Ch.5:Series Representations for Analytic Functions Ch.5:Sories Representations for Analytic Functions 5.1 Sequences and Series Chapter 5:Series Representations for Analytic 5.2 Taylor Series Functions 5.3 Power Series Li,Yongzhao 5.5 Laurent Series State Key Laboratory of Integrated Services Networks,Xidian University 5.6 Zeros and Singularities December 22,2011 5.7 The Point at Infinity Ch.5:Series Representations for Analytk Functions Ch.5:Scrics Representations for Analytic Functions L5.1 Sequences and Series 5.1 Sequences and Series Introduction Definition of a Series Definition In Ch.2 we defined what is meant by convergence of a A series is a formal expression of the form co+c1+c2+...or sequence of complex numbers;recall that the sequence (An has A as a limit if |A-Anl can be made arbitrarily equivalently j.where the termscjare complex numbers. The n-th partial sum of the series,usually denoted Sn.is the sum small by taking n large enough of the firstn+terms,that is,If the sequence of For computational convenience it is often advantageous to use partial sums has a limit S,the series is said to converge. an element An of the sequence as an approximation to A or sum toand we writeAseries that does not The use of sequences,and in particular the kind of sequences converge is said to diverge associated with series,is an important tool in both the theory and applications of analytic functions One way to demonstrate that a series converges to s is to This chapter is devoted to the development of this subject show that the reminder after summing the first n+1 terms, S-∑y-09,goes to zero as n→∞
Ch.5: Series Representations for Analytic Functions Chapter 5: Series Representations for Analytic Functions Li, Yongzhao State Key Laboratory of Integrated Services Networks, Xidian University December 22, 2011 Ch.5: Series Representations for Analytic Functions Outline 5.1 Sequences and Series 5.2 Taylor Series 5.3 Power Series 5.5 Laurent Series 5.6 Zeros and Singularities 5.7 The Point at Infinity Ch.5: Series Representations for Analytic Functions 5.1 Sequences and Series Introduction In Ch. 2 we defined what is meant by convergence of a sequence of complex numbers; recall that the sequence {An}∞n=1 has A as a limit if |A − An| can be made arbitrarily small by taking n large enough For computational convenience it is often advantageous to use an element An of the sequence as an approximation to A The use of sequences, and in particular the kind of sequences associated with series, is an important tool in both the theory and applications of analytic functions This chapter is devoted to the development of this subject Ch.5: Series Representations for Analytic Functions 5.1 Sequences and Series Definition of a Series Definition A series is a formal expression of the form c0 + c1 + c2 + ··· , or equivalently ∞j=0 cj , where the terms cj are complex numbers. The n-th partial sum of the series, usually denoted Sn, is the sum of the first n + 1 terms, that is, Sn := nj=0 cj . If the sequence of partial sums {Sn}∞n=1 has a limit S, the series is said to converge, or sum to S, and we write S = ∞j=0 cj . A series that does not converge is said to diverge One way to demonstrate that a series converges to S is to show that the reminder after summing the first n + 1 terms, S − nj=0 cj , goes to zero as n → ∞
Ch.5:Series Representations for Analyti Functions Ch.5:Series Representations for Analytic Functions 5.1 Sequences and Series L5.1 Sequences and Serics Comparison and Ratio Tests Uniform Convergence Theorem (Comparison Test)Suppose that the terms cj satisfy the If we have a sequence of functions Fi(z),F2(z).F3(z)...., inequality we must consider the possibility that for some values of z the sequence converges,while for others it diverges lS≤M Similarly,a series of complex functions()may for all integers j larger that some number J.Then if the series converge for some values of z and diverge for others ∑24 converge,so does∑=oS In applying this theory to analytic functions we need a somewhat stronger notion of convergence Theorem (Ratio Test)Suppose that the terms of the series have Figure 5.1 (page 238)shows an example of 'pointwise the property that the ratios cj+/cjl approach a limit L as convergence joo.Then the series converges if L1 Ch.5:Series Representations for Analyti Functions Ch.5:Scrics Representations