tch-prob1 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 n 0 such that

tch-prob2 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-prob3 An infinite series

tch-prob4 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-prob5 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-prob6 This is the expansion of f(z) into a Taylor series about the point z 0 Any function that is known to be analytic at a point z 0 must have a Taylor series about that point. (For, if f is analytic at z 0, it is analytic in some neighborhood  may serve as R 0 is the statement of Taylor’s Theorem) Positively orientedwithin and z is interior to it. ~ Maclaurin series. z 0 =0 的 case

tch-prob7 The Cauchy integral formula applies:

tch-prob8

9 主要 原因 (b) For arbitrary z 0 Suppose f is analytic whenand note that the composite function must be analytic when

tch-prob10 The analyticity of g(z) in the diskensures the existence of a Maclaurin series representation:

tch-prob11 45 Examples Ex1. Since is entire It has a Maclaurin series representation which is valid for all z.

tch-prob12 Ex2. Find Maclaurin series representation of Ex3.

tch-prob13 Ex4.

tch-prob14 Ex5. 為 Laurent series 預告

tch-prob Laurent Series If a function f fails to be analytic at a point z 0, 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-z 0 ). Thm. Suppose that a function f is analytic in a domain and let C denote any positively oriented simple closed contour around z 0 and lying in that domain. Then at each z in the domain

tch-prob16 where Pf: see textbook.

tch-prob 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-prob18 Ex2.

tch-prob19 Ex3. has two singular points z=1 and z=2, and is analytic in the domains Recall that (a) f(z) in D 1

tch-prob20 (b) f(z) in D 2

tch-prob21 (c) f(z) in D 3

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

tch-prob23 The greatest circle centered at z 0 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 z 2 outside that circle, according to the theorem; otherwise circle of convergence is bigger.

tch-prob24 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-prob25 Corollary. then that series is uniformly convergent in the closed disk

tch-prob 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-prob27 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-prob28 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 Uniqueness of series representation Thm 1. If a series at all points interior to some circle, then it is the Taylor series expansion for f in powers of. Thm 2. If a series converges to f(z) at all points in some annular domain about z 0, then it is the Laurent series expansion for f in powers of for that domain.

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

tch-prob31 Ex1. The Maclaurin series for is valid in disk Ex2. Zero of the entire function sinh z