Math 5522 H: Functions of Complex Variable

• Instructor Background.
• Syllabus, Class policy and exam schedule for the course.
• No homework assigned on 01-06.
• Assigned on 01-07: Pages 25-28 Text: 4.1, 4.9, 4.14.
• Assigned on 01-08: Pages 25-28 Text: 4.24, 4.28, 4.30.
• Assigned on 01-9: Pages 25-29 Text: 4.33, 4.36
• Assigned on 01-10: Pages 58-59 Text: 5.3, 5.5, 5.12
• Assigned on 01-14: Pages 58-59 Text: 5.14, 5.15
• Assigned on 01-15: Pages 58-59 Text: 5.17, 5.19
• Assigned on 01-16: Pages 60-61 Text: 5.22, 5.25.
• Assigned on 01-17: Pages 60-61 Text: 5.27, 5.29 (see (2.10)) for definition of contractive sequence).
• Assigned on 01-21: Pages 60-61 Text: 5.30, 5.31, 5.32.
• Assigned on 01-22: Pages 60-61 Text: 5.33, 5.35, Page 101: 6.1.
• Assigned on 01-23: Pages 101: 6.2, 6.7, 6.9.
• Assigned on 01-24: Pages 101: 6.16, 6.21, 6.27.
• Assigned on 01-27: Pages 104: 6.35, 6.38, 6.41.
• Assigned on 01-28: Pages 105-106: 6.49, 6.52.
• Assigned on 01-29: Pages 136-137: 4.1, 4.12, 4.15.
• Assigned on 01-30: Pages 136-137: 4.16, 4.26, 4.28.
• Assigned on 01-31: Pages 204: 8.1, 8.2.
• Assigned on 02-03: Pages 204: 8.3, 8.4, 8.5.
• Announcement: Whatever covered through Friday, Feb 7th will be on Test 1 next Wednesday, Feb. 12th.
• Assigned on 02-04: Pages 205: 8.6, 8.9.
• Assigned on 02-05: Pages 205: 8.11, 8.12.
• Assigned on 02-06: Pages 205: 8.14, 8.15, 8.20.
• Assigned on 02-07: Pages 206: 8.26, 8.32, 8.37.
• Announcement: Test Material for 2-12 exam: From beginning of classes through Maximum Principle and Corollary 2.33 on page 171 Test.
• Assignment based on 02-10 lecture: Pages 206-209: 8.44, 8.47, 8.48
• Midterm 1 review on 02-11.
• Solution to Homework collected on 02-11.
• Assigned on 02-13: Pages 206-209: 8.50, 8.52.
• Solution to Test 1.
• Assigned on 02-14: Pages 206-209: 8.56, 8.60, 8.62.
• Problem to do with p-th root of polynomial or rational function--Problem 8.56, page 210, text
• Based on 02-17 lecture: Pages 206-212: 8.63, 8.64, 8.65.
• Assigned on 02-18: Pages 206-212: 8.68, 8.74, 8.75.
• Assigned on 02-19: Pages 212-213: 8.82, 8.83.
• Assigned on 02-20: Page 238-239: 4.2, 4.6.
• Assigned on 02-21: Page 238-239: 4.9, 4.11, 4.17.
• Assigned on 02-24: Page 239-241: 4.22, 4.26, 4.29.
• Assigned on 02-25: Page 239-242: 4.32, 4.33
• Assigned on 02-26: Page 239-242: 4.37, 4.40, 4.42.
• Assigned on 02-27: Page 286-287: 5.3, 5.6, 5.11 (you may think of mean-value equality characterization).
• Assigned on 02-28: Page 286-287: 5.14, 5.22, 5.27
• Assigned on 03-02: Page 286-287: 5.31, 5.33, 5.36.
• Assigned on 03-03: Page 286-287: 5.38, 5.46, 5.47.
• Assigned on 03-04: Page 286-287: 5.51, 5.54, 5.57.
• Assigned on 03-05: Page 286-293: 5.58, 5.79.
• Assigned on 03-06: Page 286-293: 5.85, 5.87, 5.90.
