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Math 4023 Complex Analysis

创建日期：2024-04-11 16:20:38

所属分类：数学mathematics

标签： Math 4023

Complex Analysis

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Math 4023 Complex Analysis L1 Course Outline

Math 2033, Math 2043 or Math 3033 This course is for learning complex analysis, which is a useful

part of mathematics that deals with problems involving complex numbers. It is a course that prepares

students to learn more about the powerful concepts and facts that can allow us to solve problems more cleverly.

Assessment Scheme Homeworks (30%), Midterm (30%), Final Exam (40%). All records of marks or grades

will be put on Canvas. Student Learning Resources Lecture Notes and Presentation Projects: Lecture notes

(and/or presentation projects) may be downloaded from Canvas. Intended Learning Outcomes

The School of Science Intended Learning Outcomes and the Math Department Intended Learning

Out- comes will be discribed in the first class. Upon the end of the course, you should have opportunities to and

should be able to ILO A: recognize the power of abstraction and generalization, and apply logical reasoning to

investigate mathematical work with independent judgement (cf Science ILO 1, Math ILO 3P) ILO B: communicate

effectively about math to peer and teaching staffs using available equipments or presentation softwares (cf Science ILO 4,

Math ILO 4) ILO C: apply rigorous deductive reasoning to analyze and solve problems related to math profession

(cf Science ILO 6, Math ILO 2) Teaching Approach There will be lectures by the instructor on materials presented in

the lecture notes with emphasis on proofs. By understanding proofs and doing homeworks, you will gradually attain

the ability for ILO A. Also, there will be tutorials run by the teaching assistants. In these tutorials, the teaching

assistances may do more examples or assess your written works or oral presentations. For possible presentation project,

