DOCSLIB.ORG

Families of Analytic Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Families of Analytic Functions Chapter 5 Families of Analytic Functions In this chapter we consider the linear space A(Ω) of all analytic functions on an open set Ω and introduce a metric d on A(Ω) with the property that convergence in the d-metric is uniform convergence on compact subsets of Ω. We will characterize the compact subsets of the metric space (A(Ω),d) and prove several useful results on convergence of sequences of analytic functions. After these preliminaries we will present a fairly standard proof of the Riemann mapping theorem and then consider the problem of extending the mapping function to the boundary. Also included in this chapter are Runge’s theorem on rational approximations and the homotopic version of Cauchy’s theorem. 5.1 The Spaces A(Ω) and C(Ω) 5.1.1 Definitions Let Ω be an open subset of C. Then A(Ω) will denote the space of analytic functions on Ω, while C(Ω) will denote the space of all continuous functions on Ω. For n =1, 2, 3 ..., let Kn = D(0,n) ∩{z : |z − w|≥1/n for all w ∈ C \ Ω}. By basic topology of the plane, the sequence {Kn} has the following three properties: (1) Kn is compact, ⊆ o (2) Kn Kn+1 (the interior of Kn+1), (3) If K ⊆ Ω is compact, then K ⊆ Kn for n sufficiently large. Now fix a nonempty open set Ω, let {Kn} be as above, and for f,g ∈ C(Ω), define ∞ 1 f − g d(f,g)= Kn , 2n 1+ f − g n=1 Kn where {| − | ∈ } ∅ − sup f(z) g(z) : z Kn ,Kn = f g Kn = 0,Kn = ∅ 1 2 CHAPTER 5. FAMILIES OF ANALYTIC FUNCTIONS 5.1.2 Theorem The assignment (f,g) → d(f,g) defines a metric on C(Ω). Asequence {fj} in C(Ω) is d- convergent (respectively d-Cauchy) iff {fj} is uniformly convergent (respectively uniformly Cauchy) on compact subsets of Ω. Thus (C(Ω),d) and (A(Ω),d) are complete metric spaces. Proof. That d is a metric on C(Ω) is relatively straightforward. The only troublesome part of the argument is verification of the triangle inequality, whose proof uses the inequality: If a, b and c are nonnegative numbers and a ≤ b + c, then a b c ≤ + . 1+a 1+b 1+c To see this, note that h(x)=x/(1 + x) increases with x ≥ 0, and consequently h(a) ≤ b c ≤ b c { } h(b + c)= 1+b+c + 1+b+c 1+b + 1+c . Now let us show that a sequence fj is d-Cauchy iff {fj} is uniformly Cauchy on compact subsets of Ω. Suppose first that {fj} is d-Cauchy, and let K be any compact subset of Ω. By the above property (3) of the sequence {Kn}, we can choose n so large that K ⊆ Kn. Since d(fj,fk) → 0asj, k →∞, the same is true − − ≤ − { } of fj fk Kn . But fj fk K fj fk Kn , hence fj is uniformly Cauchy on K. { } Conversely, assume that fj is uniformly Cauchy on compact subsets of Ω. Let >0 and ∞ −n { } choose a positive integer m such that n=m+1 2 <. Since fj is uniformly Cauchy ≥ − on Km in particular, there exists N = N(m) such that j, k N implies fj fk Km <, hence m m 1 f − f 1 j k Kn ≤ f − f 2n 1+ f − f 2n j k Kn n=1 j k Kn n=1 m 1 ≤ f − f <. j k Km 2n n=1 It follows that for j, k ≥ N, ∞ 1 f − f d(f ,f )= j k Kn < 2. j k 2n 1+ f − f n=1 j k Kn The remaining statements in (5.1.2) follow from the above, Theorem 2.2.17, and com- pleteness of C. ♣ If {fn} is a sequence in A(Ω) and fn → f uniformly on compact subsets of Ω, then we know that f ∈ A(Ω) also. The next few theorems assert that certain other properties of the limit function f may be inferred from the possession of these properties by the fn. The first results of this type relate the zeros of f to those of the fn. 5.1.3 Hurwitz’s Theorem Suppose that {fn} is a sequence in A(Ω) that converges to f uniformly on compact subsets of Ω. Let D(z0,r) ⊆ Ω and assume that f(z) = 0 for |z − z0| = r. Then there is a positive integer N such that for n ≥ N, fn and f have the same number of zeros in D(z0,r). 