Chap 5. Series Series representations of analytic functions43. Convergence of Sequences and SeriesAn infinite sequence of complex numbers has a limit z if, for each positive , there exists a positive integer n0 such that

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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.

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An infinite series

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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.

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44. Taylor SeriesThm. Suppose that a function f is analytic throughout an open diskThen at each point z in that disk, f(z) has the seriesrepresentationThat is, the power series here converges to f(z)

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Positively orientedwithinand z is interior to it.~ Maclaurin series.z0=0case

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The Cauchy integral formula applies:

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(b) For arbitrary z0composite functionmust be analytic when

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The analyticity of g(z) in the disk ensuresthe existence of a Maclaurin series representation:

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45 ExamplesEx1.It has a Maclaurin series representation which is valid for all z.

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Ex2. Find Maclaurin series representation of Ex3.

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Ex4.

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Ex5.Laurent series

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46. Laurent SeriesIf a function f fails to be analytic at a point z0, we can not apply Taylors 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 domainand let C denote any positively oriented simple closed contour around z0 and lying in that domain. Then at each z in the domain

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wherePf: see textbook.

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47. ExamplesThe coefficients in a Laurent series are generally found by means other than by appealing directly to their integral representation.Ex1.Alterative way to calculate

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Ex2.

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Ex3.has two singular points z=1 and z=2, and is analytic in the domainsRecall that(a) f(z) in D1

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(b) f(z) in D2

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(c) f(z) in D3

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48. Absolute and uniform convergence of power seriesThm1.(1)

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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.

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Corollary.then that series is uniformly convergent in the closed disk

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49. Integration and Differentiation of power seriesHave just seen that a power seriesrepresents 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,

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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 entireBut series (4) clearly converges to f(0) when z=0. Hence f(z) is an entire function.

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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.

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50. Uniqueness of series representationThm 1. If a seriesThm 2. If a seriesconverges 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.

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51. Multiplication and Division of Power SeriesSupposethen f(z) and g(z) are analytic functions in has a Taylor series expansion Their product

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

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