for Analytic Functions L5.1 Sequences and Series 5.1 Sequences and Series Uniform Convergence(Cont'd) Uniform Convergence(Cont'd) Definition The essential feature of uniform convergence is that for a The sequence {Fn(z)is said to converge uniformly to F(z) given >0,one must be able to find an integer N that is on the set T if for any >0 there exists an integer N such that independent of z in T such that the error F(z)-Fn(z)is when n>N, less than e for n>N In contrast,for pointwise convergence,N can depend upon z. IF(2)-Fn(2)<E for all z in T Of course,uniform convergence on T implies pointwise Accordingly,the series()converges uniformly to f(z)on convergence on T T if the sequence of its partial sums converges uniformly to f(z) Example 3 and 4 show that the series (/z)converges there pointwise in the open disk z<zol and uniformly on any closed subdisk <r<zol
Ch.5: Series Representations for Analytic Functions 5.1 Sequences and Series Comparison and Ratio Tests Theorem (Comparison Test) Suppose that the terms cj satisfy the inequality |cj | ≤ Mj for all integers j larger that some number J. Then if the series ∞ j=0 Mj converge, so does ∞j=0 cj Theorem (Ratio Test) Suppose that the terms of the series ∞j=0 cj have the property that the ratios |cj+1/cj | approach a limit L as j → ∞. Then the series converges if L 1 Ch.5: Series Representations for Analytic Functions 5.1 Sequences and Series Uniform Convergence If we have a sequence of functions F1(z), F2(z), F3(z), ..., we must consider the possibility that for some values of z the sequence converges, while for others it diverges Similarly, a series of complex functions ∞j=0 fj (z) may converge for some values of z and diverge for others In applying this theory to analytic functions we need a somewhat stronger notion of convergence Figure 5.1 (page 238) shows an example of ’pointwise convergence ’ Ch.5: Series Representations for Analytic Functions 5.1 Sequences and Series Uniform Convergence (Cont’d) Definition The sequence {Fn(z)}∞n=1 is said to converge uniformly to F(z) on the set T if for any ε > 0 there exists an integer N such that when n>N, |F(z) − Fn(z)| 0, one must be able to find an integer N that is independent of z in T such that the error |F(z) − Fn(z)| is less than ε for n>N In contrast, for pointwise convergence, N can depend upon z. Of course, uniform convergence on T implies pointwise convergence on T Example 3 and 4 show that the series ∞j=0(z/z0)j converges pointwise in the open disk |z| < |z0| and uniformly on any closed subdisk |z| ≤ r < |z0|
Ch.5:Series Representations for Analytic Functions Ch.5:Series Representations for Analytic Functions L5.2 Tayfor Series L5.2 Taylor Series Introduction Definition of Taylor Series The n-th-degree polynomial that matches f.ff"...f(n) In Sec.3.1,we learned the Taylor form of the polynomial at zo is Pn(z),centered at zo Pn(2)= Suppose we want to find a polynomial pn(z)of degree at most n that approximates an analytic function f(z)in a o+ae-+re-wr++oe-wr neighborhood of a point zo Definition Naturally there are differing criteria as to how well the If f is analytic at zo,then the series polynomial approximates the function We shall construct a polynomial that "looks like"f(z)at the point zo in the sense that its derivatives match those of f at 、0)+fr0(2-20)+20(-0)2+=0(-0 20 is called the Taylor series for f around zo.When zo =0,it is known as the Maclaurin series of f Ch.5:Series Representations for Analytk Functions Ch.5:Scrics Representations for Analytic Functions 5.2 Taylor Sories -5.2 Taylor Series Convergence of Taylor Series Derivatives of Taylor Series Theorem If f is analytic in the disk z-zo<R,then the Taylor series converges to f(z)for all z in this disk.Furthermore,the Theorem convergence of the series is uniform in any closed subdisk If f is analytic at zo.the Taylor series for f'around zo can be |z-20l≤R<R obtained by termwise differentiation of the Taylor series for f around zo and converges in the same disk as the series for f The theorem implies that the Taylor series will converge to f(z)everywhere inside the largest open disk,centered at zo, over which f is analytic
