===== Math 185 LEC3: Introduction to Complex Analysis ===== //UC Berkeley, Fall 2020// * Instructor: Peng Zhou * Email: pzhou.math@berkeley.edu * Zoom Personal Meeting ID: 881-910-2324 * Lecture: TuTh 9:30-11:00, online at [[https://berkeley.zoom.us/j/97213430388?pwd=dHJCRG5mNnhsb2tzQTBQd1A4b09LZz09 | online lecture room]], see details [[math185-2020:zoom| here]] * Office Hour: by appointment. Usually free right after class. * GSI (for office hours), see [[https://math.berkeley.edu/~tdz/185/ | here]] ==== Syllabus ==== * I will loosely follow Stein-Shakarchi's first 3 chapters, then add additional topics for the remaining time. * Grading Policy: 50% homework, two midterms each 10%, take home final 30%. === References === * Stein-Shakarchi, //Complex Analayis// * Ahlfors: //Complex analysis. An introduction to the theory of analytic functions of one complex variable// * Gamelin: //Complex Analysis (Undergraduate Text in Math) // * Schaum's outlines: complex variables. //This is a practical book, with lots of examples of easy to intermediate levels. The PDF is available at Berkeley library. // === Links === * Zoom chat channel: we also have a zoom chat channel, search for "Math 185-3: complex analysis" and join. You can contact me there. * Online discussion [[https://piazza.com/berkeley/fall2020/math185lecture3 | Piazza]]. Sign up with access code "riemann". * Online homework / exam submission: [[https://www.gradescope.com/ | Gradescope ]]. entry code 982WDY * Latex Online: https://overleaf.com. Here is a {{ math185-2020:hw1-template.tex |sample template file}} to get you started. [[ https://berkeley.zoom.us/rec/share/_6tSco_Fa7O-O2GNjD4XtxGW3BrRieJGcm1HbpVkAy-82fayfhyGZkcMu5OStCe3._g_9jpSQLA2nyitQ | video tutorial ]] Passcode: c+D+B?9M ===== Lectures ===== | Date | Reading | Content | notes | video and passcode | | Aug 27 Thu | [S] 1.1.1 [A] 1.2 | Overview of the course. Complex Numbers. | {{ math185-2020:math185_27_aug_2020.pdf |note}} | | [[draft:math185-3:lectures:sep-1|Sep 1 Tue ]] | [S] 1.1.3, 1.2.2 | Review of topology and Holomorphic Functions. |{{ math185-2020:math185_1_sep_2020.pdf | note}} | [[https://berkeley.zoom.us/rec/share/3pxTM5fq23JOQ6fXtUycZrM9HqP7aaa8hCUc_KZfz0soYL_uKpVZaf2Giv8TDMNA | video ]] %% Y^?bY700 %% | | [[draft:math185-3:lectures:sep-3|Sep 3 Thu ]] | [S] 1.2.3 | Power Series |{{ math185-2020:note_3_sep_2020.pdf |note}} |[[https://berkeley.zoom.us/rec/share/V9rbbp32itBsIcEndonDqIwifkK-ocIgdwUWS5nc4HEeJH_fjboh2YpMxG3VMgp8.KtWrQMDd10vcNSze | video ]] ##cDRb5e | | Sep 8 Tue | [S] 1.3 | Integration Along Curve | {{ math185-2020:math_185_8_sep_2020.pdf | note}} | [[ https://berkeley.zoom.us/rec/share/0jJejWRAV7l3PN7VLI2uDIbs6poX87VJcjhXFzbCxkP8qhLNNbYggeElEwwn-E1Z.sgaof7uAuEZ-XRTE?startTime=1599583308000 | video ]] vT+=b2Xi | | Sep 10 Thu | [S] 1.3, 2.1 | Finish Ch 1. Begin Goursat's Thm | {{ math185-2020:math_185_09-10.pdf | note }} | [[https://berkeley.zoom.us/rec/share/Yf5pD1neRQ04RlN6AbouWBjBZFrbrgxLE_fBh3F4lINjEsYOyxa8vSfduDtRIe3k.G3a8H30ObocYr6Vr | video ]] %% ^=AhAr58 %% | | Sep 15 Tue | [S] 2.1, 2.2 | Goursat, Cauchy theorem on disk | {{ math185-2020:math_185_09-15b.pdf | note }} | [[https://berkeley.zoom.us/rec/share/nTANA5xvshdfbqCeU03NcMQKHO9ZdBNC27vTwpTFqEIvPjgxG1VYb3WLKEB8Lnid.LOg5grsj5E3sb2lo | video]] $Cd@kAe0 | | Sep 17, Thu | [S] 2.4(a), 2.3 | Cauchy Integral Formula, and Sample Calculations | {{ math185-2020:math_185_09-17b.pdf | note }}| [[ https://berkeley.zoom.us/rec/share/eUT_yR2PE0iBUG9YluvJFibr5Lgs-bR9J1qQ9moJUOCST-0itsEQ1I3TM20JlHFh.czwooKblQQdwLpoa | video ]] eA2!V7oR | | Sep 22 Tue | [S]2.3, 2.4 | More on contour integral examples. Cauchy estimate | {{ math185-2020:math185-0922.pdf |note}} |[[https://berkeley.zoom.us/rec/share/ogj5skICg6A_i0aPRCuDPlH1UkvA4VN3XAluvt8PjlQ_1e86zq6j13Elfy02HwKJ.Kmwb9HyPDNxp8fYJ | video]] +6%m*Hsp | | Sep 24 Thu | [S] 2.4, 2.5.1 | Corollary to Cauchy integral Formula | {{ math185-2020:math_185_09-24.pdf | note}} |[[https://berkeley.zoom.us/rec/share/M50r2oJaW4C_huJQYP3gs0ubuBQQ8t7D929N32s0NbyXFit7EXW-fP5et0WaQHD9.XYGTqyW0EDaacL5- | video ]] %% ZQF.q$0& %% | | Sep 29 Tue | [S] 2.5 | Schwarz Reflection Principle, | {{ math185-2020:math185_09-29.pdf |note}}| [[ https://berkeley.zoom.us/rec/share/AOjA7c8IejYnCVNjMvZJKdcO4jIM5lvVyt3FM5hYE5id5b7iFT6mq6_bCxAZktbY.W_4xMB2rhWUazt2x| video ]] %% h3=KBA21 %% | | Oct 1 Thu | | Runge Approximation Theorem| {{ math185-2020:math185_10-01.pdf | note }} | [[https://berkeley.zoom.us/rec/share/JKb3TVJZcS1Qhy9pZ6QbJhzUBHiBMg8lAPxV-sPcBBDkREx_v91lGdOIfgRX0XA.G0BlXguq0qRWMPv8 | video ]] %% FWA46%k5 %% | | Oct 6 Tue | | {{ math185-2020:math_185-midterm-1.pdf | Midterm 1}} ([[draft:math185-3:midterm-1-review| review notes]]) | {{ math185-2020:midterm1-solution.pdf |sol'n}} | {{ math185-2020:midterm1-stat.pdf |stat}}| | Oct 13 Tue | [S] 3.1 | zero, poles and residues |{{ math185-2020:math185_10-13.pdf |note}} |[[https://berkeley.zoom.us/rec/share/66bqr8-HhMo858KTwxR2ct1W0QDUMfP8vxYYSRgHfSXc2py2aqdOd9YEcAr3flTQ.0RG1F1N8q-O9ZLTJ | video]] B?*MH1bG | | Oct 15 Thu | [S] 3.2 [A] 4.2 | residues theorem, winding number |{{ math185-2020:math185_10-15.pdf |note}} |[[https://berkeley.zoom.us/rec/share/tgyvpfynOu33LnKRYhSlfHwBRbfgDxmhUNMO_5URUrc52FZi9M6QKcpE2oprgg0o.-s_D5Ehj3M-_1O8h | video]] %% @k!6@pNt%% | | Oct 20 Tue | [S] 3.3 | classification of singularities | {{ math185-2020:math185_10-20.pdf | note}} | [[ https://berkeley.zoom.us/rec/share/S-ZsJoHevHKoVBN--lfv6vNQGkVbP0RIYpAHqpA202PoK6T_OzXKiD430kk_W89X.uhl_NXpBil6f6fXu | video ]] %% f+2&L#Po %% | | Oct 22 Thu | [S] 3.3, 3.4 | global meromorphic functions are rational, argument principle | {{ math185-2020:note_oct_22_2020.pdf |note}}| [[https://berkeley.zoom.us/rec/share/pRcKjF7CqJI81_kIr297qW54Na2itGB46L0hstggxbQlUTgzvCUBHND9Yyt-r7vc.4Slmri6gFa2F79XU|video]] ih0XF3X# | | Oct 27 Tue | [S] 3.4 | Rouche theorem, open mapping theorem | {{ math185-2020:note_oct_27_2020.pdf | note}}| [[https://berkeley.zoom.us/rec/share/1vlCgBzDS8xRq9T4gee6LvjwduWueUFIkDAbtCSSwJysFAWRgBGjr5cd7hqs0nbP.ITKgo35-gtsP9hJ1 | video]] %%4Ox&345s%% | | Oct 29 Thu | [S] 3.5 | Homotopy invariance of Contour integral | {{ math185-2020:note_oct_29_2020.pdf |note}} | [[https://berkeley.zoom.us/rec/share/iDAvJ515m4kHL5EhnzFf3xIbtEbiWXPwRUguhbIdLCvFkWfiJTfJ6F0gn1kQpVkE.IqpqndDo5gvNiWrN| video]] %%^v.S7P?Z%% | | Nov 