• Announcement: As mentioned in e-mail communcations, we are moving to online lectures because of the COVID-19 virus threat. I will be posting the links to these lectures below. I will also be posting homework, and as usual they are due Tuesday. As explained earlier, you can turn in the homework each Tuesday through e-mail, or put it in my box or under my door, but I want you to let me when you have submitted it. The test scheduled on March 26th 7-8:30 has been tentatively rescheduled for following Thursday, April 2, 7-8:30 pm. It will change if the University continues to suspend regular classes.
• March 16th 1st part of online lecture about Normal Family and Analytic Functions:
• March 16th 2nd part of online lecture about Zeros of analytic functions
• Based on 03-16 online lectures above: Page 286-299: 5.91, 5.92.
• March 17th first part of online lecture L’hospital Theorem For Analytic Functions
• March 17th second part of online lecture: Discrete set, discrete map and analytic function as a discrete map
• Based on 03-17 online lectures above: Page 362-373: 5.1, 5.3, 5.6
• March 18th 1st part online lecture: Consequences of Theorem 1.5
• March 18th 2nd part online lecture: Analyticity of branch of analytic f, isolated singularities.
• Based on 03-18 online lectures above: Page 362-373: 5.8, 5.13, 5.15.
• March 19th 1st part online lecture: Isolated singularities of Removable type, Laurent Expansion
• March 19th 2nd part online lecture: Isolated Singularities of the pole type.
• Based on 03-19 online lectures above: Page 362-373: 5.18 (listen to next half lecture for residues), 5.20.
• March 20th 1st part online lecture: Poles and residue calculation exercises
• March 20th 2nd part online lecture: Characterization of poles in terms of blow up, meromorphic functions and essential singularities
• Based on 03-20 online lectures above: Page 362-373: 5.22, 5.24.
• Announcement: Since the university authorities extended spring break by a week, the online material given above is accordingly shifted by one week, i.e. online lectures on 03-16 is to be understood for 03-23 instead, and so on. First homework after spring break is due March 24th. This will be collected online on Carmen, where I want you to upload .pdf files. Test 2 exam dates is scheduled as before on April 2nd, once I get confirmation that online arrangement works for all.
• March 30th online lecture part 1: Essential Singularity and Caseroti-Weirstrass Theorem
• March 30th online lecture part 2: Isolated Singularity at infinity
• Based on 03-30 online lectures: Page 362-373: 5.33, 5.39.
• March 31st online lecture part 1: Residue Theorem
• March 31st online lecture part 2: Residue Theorem and examples
• Based on 03-31 online lectures: Page 362-373: 5.42, 5.45.
• April 1st online lecture part 1: Integration using Residue Theorem
• April 1st online lecture part 2: More residue exercises
• Based on 04-01 online lectures: Page 362-373: 5.46, 5.49 ((i),(ii) only).
• April 3rd online lecture part 1: More on residue calculus application to integrals
• April 3rd online lecture part 2: Residue Calculus application to integrals
• Based on 04-03 online lectures: Page 362-373: 5.50, 5.55.
• Test related announcement: As mentioned over e-mail, we have Test 2 scheduled for pick up in the form of pdf file on the Carmen Website at 7 p.m., Thursday April 2nd, and returned as uploaded .pdf file on Carmen by 9 p.m. sharp same evening. If for some reason you experience problems with upload-download, you immediately give me a call at 312-662-9726. I prefer e-mails unless it is an emergency. I will be next to my computer during this two hour period. The test is open book, open notes, but there should be absolutely no collaboration with fellow student or any one else. The test material will be from where we left in Test 1 through what we covered in online lecture on March 27th. To be more specific, we start Chapter V, \S 3.3 (starting with Schwarz's Lemma) ending at \S 2.4 of Chapter VIII.
• Midterm 2 review on 04-01.
• Solution to second midterm on 04-02.
• April 6th online lecture part 1: Residue Application: Argument principle and Rouche's theorem
• April 6th online lecture part 2: Application of Rouche's theorem and branch covering principle
• Based on 04-06 online lectures: Pages 362-373: 5.61, 5.65, 5.68
• April 7th online lecture part 1: Branch covering Principle, open mapping Theorem and univalent analytic functions
• April 7th online lecture part 2: Inverse Function Theorem, Hurwitz Theorem normal convergence of sequences of univalent analytic functions.