5.1. THE SPACES A(Ω) AND C(Ω) 3 Proof. Let = min{|f(z)| : |z−z0| = r} > 0. Then for sufficiently large n, |fn(z)−f(z)| < ≤|f(z)| for |z − z0| = r. By Rouch´e’s theorem (4.2.8), fn and f have the same number of zeros in D(z0,r). ♣ 5.1.4 Theorem Let {fn} be a sequence in A(Ω) such that fn → f uniformly on compact subsets of Ω. If Ω is connected and fn has no zeros in Ω for infinitely many n, then either f has no zeros in Ω or f is identically zero. Proof. Assume f is not identically zero, but f has a zero at z0 ∈ Ω. Then by the identity theorem (2.4.8), there is r>0 such that the hypothesis of (5.1.3) is satisfied. Thus for sufficiently large n, fn has a zero in D(z0,r). ♣ 5.1.5 Theorem Let {fn} be a sequence in A(Ω) such that fn converges to f uniformly on compact subsets of Ω. If Ω is connected and the fn are one-to-one on Ω, then either f is constant on Ω or f is one-to-one. Proof. Assume that f is not constant on Ω, and choose any z0 ∈ Ω. The sequence {fn −fn(z0)} satisfies the hypothesis of (5.1.4) on the open connected set Ω\{z0} (because the fn are one-to-one). Since f − f(z0) is not identically zero on Ω \{z0}, it follows from (5.1.4) that f − f(z0) has no zeros in Ω \{z0}. Since z0 is an arbitrary point of Ω, we conclude that f is one-to-one on Ω. ♣ The next task will be to identify the compact subsets of the space A(Ω) (equipped with the topology of uniform convergence on compact subsets of Ω). After introducing the appropriate notion of boundedness for subsets F⊆A(Ω), we show that each sequence of functions in F has a subsequence that converges uniformly on compact subsets of Ω. This leads to the result that a subset of A(Ω) is compact iff it is closed and bounded. 5.1.6 Definition Aset F⊆C(Ω) is bounded if for each compact set K ⊆ Ω, sup{ f K : f ∈F}< ∞, that is, the functions in F are uniformly bounded on each compact subset of Ω. We will also require the notion of equicontinuity for a family of functions. 5.1.7 Definition Afamily F of functions on Ω is equicontinuous at z0 ∈ Ω if given >0 there exists δ>0 such that if z ∈ Ω and |z − z0| <δ, then |f(z) − f(z0)| <for all f ∈F. We have the following relationship between bounded and equicontinuous subsets of A(Ω). 4 CHAPTER 5. FAMILIES OF ANALYTIC FUNCTIONS 5.1.8 Theorem Let F be a bounded subset of A(Ω). Then F is equicontinuous at each point of Ω. Proof. Let z0 ∈ Ω and choose r>0 such that D(z0,r) ⊆ Ω. Then for z ∈ D(z0,r) and f ∈F, we have − 1 f(w) − 1 f(w) f(z) f(z0)= − dw − dw. 2πi C(z0,r) w z 2πi C(z0,r) w z0 Thus 1 f(w) f(w) |f(z) − f(z0)|≤ sup − : w ∈ C(z0,r) 2πr 2π w − z w − z0 f(w) = r|z − z0| sup : w ∈ C(z0,r) . (w − z)(w − z0) But by hypothesis, there exists Mr such that |f(w)|≤Mr for all w ∈ C(z0,r) and all f ∈F. Consequently, if z ∈ D(z ,r/2) and f ∈F, then 0 | − | f(w) ∈ ≤ | − | Mr r z z0 sup : w C(z0,r) r z z0 2 , (w − z)(w − z0) (r/2) proving equicontinuity of F. ♣ We will also need the following general fact about equicontinuous families. 5.1.9 Theorem Suppose F is an equicontinuous subset of C(Ω) (that is, each f ∈F is continuous on Ω and F is equicontinuous at each point of Ω) and {fn} is a sequence from F such that fn converges pointwise to f on Ω. Then f is continuous on Ω and fn → f uniformly on compact subsets of Ω. More generally, if fn → f pointwise on a dense subset of Ω, then fn → f on all of Ω and the same conclusion holds. Proof. Let >0. For each w ∈ Ω, choose a δw > 0 such that |fn(z) − fn(w)| <for each z ∈ D(w, δw) and all n. It follows that |f(z) − f(w)|≤ for all z ∈ D(w, δw), so f is continuous. Let K be any compact subset of Ω. Since {D(w, δw):w ∈ K} is ∈ ⊆∪m an open cover of K, there are w1,... ,wm K such that K j=1D(wj,δwj ). Now choose N such that n ≥ N implies that |f(wj) − fn(wj)| <for j =1,..
