Math 5583, Fall 2021, Prof. D. Hejhal This page will be used for general announcements and the homework listing. A syllabus for the course is appended at the very end. *** You are responsible for knowing the information given there. *** ----------------------------------------------------------------------- LAST UPDATE: 12-26-21 ----------------------------------------------------------------------- Final Exam Tally: 125,123,123,120,117,116,114; 107,107,107,105,105,104,98,95,89; 86; x ; x (n=17) ----------------------------------------------------------------------- Course Grades are being submitted to registrar Monday morning 12-27. {I decided to not count the in-class exercise in the point total articulated in the Syllabus; the exercise was just an experiment for me.} ======================================================================= ======================================================================= MINI-TEST 3 tally (n=18): 20,20,20,20-,19,18,18,18; 17,16,16, 16,15,15; 14,14,12; 9; ======================================================================= The 5583 EXAM grade tally is: (n=18) 98,95,94,94,93,91,90,88; 88,86,86,76,69,69; 64,59,54; 44; Flip a coin at 88. ------------------------------------------------------------------ THERE IS NEW HOMEWORK: see below! ======================================================================= IMPORTANT NOTE: Check "Handouts" for an important handout about Cauchy-Goursat Theorem, etc. At least skim it. If you are a math major, it is strongly recommended that you study it! There are 3 beautiful proofs in it -- which can be *fully appreciated* now that we have CIT, generalized CIT, CIF, generalized CIF, and CIF for derivatives in hand (and are using same). ========================================================================= I WILL NOW START LISTING ASSIGNMENTS HERE ON THIS WEBSITE AS WELL AS ON CANVAS. PLEASE CHECK BOTH! ----------------------------------------------------------------------- See the "File" section on Canvas for the mp4 zoom recordings of the classroom lectures (as far as we have them). I temporarily discontinued the use of zoom and zoom recordings beginning October 11. Resumption of Zoom will be influenced by Covid conditions at UM and in the surrounding Twin Cities area. See also the SUPP files for important health info. They articulate the health policy of this class so long as we are in-person. (If you have any Covid related issues that arise, e.g. as to attendance, it is recommended that you get in touch with Prof H promptly.) ========================================================================= [restated from Fall 2017] GETTING ANOTHER 'FRIENDLY' BOOK IS HIGHLY RECOMMENDED One of the most highly recommended supplemental sources on complex analysis is W. Kaplan's, Advanced Calculus, the OLDER editions. Kaplan's book contains a 100-page chapter on "Functions of a Complex Variable" which is very readable. ================================================================================= DAILY REPORT and DIARY Nov 24 (wed) Proved Casorati-Weierstrass thm p.257 (thm 3). Then turned to book secs 17 (the Riemann sphere) and 77. Developed Res(f; infinity) and the CRT applied to exterior of Jordan curve C. One has Res(f; oo) = - c_{-1} or equivalently p.236(6) {this formula is only used rarely}. Did some examples with this type of CRT. Nov 29 (mon) Review of Res(f; infinity). Then did exam prob 5(a) using "cookbook" residue formulas from book. Did two prototypical examples of integrals along either (0,oo) or (-oo,oo). See, eg, sec 86 (example) and prob 3; then sec 87 example with denom (x^2 + a^2) and p.273 prob 2. Secs 86+87 are very important. I also mentioned the Weierstrass M-test for uniform conv of improper integrals; then Leibnitz's rule for improper integrals; then the oft-used trick of differentiating with respect to parameter "a" (e.g. as in x^2 + a^2). Dec 1 (wed) Additional comments on exam, especially the logic in prob 4(a) of waiting to multiply by extraneous powers of z. Began some examples from past final exams of _classifying_ point z_0 as removable, pole [of order m], or essential. See secs 78+79 (but more subtle). Use of the power series \Phi(z) in assessing orders of zeros and poles (see p.248 thm 1, p.251 thm 1). On integrals, did p.273 prob 8 using CRT; then Sec 88 (Jordan's lemma); then quickly set up p.265 prob 9. Dec 3 (fri) Finished p.265 prob 9. Then did Sec 91 (example) by letting x = y^2 and imitating Figure 109 in sec 90 with CRT. {Recommended reading sec 91 but this is optional!