Ch.5: Series Representations for Analytic Functions 5.2 Taylor Series Introduction In Sec. 3.1, we learned the Taylor form of the polynomial pn(z), centered at z0 Suppose we want to find a polynomial pn(z) of degree at most n that approximates an analytic function f(z) in a neighborhood of a point z0 Naturally there are differing criteria as to how well the polynomial approximates the function We shall construct a polynomial that ”looks like” f(z) at the point z0 in the sense that its derivatives match those of f at z0 Ch.5: Series Representations for Analytic Functions 5.2 Taylor Series Definition of Taylor Series The n-th-degree polynomial that matches f, f, f, ..., f(n) at z0 is pn(z) = f(z0)+f(z0)(z−z0)+f(z0) 2! (z−z0)2+...+f(n)(z0) n! (z−z0)n Definition If f is analytic at z0, then the series f(z0)+f(z0)(z−z0)+ f(z0) 2! (z−z0)2+... = ∞ j=0 f(j)(z0) j! (z−z0)j is called the Taylor series for f around z0. When z0 = 0, it is known as the Maclaurin series of f Ch.5: Series Representations for Analytic Functions 5.2 Taylor Series Convergence of Taylor Series Theorem If f is analytic in the disk |z − z0| < R, then the Taylor series converges to f(z) for all z in this disk. Furthermore, the convergence of the series is uniform in any closed subdisk |z − z0| ≤ R < R The theorem implies that the Taylor series will converge to f(z) everywhere inside the largest open disk, centered at z0, over which f is analytic Ch.5: Series Representations for Analytic Functions 5.2 Taylor Series Derivatives of Taylor Series Theorem If f is analytic at z0, the Taylor series for f around z0 can be obtained by termwise differentiation of the Taylor series for f around z0 and converges in the same disk as the series for f
Ch.5:Series Representations for Analytic Functions Ch.5:Series Representations for Analytic Functions L5.2 Tayfor Series L5.2 Taylor Series Linearity of Taylor Series Product of Two Taylor Series Definition The Cauchy product of two Taylor series (-z)and (is defined to be the (formal)series Theorem oSe-2o户，where5 is given by If f and g be analytic functions with Taylor series f(2)=aj(=-z0)and g(=)=(z-zo around the point zo [that is aj=f)(zo)/j!andbj=g)(zo)/j Then a,n+a5-1b1+a-2b++a1b-1+b与=24y-h =0 (i)the Taylor series for cf(=).c a constant,isc() (份the Taylor series forf(a)±g(a)is∑eo(a±b)(a-2oP Theorem Let f and g be analytic functions with Taylor series f(=)=oaj(z-zo)i and g(z)=(2-z0)around the point zo.Then the Taylor series for the product fg around zo is given by the Cauchy product of these two series 4口18。+之+1老+意0G Ch.5:Series Representations for Analyti Functions Ch.5:Scrics Representations for Analytic Functions L5.2 Taylor Sories 5.3 Power Series Comments Definition of Power Series Actually,a Taylor series for an analytic function appears to be The proof of the validity of the Taylor expansion substantiates a special instance of a certain general type of series of the the claim,made in Sec.2.3,that any analytic function can form(.Such series have a name of Power Series be displayed with a formula involving z alone,and not z. 工，ory Definition A series of the form(z)is called a power series. The constants aj are the coefficients of the power series
Ch.5: Series Representations for Analytic Functions 5.2 Taylor Series Linearity of Taylor Series Theorem If f and g be analytic functions with Taylor series f(z) = ∞j=0 aj (z − z0)j and g(z) = ∞j=0 bj (z − z0)j around the point z0 [that is aj = f(j)(z0)/j! and bj = g(j)(z0)/j!]. Then (i) the Taylor series for cf(z), c a constant, is ∞j=0 caj (z − z0)j (ii) the Taylor series for f(z) ± g(z) is ∞j=0(aj ± bj )(z − z0)j Ch.5: Series Representations for Analytic Functions 5.2 Taylor Series Product of Two Taylor Series Definition The Cauchy product of two Taylor series ∞j=0 aj (z − z0)j and ∞j=0 bj (z − z0)j is defined to be the (formal) series ∞j=0 cj (z − z0)j , where cj is given by aj b0 + aj−1b1 + aj−2b2 + ... + a1bj−1 + a0bj = j l=0 aj−lbl Theorem Let f and g be analytic functions with Taylor series f(z) = ∞j=0 aj (z − z0)j and g(z) = ∞j=0 bj (z − z0)j around the point z0. Then the Taylor series for the product fg around z0 is given by the Cauchy product of these two series Ch.5: Series Representations for Analytic Functions 5.2 Taylor Series Comments The proof of the validity of the Taylor expansion substantiates the claim, made in Sec. 2.3, that any analytic function can be displayed with a formula involving z alone, and not z, x, or y Ch.5: Series Representations for Analytic Functions 5.3 Power Series Definition of Power Series Actually, a Taylor series for an analytic function appears to be a special instance of a certain general type of series of the form ∞j=0 aj (z − z0)j . Such series have a name of Power Series Definition A series of the form ∞j=0 aj (z − z0)j is called a power series. The constants aj are the coefficients of the power series
