Chap 6 Residues and Poles

1. Chap 6 Residues and Poles if f analytic. Cauchy-Goursat Theorem: What if f is not analytic at finite number of points interior to C Residues. 53. Residues z0 is called a singular point of a function f if f fails to be analytic at z0 but is analytic at some point in every neighborhood of z0. A singular point z0 is said to be isolated if, in addition, there is a deleted neighborhood of z0 throughout which f is analytic. tch-prob

2. Ex1. Ex2. The origin is a singular point of Log z, but is not isolated Ex3. not isolated isolated When z0 is an isolated singular point of a function f, there is a R2 such that f is analytic in tch-prob

3. Consequently, f(z) is represented by a Laurent series and C is positively oriented simple closed contour When n=1, The complex number b1, which is the coefficient of in expansion (1) , is called the residue of f at the isolated singular point z0. A powerful tool for evaluating certain integrals. tch-prob

4. Ex4. 湊出z-2在分母 tch-prob

5. Ex5. tch-prob

6. 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 zk inside C, then Cauchy’s residue theorem tch-prob

7. Ex1. tch-prob

8. 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: tch-prob

9. Ex2. tch-prob

10. 55. Three Types of Isolated Singular points If f has an isolated singular point z0, then f(z) can be represented by a Laurent series tch-prob

11. (i) Type 1. Ex1. tch-prob

12. Ex2. (ii) Type 2 bn=0, n=1, 2, 3,…… is known as a removable singular point. * Residue at a removable singular point is always zero. tch-prob

13. * If we redefine f at z0 so that f(z0)=a0 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 z0 when it is assigned the value a0 there. The singularity at z0 is therefore removed. Ex3. tch-prob

14. (iii) Type 3: Infinite number of bn 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. tch-prob

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

16. an infinite number of these points clearly lie in any given neighborhood of the origin. tch-prob

17. 56. Residues at Poles identify poles and find its corresponding residues. Thm. An isolated singular point z0 of a function f is a pole of order m iff f(z) can be written as tch-prob

18. Pf: “<=“ tch-prob

19. “=>” tch-prob

20. Ex1. tch-prob

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

22. Ex4. tch-prob

23. 57. Zeros and Poles of order m Consider a function f that is analytic at a point z0. (From Sec. 40). Then f is said to have a zero of order m at z0. tch-prob

24. Ex1. Thm. Functions p and q are analytic at z0, and If q has a zero of order m at z0, then has a pole of order m there. tch-prob

25. Ex2. Corollary: Let two functions p and q be analytic at a point z0. Pf: Form Theorem in sec 56, tch-prob

26. Ex3. The singularities of f(z) occur at zeros of q, or try tan z tch-prob

27. Ex4 tch-prob

28. containing a point z0, then in any neighborhood N0 of z0 throughout which f is analytic. That is, f(z)=0 at each point z in N0. 58. Conditions under which Lemma : If f(z)=0 at each point z of a domain or arc Pf: Under the stated condition, For some neighborhood N of z0 f(z)=0 Otherwise from (Ex13, sec. 57) There would be a deleted neighborhood of z0 throughout which tch-prob

29. Since in N, an in the Taylor series for f(z) about z0 must be zero. 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. Thus in neighborhood N0 since that Taylor series also represents f(z) in N0. 圖解 Theorem. tch-prob

30. 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) tch-prob

31. 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 z0. tch-prob

32. Thm 1: Suppose that a function f is analytic and bounded in some deleted neighborhood of a point z0. If f is not analytic at z0, then it has a removable singularity there. Pf: Assume f is not analytic at z0. The point z0 is an isolated singularity of f and f(z) is represented by a Laurent series If C denotes a positively oriented circle tch-prob

33. Suppose that z0 is an essential singularity of a function f, and let w0 be any complex number. Then, for any positive number , the inequality Thm2. (a function assumes values arbitrarily close to any given number) (3) is satisfied at some point z in each deleted neighborhood tch-prob

34. Pf: Since z0 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, z0 is a removable singularity of g. We let g be defined at z0 so that it is analytic there, becomes analytic at z0 if it is defined there as But this means that z0 is a removable singularity of f, not an essential one, and we have a contradiction. tch-prob

35. tch-prob