Load more

Recommended publications
  • Class 1/28 1 Zeros of an Analytic Function
    Math 752 Spring 2011 Class 1/28 1 Zeros of an analytic function Towards the fundamental theorem of algebra and its statement for analytic functions. Definition 1. Let f : G → C be analytic and f(a) = 0. a is said to have multiplicity m ≥ 1 if there exists an analytic function g : G → C with g(a) 6= 0 so that f(z) = (z − a)mg(z). Definition 2. If f is analytic in C it is called entire. An entire function has a power series expansion with infinite radius of convergence. Theorem 1 (Liouville’s Theorem). If f is a bounded entire function then f is constant. 0 Proof. Assume |f(z)| ≤ M for all z ∈ C. Use Cauchy’s estimate for f to obtain that |f 0(z)| ≤ M/R for every R > 0 and hence equal to 0. Theorem 2 (Fundamental theorem of algebra). For every non-constant polynomial there exists a ∈ C with p(a) = 0. Proof. Two facts: If p has degree ≥ 1 then lim p(z) = ∞ z→∞ where the limit is taken along any path to ∞ in C∞. (Sometimes also written as |z| → ∞.) If p has no zero, its reciprocal is therefore entire and bounded. Invoke Liouville’s theorem. Corollary 1. If p is a polynomial with zeros aj (multiplicity kj) then p(z) = k k km c(z − a1) 1 (z − a2) 2 ...(z − am) . Proof. Induction, and the fact that p(z)/(z − a) is a polynomial of degree n − 1 if p(a) = 0. 1 The zero function is the only analytic function that has a zero of infinite order.
    [Show full text]
  • Topic 7 Notes 7 Taylor and Laurent Series
    Topic 7 Notes Jeremy Orloff 7 Taylor and Laurent series 7.1 Introduction We originally defined an analytic function as one where the derivative, defined as a limit of ratios, existed. We went on to prove Cauchy's theorem and Cauchy's integral formula. These revealed some deep properties of analytic functions, e.g. the existence of derivatives of all orders. Our goal in this topic is to express analytic functions as infinite power series. This will lead us to Taylor series. When a complex function has an isolated singularity at a point we will replace Taylor series by Laurent series. Not surprisingly we will derive these series from Cauchy's integral formula. Although we come to power series representations after exploring other properties of analytic functions, they will be one of our main tools in understanding and computing with analytic functions. 7.2 Geometric series Having a detailed understanding of geometric series will enable us to use Cauchy's integral formula to understand power series representations of analytic functions. We start with the definition: Definition. A finite geometric series has one of the following (all equivalent) forms. 2 3 n Sn = a(1 + r + r + r + ::: + r ) = a + ar + ar2 + ar3 + ::: + arn n X = arj j=0 n X = a rj j=0 The number r is called the ratio of the geometric series because it is the ratio of consecutive terms of the series. Theorem. The sum of a finite geometric series is given by a(1 − rn+1) S = a(1 + r + r2 + r3 + ::: + rn) = : (1) n 1 − r Proof.