Ch.5:Series Representations for Analytic Functions Ch.5:Series Representations for Analytic Functions L5.3 Power Series L5.3 Power Series The Goal of This Section Convergence of the Power series Theorem Consider an arbitrary power series,such as For any power series)there is real number R 60 ，z2223 G+=1+i++6+… between 0 and oo,inclusive,which depends only on the coefficients aj),such that We will answering the following questions (i)the series converges for zR. Is every power series a Taylor series? The number R is called the radius of convergence of the power series Ch.5:Series Representations for Analyti Functions Ch.5:Scrics Representations for Analytic Functions 5.3 Power Series 5.3 Power Series Convergence of the Power series(Cont'd) Convergence of the Power series(Cont'd) To see the existence of the number R in Theorem for the In particular,when R=0 the power series converges only at power series we reason informally as follows: z=z0.and when R=oo the series converges for all z Consider the set of all real numbers r such that the series converges at some point having modulus r For 0R If z is replaced by (z-zo).we deduce that the region of Lemma If the power series convergesat point having modulus convergence of the general power series) must be a disk with center zo r,then it converges at every point in the disk z<r The formula for the radius of convergence R will be given in Sec.5.4 (will not be covered)
Ch.5: Series Representations for Analytic Functions 5.3 Power Series The Goal of This Section Consider an arbitrary power series, such as ∞ j=0 zj (j + 1)2 =1+ z 4 + z2 9 + z3 16 + ··· We will answering the following questions For what values of z does the series converge? Is the sum an analytic function? Is the power series representation of a function unique? Is every power series a Taylor series? Ch.5: Series Representations for Analytic Functions 5.3 Power Series Convergence of the Power series Theorem For any power series ∞j=0 aj (z − z0)j there is a real number R between 0 and ∞, inclusive, which depends only on the coefficients {aj}, such that (i) the series converges for |z − z0| R. The number R is called the radius of convergence of the power series Ch.5: Series Representations for Analytic Functions 5.3 Power Series Convergence of the Power series (Cont’d) In particular, when R = 0 the power series converges only at z = z0, and when R = ∞ the series converges for all z For 0 R If z is replaced by (z − z0), we deduce that the region of convergence of the general power series ∞j=0 aj (z − z0)j must be a disk with center z0 The formula for the radius of convergence R will be given in Sec. 5.4 (will not be covered)
Ch.5:Series Representations for Analytic Functions Ch.5:Series Representations for Analytic Functions L5.3 Power Series L5.3 Power Series Uniform Convergence Uniform Convergence(Cont'd) Knowing that the uniform limit of a sequence of continuous Uniform convergence is a powerful feature of a sequence,as functions is continuous,we can integrate this limit.In fact the the next three results show integral of the limit is the limit of integrals The first says that the uniform limit of continuous functions is Theorem itself continuous Let fn be a sequence of functions continuous on a set TCC Lemma containing the contour T,and suppose that fn converges uniformly Let fn be a sequence of functions continuous on a set T CC and to f on T.Then the sequencefn(z)dz converges tof(z)dz converging uniformly to f on T.Then f is also continuous on T Combining these results with Morera's theorem (page 210). we can prove the following theorem in the next slide 白·0+之。，急，是2风C Ch.5:Series Representations for Analyti Functions Ch.5:Scrics Representations for Analytic Functions L5.3 Power Series 5.3 Power Series Uniform Convergence(Cont'd) Uniform Convergence(Cont'd) Theorem Let fn be a sequence of functions analytic in a simple connected domain D and converging uniformly to f in D.Then f is analytic But any point within the circle of convergence lies inside every in D such a subdisk,so we can state the following Since the partial sums of a power series are analytic functions Theorem (indeed,polynomials)and since they converge uniformly in A power series sums to a function that is analytic at every point any closed subdisk interior to the circle of convergence,we inside its circle of convergence know that the limit function is analytic inside every such subdisk