• Based on 04-07 online lectures: Pages 362-373: 5.69, 5.70.
• April 8th online lecture part 1: Extended Complex Plane, Riemann Sphere.
• April 8th online lecture part 2: Functions in Extended Complex Plane and Topology.
• Based on 04-07, 04-08 lectures: Page 362-373: 5.77, 5.78, 5.81.
• April 9th online lecture part 1: Topology in extended complex plane and sequences.
• April 9th online lecture part 2: Continuous and Meromorphic functions in extended complex plane.
• Based on 04-09 lectures: Page 362-373: 5.82, 5.85.
• April 10th online lecture part 1: Discrete Subset and Meromorphic functions in the extended complex plane.
• April 10th online lecture part 2: Discrete Mapping and Open mapping Theorem
• Based on 04-10 lectures: Page 362-373: 5.84, 5.86.
• April 13th online lecture part 1: Angle between arcs and diffeomorphism
• April 13th online lecture part 2: Diffeomorphism, isogonality, conformality and anti-conformality at a point
• Based on 04-13 lecture: Pages 466-467. 6.1, 6.4 (note orientation preserving means angle theta is preserved).
• April 14th online lecture part 1: Equivalence of Diffeomorphic map between conformality and existence of complex derivative at a point.
• April 14th online lecture part 2: Equivalence of Conformal map and univalent analytic functions and examples.
• Based on 04-14 lectures: Pages 466-467. 6.8, 6.9, 6.11
• April 15th part 1: (prior prepared online video, which I could not upload earlier): Examples on conformal mapping properties of some functions
• April 15th part 2: (prior prepared online video, which I could not upload earlier)): Examples for determining mapping function between specified domains, and general conformal map from C to C
• Based on 04-15 online zoom lecture (See videos above as well): Pages 466-467. 6.12, 6.15, 6.16
• Access Password: B9+22339, April 16th Zoom online live lecture recording (Characterization of conformal maps in the Extended complex plane).
• Assigned on 04-16: Pages 466-467. 6.18, 6.20, 6.22.
• (Access Password: U0??j%6.) (Please be aware of an audio gap towards the end; thanks to some of you, it has been made up later) April 17th Zoom online live lecture recording (Mobius map, fixed points, and invariance of cross ratios).
• Assigned on 04-17: Pages 466-467. 6.23, 6.25, 6.27
• Access Password:1D^kF0#* April 20th Zoom online live lecture recording: More on Mobius map and applications; conformal invariance of harmonic functions.
• Assigned on 04-20: Pages 466-467. 6.33, 6.59
• Access Password: 8F&20T!9 April 21st Zoom online live lecture recording: Theorem on Univalence on boundary implying univalence in the interior, preliminaryLemmas by way of Riemann mapping theorem
• Assigned for 04-21: Pages 466-470. 6.60, 6.61, 6.63.
• Access Password: 4Y%$!179 April 22nd Zoom online live lecture recording: Riemann Mapping theorem between simply connected domains.
• Suggested problems based on 04-22 (Do not have to turn it in, but a good idea to look at it)): Pages 466-470: 6.64, 6.65, 6.67 (Hint: appeal to uniqueness of mapping applied to suitable subdomains)
• Access Password: 7a#8z3^E April 23nd Zoom online live lecture recording: Tying lose ends and Finals review.
• Access Password:6H#2R7$2 April 24th Zoom online live lecture recording: More Finals Review.
• Announcement: As mentioned over e-mail, we have Finals on April 28th evening, 6-8:15 p.m. It will operate under the same rules as Test 2. You will download the exam as an assignment on Carmen and then upload it back on time before 8:15 p.m. Note, you have to try uploading it at least 5 minutes earlier so that if you have a problem, you can communicate with me before the deadline for upload is gone. Late exam turn ins will not be allowed. It is a comprehensive exam covering everything we had done until and including Riemann Mapping and related theorems.

  Last updated: 04/24/20

  Questions? tanveer@math.ohio-state.edu