    [Show full text]
  • Complex Analysis Qual Sheet
    Complex Analysis Qual Sheet Robert Won \Tricks and traps. Basically all complex analysis qualifying exams are collections of tricks and traps." - Jim Agler 1 Useful facts 1 X zn 1. ez = n! n=0 1 X z2n+1 1 2. sin z = (−1)n = (eiz − e−iz) (2n + 1)! 2i n=0 1 X z2n 1 3. cos z = (−1)n = (eiz + e−iz) 2n! 2 n=0 1 4. If g is a branch of f −1 on G, then for a 2 G, g0(a) = f 0(g(a)) 5. jz ± aj2 = jzj2 ± 2Reaz + jaj2 6. If f has a pole of order m at z = a and g(z) = (z − a)mf(z), then 1 Res(f; a) = g(m−1)(a): (m − 1)! 7. The elementary factors are defined as z2 zp E (z) = (1 − z) exp z + + ··· + : p 2 p Note that elementary factors are entire and Ep(z=a) has a simple zero at z = a. 8. The factorization of sin is given by 1 Y z2 sin πz = πz 1 − : n2 n=1 9. If f(z) = (z − a)mg(z) where g(a) 6= 0, then f 0(z) m g0(z) = + : f(z) z − a g(z) 1 2 Tricks 1. If f(z) nonzero, try dividing by f(z). Otherwise, if the region is simply connected, try writing f(z) = eg(z). 2. Remember that jezj = eRez and argez = Imz. If you see a Rez anywhere, try manipulating to get ez. 3. On a similar note, for a branch of the log, log reiθ = log jrj + iθ.
    [Show full text]
  • Smooth Versus Analytic Functions
    Smooth versus Analytic functions Henry Jacobs December 6, 2009 Functions of the form X i f(x) = aix i≥0 that converge everywhere are called analytic. We see that analytic functions are equal to there Taylor expansions. Obviously all analytic functions are smooth or C∞ but not all smooth functions are analytic. For example 2 g(x) = e−1/x Has derivatives of all orders, so g ∈ C∞. This function also has a Taylor series expansion about any point. In particular the Taylor expansion about 0 is g(x) ≈ 0 + 0x + 0x2 + ... So that the Taylor series expansion does in fact converge to the function g˜(x) = 0 We see that g andg ˜ are competely different and only equal each other at a single point. So we’ve shown that g is not analytic. This is relevent in this class when finding approximations of invariant man- ifolds. Generally when we ask you to find a 2nd order approximation of the center manifold we just want you to express it as the graph of some function on an affine subspace of Rn. For example say we’re in R2 with an equilibrium point at the origin, and a center subspace along the y-axis. Than if you’re asked to find the center manifold to 2nd order you assume the manifold is locally (i.e. near (0,0)) defined by the graph (h(y), y). Where h(y) = 0, h0(y) = 0. Thus the taylor approximation is h(y) = ay2 + hot. and you must solve for a using the invariance of the manifold and the dynamics.