Ch.5: Series Representations for Analytic Functions 5.3 Power Series Uniform Convergence Uniform convergence is a powerful feature of a sequence, as the next three results show The first says that the uniform limit of continuous functions is itself continuous Lemma Let fn be a sequence of functions continuous on a set T ⊂ C and converging uniformly to f on T. Then f is also continuous on T Ch.5: Series Representations for Analytic Functions 5.3 Power Series Uniform Convergence (Cont’d) Knowing that the uniform limit of a sequence of continuous functions is continuous, we can integrate this limit. In fact the integral of the limit is the limit of integrals Theorem Let fn be a sequence of functions continuous on a set T ⊂ C containing the contour Γ, and suppose that fn converges uniformly to f on T. Then the sequence Γ fn(z)dz converges to Γ f(z)dz Combining these results with Morera’s theorem (page 210), we can prove the following theorem in the next slide Ch.5: Series Representations for Analytic Functions 5.3 Power Series Uniform Convergence (Cont’d) Theorem Let fn be a sequence of functions analytic in a simple connected domain D and converging uniformly to f in D. Then f is analytic in D Since the partial sums of a power series are analytic functions (indeed, polynomials) and since they converge uniformly in any closed subdisk interior to the circle of convergence, we know that the limit function is analytic inside every such subdisk Ch.5: Series Representations for Analytic Functions 5.3 Power Series Uniform Convergence (Cont’d) But any point within the circle of convergence lies inside every such a subdisk, so we can state the following Theorem A power series sums to a function that is analytic at every point inside its circle of convergence
Ch.5:Series Representations for Analytic Functions Ch.5:Series Representations for Analytic Functions L5.3 Power Series L5.5 Laurent Series Relationship Between Power Series and Taylor Series Introduction Theorem Ifj(-z)i converges to f(z)in some neighborhood ofzo In this section.we wish to investigate the possibility of a series (that is,the radius of its circle of convergence is nonzero),then representation of a function f near a singularity After all,if the occurrence of a singularity is merely due to a aj= f)(zo) (0=0,1,2,) vanishing denominator,might it not be possible to express the function as something like A/(z-z0)P+g(z),where g is Consequently,()is the Taylor expansion of f(=) analytic and has a Taylor series around zo? around zo To be sure,not all singularities are of this type(recall Logz at 20=0) If a power series converges inside some circle,it is the Taylor If the function is analytic in an annulus surrounding one or series of its (analytic)limit function and can be integrated more of its singularities(note that Logz does not have this and differentiated term by term inside this circle;moreover, property,due to its branch cut).we can display its "singular this limit function must fail to be analytic somewhere on the part"according to the following theorem circle of convergence Ch.5:Series Representations for Analytk Functions Ch.5:Scrics Representations for Analytic Functions 七5.5 Laurent Serics 5.5 Laurent Series Definition of Laurent Series Definition of Laurent Series (Cont'd) Theorem Let f be analytic in the annulus r<z-zo<R.Then f can be expressed there as the sum of two series Such an expansion,containing negative as well as positive powers of (z-z0),is called the Laurent series for f in this annulus.It is usually abbreviated both series converging in the annulus,and converging uniformly in any closed ar(:-y j=-0 subannulusr p <-zo<pa R.The coefficients aj are given by Note that if f is analytic throughout the disk z-zo<R, 1f) aj= 2m1-20* G=0,±1，±2，) the coefficients with negative subscripts are zero by Cauchy's theorem,and the others reproduce the Taylor series for f where C is any positively oriented simple closed contour lying in the annulus and containing zo in its interior