you may do it via ZOOM. These will train you for ILO B. For ILO C, there will be written assignments and tests. Tentative Course Schedule (2 hours) Syllabus. Complex Sequences and Series (3 hours) Set Descriptions and Terminologies (3 hours) Continuity and Uniform Convergence (3 hours) Stereographic Projection (3 hours) Power Series. Cauchy-Riemann Equations (4 hours) Conformal Mappings, Contour Integrals, Cauchy Theory (4 hours) Harmonic Functions and Conjugates (4 hours) Morera’s Theorem, Isolated Singularities (4 hours) Residue Theory, Infinite Products. Lecture Notes for Math 4023 (Complex Analysis) c© Department of Mathematics, HKUST 1. Complex Sequences and Series Let N,Z,R denote the set of all positive integers, integers, real numbers respectively. For (x, y) ∈ R2, we will also write it as x + iy, where i = (0, 1) is to be defined as square root of −1. The set of all complex numbers is C = {x+ iy:x, y ∈ R} with addition, subtraction, multiplication and division defined as follows. For z = (x, y) = x+ iy, the real part, imaginary part, conjugate and absolute value of z are Re z = x, Im z = y, z = x− iy and |z| = √ x2 + y2 respectively. For z = x+ iy and w = u+ iv, define z + w = (x+ u) + i(y + v), z − w = (x− u) + i(y − v), zw = (xu− yv) + i(xv + yu). Note i2 = (0 + i)(0 + i) = −1 and ww = u2 + v2 = |w|2. So for w 6= 0, we may define z/w = zw/|w|2. Next, the distance between z and w is given by |z − w|. For z = x+ iy 6= 0, the argument of z, denoted by arg z, is the angle from the positive x-axis to the ray from (0, 0) to (x, y) in radian (modulo 2pi). For example, arg i is any of the numbers pi/2+2npi, where n ∈ Z . The expressions cis θ or eiθ denote cos θ+ i sin θ. Every nonzero complex number z has a polar representation z = r cis θ, where r = |z|, θ = arg z. Let z, w be complex numbers and n be an integer. The following properties can easily be verified: (1) z + z = 2 Re z, z − z = 2i Im z, zz = |z|2, z/z = cis(2 arg z) for z 6= 0, (2) z + w = z + w, z − w = z − w, zw = z w, z/w = z/w for w 6= 0, (3) |z +w| ≤ |z|+ |w|, |z − w| ≥ ∣∣|z| − |w|∣∣, |zw| = |z||w|, |z/w| = |z|/|w| for w 6= 0, (4) for z, w 6= 0, arg(zw) = arg z + argw, arg( z w ) = arg z − argw, arg(zn) = n arg z. In analysis, reasoning involving limits are very common and important. We will begin with the concept of the limit of a sequence. (For convenience, we will abbreviate “if and only if” by “iff” or “ ⇐⇒ ” in the sequel.) Definition.A sequence {z1, z2, z3, . . .} (or in short, {zn}) converges to z ∈ C (denoted by zn → z) iff for each ε > 0, there exists Nε ∈ N such that n ≥ Nε implies |zn − z| ≤ ε (in short, lim n→∞ |zn − z| = 0.) Otherwise, the sequence is said to diverge. Examples. (1) zn = zn  converges to 0 if |z| < 1,converges to 1 if z = 1,diverges otherwise. (For |z| < 1, |zn−0| = |z|n→ 0 as n→∞. For z = 1, it is clear. For |z| ≥ 1 and z 6= 1, assume zn → w. Let ε = |z − 1|/2 > 0. Then there exists N ∈ N such that n ≥ N implies |zn−w| < ε. Now 2ε = |z− 1| ≤ |z|n|z− 1| = |zn+1− zn| ≤ |zn+1−w|+ |w− zn| < 2ε, which is a contradiction.) (2) lim n→∞ n n + z = 1, since ∣∣∣∣ nn+ z − 1 ∣∣∣∣ = |z||n+ z| = √ x2 + y2√ (n + x)2 + y2 → 0 as n→∞. Often the limit of a sequence is difficult or impossible to find. We now introduce a criterion that allows us to conclude a sequence is convergent without having to identify the limit explicitly. From math analysis course, we have the following 1 Definition. A sequence {zn} in C is a Cauchy sequence iff for each ε > 0, there exists Nε ∈ N such that m,n ≥ Nε implies |zm − zn| ≤ ε. Theorem. A sequence {zn} converges in C iff {Re zn}, {Imzn} converge in R iff {zn} is a Cauchy sequence. Proof. The case zn’s are real is proved in math analysis course. For the case zn = xn + iyn’s are complex, suppose {zn} converges to z = x + iy. Since |xn − x|, |yn − y| ≤ √|xn − x|2 + |yn − y|2 = |zn − z|, by the sandwich theorem, {xn}, {yn} converge to x, y respectively. Next suppose {xn}, {yn} converge (hence are Cauchy sequences). As |zn−zm|= √|xn − xm|2 + |yn − ym|2 ≤ |xn − xm|+ |yn − ym|, it follows {zn} is a Cauchy sequence. Finally, suppose {zn} is a Cauchy sequence. Since |xn − xm|, |yn − ym| ≤ |zn − zm|, both {xn}, {yn} are Cauchy sequences. Then {xn}, {yn} converge to some x, y respectively. Since |(xn + iyn) − (x + iy)| ≤ |xn − x|+ |yn − y|, by the sandwich theorem, {zn} converges to x+ iy. QED Definitions.A series ∞∑ k=1 zk = z1 + z2 + z3 + . . . converges iff the sequence z1, z1+z2, z1+z2+z3, . . . converges (iff Sn = z1 + z2 + . . .+ zn is a Cauchy sequence). Otherwise, the series is said to diverge. There are two simple tests for checking convergence of series, namely the term test and the absolute convergence test. The former provides a necessary condition for convergence and the latter provides a sufficient condition for convergence. Term Test. If ∞∑ k=1 zk converges, then lim n→∞ zn = limn→∞(Sn − Sn−1) = limn→∞Sn − limn→∞Sn−1 = 0. Absolute Convergence Test. If ∞∑ k=1 |zk| converges, then ∞∑ k=1 zk converges (because Tn = |z1|+|z2|+. . .+|zn| is a Cauchy sequence and for m > n, |Sm − Sn| = |zn+1 + zn+2 + . . .+ zm| ≤ |zn+1|+ |zn+2|+ . . .+ |zm| = Tm − Tn = |Tm − Tn|, forcing Sn to be a Cauchy sequence.) Examples. (1) ∞∑ k=1 1 k2 + i converges because ∞∑ k=1 ∣∣∣∣ 1k2 + i ∣∣∣∣ = ∞∑ k=1 1√ k4 + 1 ( ≤ ∞∑ k=1 1 k2 ) converges. (2) ∞∑ k=1 1 k + i diverges because Re ( ∞∑ k=1 1 k + i ) = Re ( ∞∑ k=1 k − i k2 + 1 ) = ∞∑ k=1 k k2 + 1 ( ≥ ∞∑ k=1 1 2k ) diverges. Exercises 1. For what complex values z will the following series converge (a) ∞∑ n=0 ( z 1 + z )n ; (b) ∞∑ n=1 nzn; (c) ∞∑ n=0 zn 1 + z2n ? 2. When will equality occur in the triangle inequality |w+ z| ≤ |w|+ |z|? That is, under what conditions on w and z will |w + z| = |w|+ |z|? 2 3. Establish the identity ∣∣∣∣∣ n∑ k=1 αkβk ∣∣∣∣∣ 2 = n∑ k=1 |αk|2 n∑ k=1 |βk|2 − ∑ 1≤k

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MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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