    [Show full text]
  • Chapter 2 Complex Analysis
    Chapter 2 Complex Analysis In this part of the course we will study some basic complex analysis. This is an extremely useful and beautiful part of mathematics and forms the basis of many techniques employed in many branches of mathematics and physics. We will extend the notions of derivatives and integrals, familiar from calculus, to the case of complex functions of a complex variable. In so doing we will come across analytic functions, which form the centerpiece of this part of the course. In fact, to a large extent complex analysis is the study of analytic functions. After a brief review of complex numbers as points in the complex plane, we will ¯rst discuss analyticity and give plenty of examples of analytic functions. We will then discuss complex integration, culminating with the generalised Cauchy Integral Formula, and some of its applications. We then go on to discuss the power series representations of analytic functions and the residue calculus, which will allow us to compute many real integrals and in¯nite sums very easily via complex integration. 2.1 Analytic functions In this section we will study complex functions of a complex variable. We will see that di®erentiability of such a function is a non-trivial property, giving rise to the concept of an analytic function. We will then study many examples of analytic functions. In fact, the construction of analytic functions will form a basic leitmotif for this part of the course. 2.1.1 The complex plane We already discussed complex numbers briefly in Section 1.3.5.
    [Show full text]
  • Lecture Note for Math 220A Complex Analysis of One Variable
    Lecture Note for Math 220A Complex Analysis of One Variable Song-Ying Li University of California, Irvine Contents 1 Complex numbers and geometry 2 1.1 Complex number field . 2 1.2 Geometry of the complex numbers . 3 1.2.1 Euler's Formula . 3 1.3 Holomorphic linear factional maps . 6 1.3.1 Self-maps of unit circle and the unit disc. 6 1.3.2 Maps from line to circle and upper half plane to disc. 7 2 Smooth functions on domains in C 8 2.1 Notation and definitions . 8 2.2 Polynomial of degree n ...................... 9 2.3 Rules of differentiations . 11 3 Holomorphic, harmonic functions 14 3.1 Holomorphic functions and C-R equations . 14 3.2 Harmonic functions . 15 3.3 Translation formula for Laplacian . 17 4 Line integral and cohomology group 18 4.1 Line integrals . 18 4.2 Cohomology group . 19 4.3 Harmonic conjugate . 21 1 5 Complex line integrals 23 5.1 Definition and examples . 23 5.2 Green's theorem for complex line integral . 25 6 Complex differentiation 26 6.1 Definition of complex differentiation . 26 6.2 Properties of complex derivatives . 26 6.3 Complex anti-derivative . 27 7 Cauchy's theorem and Morera's theorem 31 7.1 Cauchy's theorems . 31 7.2 Morera's theorem . 33 8 Cauchy integral formula 34 8.1 Integral formula for C1 and holomorphic functions . 34 8.2 Examples of evaluating line integrals . 35 8.3 Cauchy integral for kth derivative f (k)(z) . 36 9 Application of the Cauchy integral formula 36 9.1 Mean value properties .
    [Show full text]
  • Notes on Analytic Functions
    Notes on Analytic Functions In these notes we define the notion of an analytic function. While this is not something we will spend a lot of time on, it becomes much more important in some other classes, in particular complex analysis. First we recall the following facts, which are useful in their own right: Theorem 1. Let (fn) be a sequence of functions on (a; b) and suppose that each fn is differentiable. 0 If (fn) converges to a function f uniformly and the sequence of derivatives (fn) converges uniformly to a function g, then f is differentiable and f 0 = g. So, the uniform limit of differentiable functions is itself differentiable as long as the sequence of derivatives also converges uniformly, in which case the derivative of the uniform limit is the uniform limit of the derivatives. We will not prove this here as the proof is not easy; however, note that the proof of Theorem 26.5 in the book can be modified to give a proof of the above theorem in the 0 special case that each derivative fn is continuous. Note that we need the extra assumption that the sequence of derivatives converges uniformly, as the following example shows: q 2 1 Example 1. For each n, let fn :(−1; 1) be the function fn(x) = x + n . We have seen before that (fn) converges uniformly to f(x) = jxj on (−1; 1). Each fn is differentiable at 0 but f is not, so the uniform limit of differentiable functions need not be differentiable. We shouldn't expect that the uniform limit of differentiable functions be differentiable for the fol- lowing reason: differentiability is a local condition which measures how rapidly a function changes, while uniform convergence is a global condition which measures how close functions are to one another.