Ch.5: Series Representations for Analytic Functions 5.3 Power Series Relationship Between Power Series and Taylor Series Theorem If ∞j=0 aj (z − z0)j converges to f(z) in some neighborhood of z0 (that is, the radius of its circle of convergence is nonzero), then aj = f(j)(z0) j! (j = 0, 1, 2,...) Consequently, ∞j=0 aj (z − z0)j is the Taylor expansion of f(z) around z0 If a power series converges inside some circle, it is the Taylor series of its (analytic) limit function and can be integrated and differentiated term by term inside this circle; moreover, this limit function must fail to be analytic somewhere on the circle of convergence Ch.5: Series Representations for Analytic Functions 5.5 Laurent Series Introduction In this section, we wish to investigate the possibility of a series representation of a function f near a singularity After all, if the occurrence of a singularity is merely due to a vanishing denominator, might it not be possible to express the function as something like A/(z − z0)p + g(z), where g is analytic and has a Taylor series around z0? To be sure, not all singularities are of this type (recall Logz at z0 = 0) If the function is analytic in an annulus surrounding one or more of its singularities (note that Logz does not have this property, due to its branch cut), we can display its ”singular part” according to the following theorem Ch.5: Series Representations for Analytic Functions 5.5 Laurent Series Definition of Laurent Series Theorem Let f be analytic in the annulus r < |z − z0| < R. Then f can be expressed there as the sum of two series f(z) = ∞ j=0 aj (z − z0)j + ∞ j=1 a−j (z − z0)−j both series converging in the annulus, and converging uniformly in any closed subannulus r<ρ1 ≤ |z − z0| ≤ ρ2 < R. The coefficients aj are given by aj = 1 2πi C f(ξ) (ξ − z0)j+1 dξ (j = 0, ±1, ±2,...) where C is any positively oriented simple closed contour lying in the annulus and containing z0 in its interior Ch.5: Series Representations for Analytic Functions 5.5 Laurent Series Definition of Laurent Series (Cont’d) Such an expansion, containing negative as well as positive powers of (z − z0), is called the Laurent series for f in this annulus. It is usually abbreviated ∞ j=−∞ aj (z − z0)j Note that if f is analytic throughout the disk |z − z0| < R, the coefficients with negative subscripts are zero by Cauchy’s theorem, and the others reproduce the Taylor series for f
Ch.5:Series Representations for Analytic Functions Ch.5:Series Representations for Analytic Functions L5.5 Laurent Series L5.5 Laurent Series Definition of Laurent Series (Cont'd) Definition of Laurent Series(Cont'd) Theorem Replacing (z-zo)with 1/(z-zo)in Theorem 7(page 253). one easily sees that any formal series of the form Let c(z-zo)i and (z-z)-i be any two series with the following properties: (2-zo)will converge outside some"circle of convergence"z-zo =r whose radius depends on the (i)c(-)comverges for R coefficients,with uniform convergence holding in each region 01 l2-20l2r>T 间2c-je-o)时comverges for-刘l>rand Thus termwise integration is justified by Theorem 8(page =1 255).and proceeding in a manner analogous to that of Sec (iii)r<R 5.3 we can prove the theorem in the next slide Then there is a function f(z),analytic for r<z-zol<R,whose Laurent series in this annulus is give by Ch.5:Series Representations for Analyti Functions Ch.5:Scrics Representations for Analytic Functions L5.6 Zeros and Singularities 5.6 Zeros and Singularitics Introduction Zeros of Complex-Valued Functions Definition A point zo is called a zero of order m for the function f if f is This section focuses on using the Laurent expansion to analytic at zo and f and its first m-1 derivatives vanish at z0. classify the behavior of an analytic function near its zeros and but f(m)(z0）≠0 isolated singularities A zero of a function is a point zo where f is analytic and In other words,we have f(2o)=0 f(2o)=f'(2o)=f"(2o)=…=fm-(2o)=0≠fm(2o) An isolated singularity of f is a point zo such that f is In this case the Taylor series for f around z0 takes the form analytic in some punctured disk 0<|z-zo<R but not f(z）=am(2-20)m+am+1(2-20)m+1+am+2(2-20)m+2+ analytic at zo itself or f(2）=(2-20)m[am+am+1(2-20）+am+2(z-20)2+…] where am=fm(zo)/m!≠0