} Also did integral of (log x)/(x^2 + a^2) on (0,oo) using the same Figure plus CRT. Showed a slick trick that made this last integral almost trivial. Dec 6 (mon) Did [important] Sec 92; one example; then Sec 89, p.276 [Integral of (sin x)/x]. Started argument principle in Sec 93; see p.290(8) with no "P" term. Then stated Rouche's Theorem in Sec 94; did one example like p.291 example 1. {Will prove the argument principle, in both forms A and B, as well as Rouche's theorem on Wed Dec 8.} ================================================================================== HOMEWORK ASSIGNMENTS "H" problems are to be handed in. The other problems are just recommended ones that you should know how to do ("Rec"). Do not hand those in! Ass. #1 (Due: WED, Sept 22, 4pm -- as ONE, legible, pdf file on Canvas) "H" sec 6 #9; sec 9 #9; sec 11 #6; sec 12 #4(bc); sec 20 #4; sec 24 #3(a), #3(b). "Rec" sec 5 3,4; sec 6 7; sec 9 5,10; sec 11 3; sec 12 1--3,5,7; sec 14 2,3; sec 18 5; sec 20 2,6,7; sec 24 1,2. Ass. #2 (Due: MON, Oct 11, 4PM -- on Canvas, like Ass. #1) "H" sec 26 #6; sec 30 #13 ; sec 33 #10; sec 38 (variant of #3) Derive the cos(z_1 + z_2) formula by multiplying everything out the long way; sec 39 #17 [show all work, do not refer to sec 38]; sec 40 #1(b) [do not use 113(4); show all work]; sec 40 #5. "Rec" sec 18 10-11; sec 26 1-5,7; sec 27 1-3; sec 30 1-10; sec 33 1-8; sec 36 2; sec 38 5-9; sec 39 6; sec 40 3. Ass. #3 (Due: WED, Oct 27, 11:30pm -- on Canvas, as one big PDF file) "H" sec 46 #8 (feel free to use: 'any definite integral can be treated as improper'); sec 47 #5; sec 49 #5 (review idea of sec 49 #2 and use 'definite integral treated as improper'); sec 53 #1(c)(f), 4; sec 57 #4, 10. "Rec" sec 27 e.g. 1; sec 29 e.g. 5; sec 33 e.g. 1-3; sec 34 e.g. 3; sec 36 e.g. 2; sec 38 e.g. 9; sec 39 e.g. 6,10,11; sec 40 e.g. 1(a)(d),2; sec 42 e.g. 4; sec 43 e.g. 5,6; sec 46 e.g. 2,3,9,12(a); sec 47 e.g. 4; sec 49 e.g. 2; sec 53 e.g. 2; sec 57 e.g. 1-3. Ass. #4 (Due: WED, Nov 10, 3pm -- on Canvas, as a single PDF file) {Very short hand-in HW assignment.} 1. Let f(z) = z / (1 + z^3)^{2}. Let z_0 = 0. (a) Draw the domain on which f(z) is analytic. (b) Expand f(z) as a Taylor series about z = z_0 and state the series' radius of convergence. 2. Let g(z) = Log ( 8 + z^3 ). Let z_0 = 0. (a) Draw the domain on which g(z) is analytic. (b) Expand g(z) as a Taylor series about z = z_0 and state the series' radius of convergence. Ass. #5 (Due: MON, Nov 22, 11pm -- on Canvas, as a single PDF file) Good practice for exam. Two hand-in problems, plus 1 extra credit. 1. Consider the curve |z| = 4 counterclockwise. Let f(z) = sin(2z) over [(z-1)^2 times (z+3)]. Find the integral of f(z)dz using the Cauchy residue theorem. 2. See problem 5(a) on p.254 of textbook. Take the curve to be |z-2| = 3 counterclockwise. Calculate the integral of tan(z)dz using the Cauchy residue theorem. 2(*) [2 pts, extra credit] Same curve, but find the integral of [tan(z)]^2 dz. "Rec" (do not hand in) sec 77 #1, #2 Ass. #6 (Due: FRI, Dec 10, 3:30PM -- on Canvas, as a single PDF file) 3 problems from the textbook. 1) p.273 (sec 88) Prob #6. Show your work; use of Jordan's lemma is not allowed. 2) p.287 (sec 92) Prob #5. Show your work; use CRT. Do not use Leibnitz's rule. 3) p.294 (sec 94) Prob #8. Use Rouche's theorem; explain your reasoning. ================================================================================= COURSE SYLLABUS Math 5583; Fall 2021; Prof. Hejhal ----------------------------------- Due to ongoing Covid-19 issues, Fall semester can be expected to present some difficulties and surprises (e.g, with class modality). The course structure will therefore need to retain a certain flexibility to, in the words of Provost Croson, gracefully pivot. The syllabus will be kept reasonable and will, for this semester, correspond to what I would call an "expedited course" in complex analysis. The instructor will strive -- through any chaos -- to run a class that is both enjoyable and valuable for students. Due to participant health concerns, SAFETY and PRUDENCE will need to be kept front and center as we move forward. (Please see the Covid-19 link given below and the additional files on Canvas.) To address the possible need for student absences/accommodations, the [initial] plan will be for most of the lectures to be simultaneously Zoom-recorded for later viewing on Canvas. (The room assignment for Math 5583 has been made with this recording capability in mind.) -------------------------------------------------------------------------------- Our Initial Syllabus ==================== WELCOME to COMPLEX ANALYSIS, Math 5583, Fall 2021 (in-person modality) Instructor: Prof. D. A. Hejhal Office: Vincent Hall 220 Phone: 625-4557; Email: hejhal@umn.edu Time of Class: MWF 12:20 - 1:10 pm Room: Tate B65 (basement level of bldg) Initial course website: www.math.umn.edu/~hejhal/ (click on 5583) Important Announcements are posted on this website. Office Hours: on Zoom at least initially; time to be announced