    [Show full text]
  • Introduction to Complex Analysis Michael Taylor
    Introduction to Complex Analysis Michael Taylor 1 2 Contents Chapter 1. Basic calculus in the complex domain 0. Complex numbers, power series, and exponentials 1. Holomorphic functions, derivatives, and path integrals 2. Holomorphic functions defined by power series 3. Exponential and trigonometric functions: Euler's formula 4. Square roots, logs, and other inverse functions I. π2 is irrational Chapter 2. Going deeper { the Cauchy integral theorem and consequences 5. The Cauchy integral theorem and the Cauchy integral formula 6. The maximum principle, Liouville's theorem, and the fundamental theorem of al- gebra 7. Harmonic functions on planar regions 8. Morera's theorem, the Schwarz reflection principle, and Goursat's theorem 9. Infinite products 10. Uniqueness and analytic continuation 11. Singularities 12. Laurent series C. Green's theorem F. The fundamental theorem of algebra (elementary proof) L. Absolutely convergent series Chapter 3. Fourier analysis and complex function theory 13. Fourier series and the Poisson integral 14. Fourier transforms 15. Laplace transforms and Mellin transforms H. Inner product spaces N. The matrix exponential G. The Weierstrass and Runge approximation theorems Chapter 4. Residue calculus, the argument principle, and two very special functions 16. Residue calculus 17. The argument principle 18. The Gamma function 19. The Riemann zeta function and the prime number theorem J. Euler's constant S. Hadamard's factorization theorem 3 Chapter 5. Conformal maps and geometrical aspects of complex function the- ory 20. Conformal maps 21. Normal families 22. The Riemann sphere (and other Riemann surfaces) 23. The Riemann mapping theorem 24. Boundary behavior of conformal maps 25. Covering maps 26.
    [Show full text]
  • Cauchy's Theorem
    ® Cauchy’s Theorem 26.5 Introduction In this Section we introduce Cauchy’s theorem which allows us to simplify the calculation of certain contour integrals. A second result, known as Cauchy’s integral formula, allows us to evaluate some I f(z) integrals of the form dz where z0 lies inside C. C z − z0 • be familiar with the basic ideas of functions Prerequisites of a complex variable Before starting this Section you should ... • be familiar with line integrals Learning Outcomes • state and use Cauchy’s theorem • state and use Cauchy’s integral formula On completion you should be able to ... HELM (2008): 39 Section 26.5: Cauchy’s Theorem 1. Cauchy’s theorem Simply-connected regions A region is said to be simply-connected if any closed curve in that region can be shrunk to a point without any part of it leaving a region. The interior of a square or a circle are examples of simply connected regions. In Figure 11 (a) and (b) the shaded grey area is the region and a typical closed curve is shown inside the region. In Figure 11 (c) the region contains a hole (the white area inside). The shaded region between the two circles is not simply-connected; curve C1 can shrink to a point but curve C2 cannot shrink to a point without leaving the region, due to the hole inside it. C2 C1 (a) (b) (c) Figure 11 Key Point 2 Cauchy’s Theorem The theorem states that if f(z) is analytic everywhere within a simply-connected region then: I f(z) dz = 0 C for every simple closed path C lying in the region.