Ch.5: Series Representations for Analytic Functions 5.5 Laurent Series Definition of Laurent Series (Cont’d) Replacing (z − z0) with 1/(z − z0) in Theorem 7 (page 253), one easily sees that any formal series of the form ∞j=1 c−j (z − z0)−j will converge outside some ”circle of convergence” |z − z0| = r whose radius depends on the coefficients, with uniform convergence holding in each region |z − z0| ≥ r > r Thus termwise integration is justified by Theorem 8 (page 255), and proceeding in a manner analogous to that of Sec 5.3 we can prove the theorem in the next slide Ch.5: Series Representations for Analytic Functions 5.5 Laurent Series Definition of Laurent Series (Cont’d) Theorem Let ∞j=0 cj (z − z0)j and ∞j=1 c−j (z − z0)−j be any two series with the following properties: (i) ∞ j=0 cj (z − z0)j converges for |z − z0| r and (iii) r<R Then there is a function f(z), analytic for r < |z − z0| < R, whose Laurent series in this annulus is give by ∞j=−∞ cj (z − z0)j Ch.5: Series Representations for Analytic Functions 5.6 Zeros and Singularities Introduction This section focuses on using the Laurent expansion to classify the behavior of an analytic function near its zeros and isolated singularities A zero of a function is a point z0 where f is analytic and f(z0)=0 An isolated singularity of f is a point z0 such that f is analytic in some punctured disk 0 < |z − z0| < R but not analytic at z0 itself Ch.5: Series Representations for Analytic Functions 5.6 Zeros and Singularities Zeros of Complex-Valued Functions Definition A point z0 is called a zero of order m for the function f if f is analytic at z0 and f and its first m − 1 derivatives vanish at z0, but f(m)(z0) = 0 In other words, we have f(z0) = f(z0) = f(z0) = ··· = f(m−1)(z0)=0 = f(m)(z0) In this case the Taylor series for f around z0 takes the form f(z) = am(z−z0)m+am+1(z−z0)m+1+am+2(z−z0)m+2+··· or f(z)=(z−z0)m am + am+1(z − z0) + am+2(z − z0)2 + ··· where am = f(m)(z0)/m! = 0
Ch.5:Series Representations for Analytic Functions Ch.5:Series Representations for Analytic Functions L5.6 Zeros and Singularities L5.6 Zeros and Singularitic Zeros of Complex-Valued Functions(Cont'd) Zeros of Complex-Valued Functions(Cont'd) Theorem Let f be analytic at zo.Then f has a zero of order m at zo if and only if f can be written as Notice that if f is nonconstant,analytic,and zero at zo.the f（2)=(z-20)mg(z) order of the zero must be a whole number where g is analytic at zo and g(zo)0 The function 21/2 could be said to have a zero of order 1/2 at z=0,but of course it is not analytic there Corollary If f ia an analytic function such that f(zo)=0,then either f is identically zero in a neighborhood of zo or there is a punctured disk about zo in which f has no zeros Ch.5:Series Representations for Analyti Functions Ch.5:Scrics Representations for Analytic Functions 5.6 Zeros and Singularities 5.6 Zeros and Singularitics Singularities of Complex-Valued Functions Singularities of Complex-Valued Functions (Cont'd) We know that f has a Laurent expansion around any isolated Definition singularity zo; Let f have an isolated singularity at zo,and let(1)be the Laurent expansion of f in 0<z-zo<R.Then fa)=∑a(e-2oP (1) (i)If aj =0 for all j<0,we say that zo is a removable singularity of f for,say 0<z-zo<R.(The r is zero for an isolated (ii)If a-m0 for some positive integer m but aj =0 for all singularity) j<-m,we say that zo is a pole of order m for f We can classify zo into one of the following three categories in (iii)If a-m0 for an infinite number of negative values of j,we the next slide say that zo is an essential singularity of f
Ch.5: Series Representations for Analytic Functions 5.6 Zeros and Singularities Zeros of Complex-Valued Functions (Cont’d) Theorem Let f be analytic at z0. Then f has a zero of order m at z0 if and only if f can be written as f(z)=(z − z0)mg(z) where g is analytic at z0 and g(z0) = 0 Corollary If f ia an analytic function such that f(z0)=0, then either f is identically zero in a neighborhood of z0 or there is a punctured disk about z0 in which f has no zeros Ch.5: Series Representations for Analytic Functions 5.6 Zeros and Singularities Zeros of Complex-Valued Functions (Cont’d) Notice that if f is nonconstant, analytic, and zero at z0, the order of the zero must be a whole number The function z1/2 could be said to have a zero of order 1/2 at z = 0, but of course it is not analytic there Ch.5: Series Representations