at ② analytic on D (domain) if analytic at any . : singular point of , if is not analytic at

13. Ex. 2.1.4 for for , not exists not differentiable ∴ analytic on . Operations hold on.(＋－ ×÷)

14. §2.2 Necessary and sufficient conditions of analytic functions. Let analytic on or C-R equation Ex. 1

15. Ex.2 C-R equation holds only at is singular point. Note: exists. ① All partial derivatives are continuous & C-R equation exist. ② diff. at & C-R equation exist.

16. Ex. then but not exist. However, C-R equation holds at nbd of exist analytic

20. Theorem 2.2.1 differentiable at any , and i.e. C-R equation Corollary.

24. Ex.2.2.3

25. Ex.2.2.5

26. Ex:2.2.6

28. Homework: P32-33: A1,A2,A3,A5,A6 *A7

29. §2.3 Elementary functions 1.Exponential function Def.

30. Property 1. i.e. ez is periodic function with elemental period - it’s one of the differences with real exponential function Property 2. however Ex. - another differences with real exponential function Property 3.

31. Property 4. If ,then ,and so for some integer n. But because both lie in where the difference between the imaginary parts of any points is less than , we have . is one -to-one. Let with , we claim the equation has a solution

32. The equation is then equivalent to the two equations This y is merely arg w, see Fig.1

34. 2. Logarithmic function w is logarithmic function of z if denoted by Let multi-value, is principal value. when

35. ∵ not continuous at (0,0) & arg(z) not continuous at (0,0) & . • analytic on . • Note: Equation of set Principal value equation. ? ?

36. Ex:

37. 3. The Power Function For in general multi-value function

39. Ex:

40. Properties (1) Periodic odd even 4. Trigonometric function & Hyperbolic function (2) (3)

41. Def. (4)

42. Hyperbolic function: Properties ①. sh z, ch z analytic on C, ②. sh z, ch z periodic with period ch z iseven, sh z isodd. ③. unbounded unbounded

43. Ex:

44. 5. Inverse Trigonometric & Hyperbolic

46. Inverse Hyperbolic

47. Homework: P33-34:A9-A13