Textbook: "Complex Variables and Applications", McGraw-Hill, 9th edition, by J.W. Brown and R.V. Churchill (= a very standard text) -------- Prerequisite: two semesters of second year calculus, including the basics of infinite series AND Taylor series as presented, e.g., in Math 2283 or 3283. A familiarity with trigonometry and the algebra of complex numbers is presupposed from either high school or earlier UM courses. Good knowledge of line integrals and Green's theorem from multivariable calculus will prove vital after week 4. EVERY YEAR(!!) some students find that they have gotten in over their heads by enrolling in Math 5583 -- due to a lack of proper knowledge of the *prerequisite* material. It is best to recognize such a situation EARLY and to _seriously_ consider dropping the class if it cannot be promptly addressed. [The level of difficulty in this course is by-and-large strictly increasing.] Math 5583 is a standard introductory course on complex analysis, i.e., functions of one complex variable. It includes: algebraic aspects of complex numbers; analytic functions; corresponding examples of elementary functions; the Cauchy integral theorem and Cauchy integral formula; maximum modulus principle; Taylor and Laurent series; isolated singularities; residues and their applications; and some geometric aspects of mappings [i.e., transformations] w = f(z). Math 5583 is a 5000-level (advanced undergraduate) course; it is _not_ a calculus class. Complex analysis is a subject in mathematics characterized by its logical unity and palpable aesthetics. The lectures will present 'core' material in an optimized fashion which will NOT necessarily correspond to successive sections in the textbook. It is assumed that students will read along in the text (correlating things, based on pointers given by the instructor). For exams and mini-tests, students will need to demonstrate an ability to work with concepts, e.g., in APPLYING theorems to specific problems. This semester, there will be a de-emphasis on "giving rigorous proofs for everything". Current plans are for ALL exams and mini-tests to be take-home and open book. The textbook, which is a classic and generally quite readable, will be seen to provide an important supplement to what is presented in lecture. We will occasionally omit certain sections; and, use handouts from other sources. Preliminary Schedule (approximate only!!) =========================================== Week Dates Material 1 8 Sep - 10 Sep chap 1 + some preliminaries 2 13 Sep - 17 Sep setting the stage + start chap 2 3 20 Sep - 24 Sep in chap 2 4 27 Sep - 1 Oct end chap 2 (see also sec 115) + chap 3 5 4 Oct - 8 Oct chap 3 + start chap 4 6 11 Oct - 15 Oct chap 4 7 18 Oct - 22 Oct chap 4 8 25 Oct - 29 Oct chap 4 (conclude), chap 5 9 1 Nov - 5 Nov chap 5 10 8 Nov - 12 Nov last remarks on chap 5; chap 6 11 15 Nov - 19 Nov chap 6 12 22 Nov - 24 Nov (Wed.) chap 7 (Thanksgiving week) 13 27 Nov - 3 Dec chap 7 14 6 Dec - 10 Dec end chap 7; selected topics from ch. 8+9 15 13 Dec - 15 Dec (Wed.) selected topics from ch. 8+9 Homework: "H" problems (hand-in) + recommended ones (do not hand in) Master List at http://www.math.umn.edu/~hejhal/ Final Exam: the official time is Mon, 20 Dec, 8-10 AM, according to Onestop. If the final exam is take-home and open book, the timeframe for doing the exam will be revised (and announced well in advance). COURSE GRADING STRUCTURE: Hand-In Homework 15% {there will also be recommended HW problems!} Assessments (several mini-tests and 1 bigger test) 60% Final Exam (with possible oral supplement) 25% -------------------------------------------------------------- Total: 100% NOTES: In the absence of a compelling reason, there will be no make-up Assessments. Partly depending on if we get a HW grader, the 15 and 25 [above] may be revised up/down by several points. Having several mini-tests during the semester has proven to be an important student diagnostic as to what level of performance is required on the two 'full-fledged' exams. Professor Hejhal adheres to all departmental and CSE rules concerning incompletes and academic dishonesty. Consistent with this, grades of INC will be given only very sparingly. Also, any 'take-home' work on exams or mini-tests that is submitted under the Honor Code may need to be explained to Prof Hejhal, typically orally, with a possible revised grade then assigned. SPECIAL LINK: the following supplement for Fall Semester was prepared by the FCC in the University Senate and is recommended by the Provost's office. Please read it, since we will be adhering to it. https://docs.google.com/document/d/1sc_wcOe3fmhVcAvaoyoJaKbTxKL7rdh699BlbWrGYBA/edit See also: https://safe-campus.umn.edu/return-campus/get-the-vax/ and the "SUPP" files in the File Section of Canvas.