1. 1 Chap 6 Residues and Poles Cauchy-Goursat Theorem: if f analytic. What if f is not analytic at finite number of points interior to C Residues. 53. Residues z 0 is called a singular point of a function f if f fails to be analytic at z 0 but is analytic at some point in every neighborhood of z 0. A singular point z 0 is said to be isolated if, in addition, there is a deleted neighborhood of z 0 throughout which f is analytic. Singular points C 殘值 除了那點 Z 0 之外的小圈圈 ( 半徑為  ) 之內 f 都是可解析的

2. 2 Ex1. Ex2.The origin is a singular point of Log z, but is not isolated Ex3. not isolatedisolated When z 0 is an isolated singular point of a function f, there is a R 2 such that f is analytic in 0

3. 3 Consequently, f(z) is represented by a Laurent series and C is positively oriented simple closed contour When n=1, The complex number b 1, which is the coefficient of in expansion (1), is called the residue of f at the isolated singular point z 0. A powerful tool for evaluating certain integrals. R.O.C.

4. 4 湊出 z-2 在分母 Ex4. 4

5. 5 Ex5. 0

6. 6 More on Cauchy Integral Formula (1) Simply-Connected and Multiply-Connected Simply Connected Multiply Connected C

7. 7 More on Cauchy Integral Formula (2) Simply-Connected and Multiply-Connected

8. 8 More on Cauchy Integral Formula (3) Simply-Connected and Multiply-Connected Then connection to Taylor Series…….

9. 9 Why (chap4)

10. 10 More on Cauchy Integral Formula (4) Simply-Connected and Multiply-Connected C C C1C1 C C1C1 C2C2

11. 11 54. Residue Theorems Thm1. Let C be a positively oriented simple closed contour. If f is analytic inside and on C except for a finite number of (isolated) singular points z k inside C, then Cauchy’s residue theorem

12. 12 Ex1.

13. 13 分解大突破 展開法 係數比較法 因式分解法

14. 14 Thm2: If a function f is analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contour C, then 展開法 係數比較法 因式分解法 分別找 k 個 singular points 的 c-1(Residue) 1 個 z=0 的 residue g(z) 2 Z -1 Z1Z1 0

15. 15 Thm2:If a function f is analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contour C, then Pf:

16. 16 Ex2.

17. 17 55. Three Types of Isolated Singular points If f has an isolated singular point z 0, then f(z) can be represented by a Laurent series

18. 18 (i) Type 1. Ex1.

19. 19 Ex2. (ii) Type 2 b n =0, n=1, 2, 3,…… is known as a removable singular point. * Residue at a removable singular point is always zero.

20. 20 * If we redefine f at z 0 so that f(z 0 )=a 0 define Above expansion becomes valid throughout the entire disk * Since a power series always represents an analytic function Interior to its circle of convergence (sec. 49), f is analytic at z 0 when it is assigned the value a 0 there. The singularity at z 0 is therefore removed. Ex3.

21. 21 (iii) Type 3: Infinite number of b n is nonzero. is said to be an essential singular point of f. In each neighborhood of an essential singular point, a function assumes every finite value, with one possible exception, an infinite number of times. ~ Picard’s theorem.

22. 22 has an essential singular point at where the residue an infinite number of these points clearly lie in any given neighborhood of the origin. Ex4.

23. 23 an infinite number of these points clearly lie in any given neighborhood of the origin.

24. 24 56. Residues at Poles identify poles and find its corresponding residues. Thm.An isolated singular point z 0 of a function f is a pole of order m iff f(z) can be written as

25. 25 Pf: “<=“

26. 26 “=>”

27. 27 Ex1.

28. 28 Ex3. Need to write out the Laurent series for f(z) as in Ex 2. Sec. 55.

29. 29 Ex4.

30. 30 57. Zeros and Poles of order m Consider a function f that is analytic at a point z 0. (From Sec. 40). Then f is said to have a zero of order m at z 0.

31. 31 Ex1. Thm. Functions p and q are analytic at z 0, and If q has a zero of order m at z 0, then has a pole of order m there.

32. 32 Ex2. Corollary: Let two functions p and q be analytic at a point z 0. Pf: Form Theorem in sec 56,

33. 33 Ex3. The singularities of f(z) occur at zeros of q, or try tan z

34. 34 58. Conditions under which Lemma : If f(z)=0 at each point z of a domain or arc containing a point z 0, then in any neighborhood N 0 of z 0 throughout which f is analytic. That is, f(z)=0 at each point z in N 0. Pf:Under the stated condition, For some neighborhood N of z 0 f(z)=0 Otherwise from (Ex13, sec. 57) There would be a deleted neighborhood of z 0 throughout which arc Z0 N N0 0 f(z) 連續周圍 2D, 3D

35. 35 f(z 0 ) 0 z0z0 0 z0z0 0 z0z0 z 0 z0z0 z 0

36. 36 Since in N, a n in the Taylor series for f(z) about z 0 must be zero. Thus in neighborhood N 0 since that Taylor series also represents f(z) in N 0. 圖解 Theorem.If a function f is analytic throughout a domain D and f(z)=0 at each point z of a domain or arc contained in D, then in D.

37. 37 Corollary:A function that is analytic in a domain D is uniquely determined over D by its values over a domain, or along an arc, contained in D. Example: along real x-axis (an arc)

38. 38 Sect. 59 Reference, study for strengthen your theory background Is not covered in final exam

39. 39 59. Behavior of f near Removable and Essential Singular Points Observation : A function f is always analytic and bounded in some deleted neighborhood of a removable singularity z 0.

40. 40 Thm 1:Suppose that a function f is analytic and bounded in some deleted neighborhood of a point z 0. If f is not analytic at z 0, then it has a removable singularity there. Pf:Assume f is not analytic at z 0. The point z 0 is an isolated singularity of f and f(z) is represented by a Laurent series If C denotes a positively oriented circle

41. 41 Thm2.Suppose that z 0 is an essential singularity of a function f, and let w 0 be any complex number. Then, for any positive number, the inequality is satisfied at some point z in each deleted neighborhood (a function assumes values arbitrarily close to any given number) (3)

42. 42 Pf:Since z 0 is an isolated singularity of f. There is a throughout which f is analytic. Suppose (3) is not satisfied for any z there. Thus is bounded and analytic in According to Thm 1, z 0 is a removable singularity of g. We let g be defined at z 0 so that it is analytic there, becomes analytic at z 0 if it is defined there as But this means that z 0 is a removable singularity of f, not an essential one, and we have a contradiction.

43. 43 Cauchy Integral 補充 1. Cauchy-Goursat Theorem: if f analytic. 12 2. 3 z z X z0z0 f(z) 3.

44. 44 湊出 z-2 在分母 Ex4. 4

45. 45 for C: Ans:2πi·3 Ans:0