for Analytic Functions 5.6 Zeros and Singularities Singularities of Complex-Valued Functions We know that f has a Laurent expansion around any isolated singularity z0; f(z) = ∞ j=−∞ aj (z − z0)j (1) for, say 0 < |z − z0| < R. (The r is zero for an isolated singularity) We can classify z0 into one of the following three categories in the next slide Ch.5: Series Representations for Analytic Functions 5.6 Zeros and Singularities Singularities of Complex-Valued Functions (Cont’d) Definition Let f have an isolated singularity at z0, and let (1) be the Laurent expansion of f in 0 < |z − z0| < R. Then (i) If aj = 0 for all j < 0, we say that z0 is a removable singularity of f (ii) If a−m = 0 for some positive integer m but aj = 0 for all j < −m, we say that z0 is a pole of order m for f (iii) If a−m = 0 for an infinite number of negative values of j, we say that z0 is an essential singularity of f
Ch.5:Series Representations for Analytic Functions Ch.5:Series Representations for Analytic Functions L5.6 Zeros and Singularities L5.6 Zeros and Singularitics Removable Singularities Removable Singularities(Cont'd) When f has a removable singularity at zo.its Laurent series takes the form f(z)=a0+a1(2-20）+a2(z-20)2+· (0<z-20l<B) Conversely,if a function is bounded in some punctured neighborhood of an isolated singularity,that singularity is Lemma removable If f has a removable singularity at zo.then Clearly,removable singularities are not too important in the (i)f(z)is bounded in some punctured circular neighborhood of theory of analytic functions 20 The concept is occasionally helpful in providing compact (ii)f(z)has a(finite)limit as z approaches zo.and descriptions of the other kinds of singularities (iii)f(z)can be redefined at zo so that the new function is analytic at zo Ch.5:Series Representations for Analyti Functions Ch.5:Scrics Representations for Analytic Functions 5.6 Zeros and Singularities 5.6 Zeros and Singularities A Pole of Order m A Pole of Order m(Cont'd) The Laurent series for a function with a pole of order m looks like a-m a-(m-1) f儿e=-0m+-0++ a-1ao+ Lemma z-20 A function f has a pole of order m at zo if and only if in some a1(2-20)+a2(z-20)2+…(a-m≠0) punctured neighborhood of zo valid in some punctured neighborhood of z0 g(z) A pole of order 1 is called a simple pole fa=2-20m Lemma where g is analytic at zo and g(zo)0 If the function f has a pole of order m at zo,then (See Example 1 on page 281) (z-z0)f(z)oo as zzo for all integers I<m,while (z-zo)"f(z)has a removable singularity at zo.In particular, f(z)oo as z approaches a pole
Ch.5: Series Representations for Analytic Functions 5.6 Zeros and Singularities Removable Singularities When f has a removable singularity at z0, its Laurent series takes the form f(z) = a0+a1(z−z0)+a2(z−z0)2+··· (0 < |z−z0| < R) Lemma If f has a removable singularity at z0, then (i) f(z) is bounded in some punctured circular neighborhood of z0 (ii) f(z) has a (finite) limit as z approaches z0, and (iii) f(z) can be redefined at z0 so that the new function is analytic at z0 Ch.5: Series Representations for Analytic Functions 5.6 Zeros and Singularities Removable Singularities (Cont’d) Conversely, if a function is bounded in some punctured neighborhood of an isolated singularity, that singularity is removable Clearly, removable singularities are not too important in the theory of analytic functions The concept is occasionally helpful in providing compact descriptions of the other kinds of singularities Ch.5: Series Representations for Analytic Functions 5.6 Zeros and Singularities A Pole of Order m The Laurent series for a function with a pole of order m looks like f(z) = a−m (z − z0)m + a−(m−1) (z − z0)m−1 + ··· + a−1 z − z0 + a0 + a1(z − z0) + a2(z − z0)2 + ··· (a−m = 0) valid in some punctured neighborhood of z0 A pole of order 1 is called a simple pole Lemma If the function f has a pole of order m at z0, then |(z − z0)lf(z)|→∞ as z → z0 for all integers l<m, while (z − z0)mf(z) has a removable singularity at z0. In particular, |f(z)|→∞ as z approaches a pole Ch.5: Series Representations for Analytic Functions 5.6 Zeros and Singularities A Pole of Order m (Cont’d) Lemma A function f has a pole of order m at z0 if and only if in some punctured neighborhood of z0 f(z) = g(z) (z − z0)m where g is analytic at z0 and g(z0) = 0 (See Example 1 on page 281)