    [Show full text]
  • Riemann Mapping Theorem (4/10 - 4/15)
    Math 752 Spring 2015 Riemann Mapping Theorem (4/10 - 4/15) Definition 1. A class F of continuous functions defined on an open set G is called a normal family if every sequence of elements in F contains a subsequence that converges uniformly on compact subsets of G. The limit function is not required to be in F. By the above theorems we know that if F ⊆ H(G), then the limit function is in H(G). We require one direction of Montel's theorem, which relates normal families to families that are uniformly bounded on compact sets. Theorem 1. Let G be an open, connected set. Suppose F ⊂ H(G) and F is uniformly bounded on each compact subset of G. Then F is a normal family. ◦ Proof. Let Kn ⊆ Kn+1 be a sequence of compact sets whose union is G. Construction for these: Kn is the intersection of B(0; n) with the comple- ment of the (open) set [a=2GB(a; 1=n). We want to apply the Arzela-Ascoli theorem, hence we need to prove that F is equicontinuous. I.e., for " > 0 and z 2 G we need to show that there is δ > 0 so that if jz − wj < δ, then jf(z) − f(w)j < " for all f 2 F. (We emphasize that δ is allowed to depend on z.) Since every z 2 G is an element of one of the Kn, it suffices to prove this statement for fixed n and z 2 Kn. By assumption F is uniformly bounded on each Kn.
    [Show full text]
  • 18.04 S18 Topic 2: Analytic Functions
    Topic 2 Notes Jeremy Orloff 2 Analytic functions 2.1 Introduction The main goal of this topic is to define and give some of the important properties of complex analytic functions. A function f .z/ is analytic if it has a complex derivative f ¨.z/. In general, the rules for computing derivatives will be familiar to you from single variable calculus. However, a much richer set of conclusions can be drawn about a complex analytic function than is generally true about real differentiable functions. 2.2 The derivative: preliminaries In calculus we defined the derivative as a limit. In complex analysis we will do the same. Δf f .z + Δz/* f .z/ f ¨.z/ = lim = lim : Δz→0 Δz Δz→0 Δz Before giving the derivative our full attention we are going to have to spend some time exploring and understanding limits. To motivate this we’ll first look at two simple examples – one positive and one negative. Example 2.1. Find the derivative of f .z/ = z2. Solution: We compute using the definition of the derivative as a limit. .z + Δz/2 * z2 z2 + 2zΔz + (Δz/2 * z2 lim = lim = lim 2z + Δz = 2z: Δz→0 Δz Δz→0 Δz Δz→0 That was a positive example. Here’s a negative one which shows that we need a careful understanding of limits. Example 2.2. Let f .z/ = z. Show that the limit for f ¨.0/ does not converge. Solution: Let’s try to compute f ¨.0/ using a limit: f (Δz/* f .0/ Δz Δx * iΔy f ¨.0/ = lim = lim = : Δz→0 Δz Δz→0 Δz Δx + iΔy Here we used Δz = Δx + iΔy.
    [Show full text]
  • Lecture 8 Taylor and Laurent Series
    Lecture 8 Taylor and Laurent series Dr. Moataz El-Zekey Complex sequences An infinite sequence of complex numbers , , , , , denoted by , can be considered as a function defined on a set of positive integers into the complex plane. For example, we take = so that the complex sequence is Convergence of complex sequences ° Given a complex sequence , if for each positive quantity , there exists a positive integer N such that then the sequence is said to converge to the limt z. We write If there is no limit, we say that the sequence diverges . Complex sequences It is easy to show that Therefore, the study of the convergence of a complex sequence is equivalent to the consideration of two real sequences. The above theorem enables us to write whenever we know that both limits on the right exist or the one on the left exists. Infinite series of complex numbers ° An infinite series of complex numbers , , , , is the infinite sum of the sequence given by ° To study the properties of an infinite series, we define the sequence of partial sums by ° If the limit of the sequence converges to S, then the series is said to be convergent and S is its sum ; otherwise, the series is divergent . ° The sum, when it exists, is unique. ° The consideration of an infinite series is relegated to that of an infinite sequence of partial sums. Infinite series of complex numbers Hence, Σ is convergent if and only if Σ and Σ are convergent. There are many parallels with real series. Now if Σ and Σ are convergent, then → 0, → 0.
    [Show full text]