Chap 5. Series Series representations of analytic functions

1. Chap 5. Series Series representations of analytic functions 43. Convergence of Sequences and Series An infinite sequence數列 of complex numbers has a limit z if, for each positive , there exists a positive integer n0 such that tch-prob

2. The limit z is unique if it exists. (Exercise 6). When the limit exists, the sequence is said to converge to z. Otherwise, it diverges. Thm 1. tch-prob

3. An infinite series tch-prob

4. A necessary condition for the convergence of series (6) is that The terms of a convergent series of complex numbers are, therefore, bounded, Absolute convergence: Absolute convergence of a series of complex numbers implies convergence of that series. tch-prob

5. 44. Taylor Series Thm. Suppose that a function f is analytic throughout an open disk Then at each point z in that disk, f(z) has the series representation That is, the power series here converges to f(z) tch-prob

6. This is the expansion of f(z) into a Taylor series about the point z0 Any function that is known to be analytic at a point z0 must have a Taylor series about that point. (For, if f is analytic at z0, it is analytic in some neighborhood  may serve as R0 is the statement of Taylor’s Theorem) ~ Maclaurin series. z0=0的case Positively oriented within and z is interior to it. tch-prob

7. The Cauchy integral formula applies: tch-prob

8. tch-prob

9. Suppose f is analytic when and note that the 主要 原因 (b) For arbitrary z0 composite function must be analytic when tch-prob

10. The analyticity of g(z) in the disk ensures the existence of a Maclaurin series representation: tch-prob

11. Since is entire 45 Examples Ex1. It has a Maclaurin series representation which is valid for all z. tch-prob

12. Ex2. Find Maclaurin series representation of Ex3. tch-prob

13. Ex4. tch-prob

14. Ex5. 為Laurent series 預告 tch-prob

15. 46. Laurent Series If a function f fails to be analytic at a point z0, we can not apply Taylor’s theorem at that point. However, we can find a series representation for f(z) involving both positive and negative powers of (z-z0). Thm. Suppose that a function f is analytic in a domain and let C denote any positively oriented simple closed contour around z0 and lying in that domain. Then at each z in the domain tch-prob

16. where Pf: see textbook. tch-prob

17. 47. Examples The coefficients in a Laurent series are generally found by means other than by appealing directly to their integral representation. Ex1. Alterative way to calculate tch-prob

18. Ex2. tch-prob

19. Ex3. has two singular points z=1 and z=2, and is analytic in the domains Recall that (a) f(z) in D1 tch-prob

20. (b) f(z) in D2 tch-prob

21. (c) f(z) in D3 tch-prob

22. 48. Absolute and uniform convergence of power series Thm1. (1) tch-prob

23. The greatest circle centered at z0 such that series (1) converges at each point inside is called the circle of convergence of series (1). • The series CANNOT converge at any point z2 outside that circle, according to the theorem; otherwise circle of convergence is bigger. tch-prob

24. When the choice of depends only on the value of and is independent of the point z taken in a specified region within the circle of convergence, the convergence is said to be uniform in that region. tch-prob

25. then that series is uniformly convergent in the closed disk Corollary. tch-prob

26. 49. Integration and Differentiation of power series Have just seen that a power series represents continuous function at each point interior to its circle of convergence. We state in this section that the sum S(z) is actually analytic within the circle. Thm1. Let C denote any contour interior to the circle of convergence of the power series (1), and let g(z) be any function that is continuous on C. The series formed by multiplying each term of the power series by g(z) can be integrated term by term over C; that is, tch-prob

27. Corollary. The sum S(z) of power series (1) is analytic at each point z interior to the circle of convergence of that series. Ex1. is entire But series (4) clearly converges to f(0) when z=0. Hence f(z) is an entire function. tch-prob

28. Thm2. The power series (1) can be differentiated term by term. That is, at each point z interior to the circle of convergence of that series, Ex2. Diff. tch-prob

29. at all points interior to some circle , then it is the Taylor series expansion for f in powers of . 50. Uniqueness of series representation Thm 1. If a series Thm 2. If a series converges to f(z) at all points in some annular domain about z0, then it is the Laurent series expansion for f in powers of for that domain. tch-prob

30. 51. Multiplication and Division of Power Series Suppose then f(z) and g(z) are analytic functions in Their product has a Taylor series expansion tch-prob

31. Ex1. The Maclaurin series for is valid in disk Ex2. Zero of the entire function sinh z tch-prob