3 Tue | [S] 3.6 | Multivalued Function and Log | {{ math185-2020:note_nov_3_2020.pdf | note}} | [[https://berkeley.zoom.us/rec/share/nmtuPZ3NQM-ILfPbo-vYV_NH0FpJEjNoqaYYtq8GIqgWxT5g0BNqpgY_7HrBtAXN.BgxvyOw56Hxz6qJa | video ]]Tt0T=D#8 | | Nov 5 Thu | [S] 3.7, [A] 4.6 | Harmonic Functions and Summary | {{ math185-2020:note_nov_5_2020.pdf | note}} | [[https://berkeley.zoom.us/rec/share/4pXUP0IpPu1wA_g_82L3y6WPmOR3Q7NuvyBDwsiv4JTlT6_YCo9pSlX1MYwbKI-c.0unuuK5TZ5hTSSse | video ]] %%PY+0MQ*c%% | | Nov 10 Tue | | Midterm 2 | | [[draft:math185-3:mt2-stat|stat]] | | Nov 12 Thu | | Review Midterm 2 | {{ math185-2020:note_nov_12_2020.pdf | note }} | [[https://berkeley.zoom.us/rec/share/A5-0T4ZaPplcYVsEudOfnmH0Md7DuA9fyapFGJZT5I-28ca6_wp32kIlvykoddtt.gkMMlS4LRY9HNAFj | video]] %%8#W#6Z0O%% | | Nov 17 Tue | [A] Ch5 | section 1 and 2, partial fraction, Mittag-Leffler problem | {{ math185-2020:note_nov_17_2020.pdf | note}} | [[https://berkeley.zoom.us/rec/share/9r0UNmVw4HiGGD0TX8x62uCSyoE7FNlWR8Ay4SiuPCNxcfv0QUX1K1wXLF86Nr5k.FGtS19g-vpSJq492 | video ]] 0WxX%$K7 | | Nov 19 Thu | [A] Ch5 | section 2.1, 2.2 Infinite Product| {{ math185-2020:note_nov_19_2020.pdf |note}} | [[https://berkeley.zoom.us/rec/share/236rJpcE-SBIvU48QAf2LYyZ7wVIbFFr4xzySHEpBhSnF-gk6mC0inWdlZHf8feM.ojgWfh0F9TySmsuw | video ]] %% LV&5rj$6 %% | | Nov 24 Tue | [A] Ch 5 | section 5, Normal Family | {{ math185-2020:note_nov_24_2020.pdf | note}}| [[https://berkeley.zoom.us/rec/share/bA-ZChxmwDTubPV0UNssBZZYDBvfZRgqOxhG5wKW_NIE6KkJghCxEIwX5DYmjGiU.9irB_hBoddB6eE0R | video ]] A9Ce%=yR | | Dec 1 Tue | [A] Ch 5 | section 5, Normal Family, Arzela-Ascoli Thm | {{ math185-2020:note_dec_1_2020.pdf |note}}| [[ https://berkeley.zoom.us/rec/share/sf7BDGCyhkovTbm21W1AJ4D7LXTgANfxxzwp26rbuAA361rSSViqie6B-F_u2xPz.Z9_PGWLux-6-Eg0_ | video ]] aSk5?Sb2 | | Dec 3 Thu | [A] Ch 6.1 [S] Ch 8 | Riemann Mapping Theorem | {{ math185-2020:note_dec_2_2020.pdf | note}} | [[https://berkeley.zoom.us/rec/share/aXlcL0VA88PFF4kR4DTCsXyk3aJgZRozCdfgGxpmMviVoWNmmhjG7ZJJdhRFKazj.64fBTRc3dJgrWrNI | video ]] %% ^?a71a4M %% | | {{ math185-2020:math_185_final.pdf |Final Exam}} | [[draft:math185-3:final-review| review]] | Dec 15(Tue) 12:00noon - Dec 17(Thu) 12:00 noon | {{ math185-2020:math_185_final-solution.pdf |solution}}| ==== Homework ==== | HW 1 | Due 09/10 11:59pm | 2, 7, 16(a,c,e), 17, 22 | {{ math185-2020:ch_1_exercise.pdf | in Ch 1}}. | {{ math185-2020:hw1_solution.pdf |solution}} [[draft:math185-3:hw1-tex| tex ]]| | HW 2 | Due 09/17 11:00pm | 10, 11, 13, 18, 25 | {{ math185-2020:ch_1_exercise.pdf | in Ch 1}}. | {{ math185-2020:hw2-sol.pdf | solution}} [[draft:math185-3:hw2-tex| tex ]]| | HW 3 | Due 09/25 10:00pm | 1, 2, 4, 5, 6 | {{ math185-2020:ch2_exercises.pdf | Ch 2}}| {{ math185-2020:hw3-sol.pdf | solution}} | | HW 4 | Due 10/2 10:00pm | 7,8,9,11,12 | {{ math185-2020:ch2_exercises.pdf | Ch 2}}| {{ math185-2020:hw4-sol.pdf |solution}}| | HW 5 | Due 10/23 10:00pm | 1,2,3,7 and [[draft:math185-3:hw5-extra|this]] | {{ math185-2020:ch3-exercises.pdf | Ch 3}}| {{ math185-2020:hw5_solution.pdf | solution}}| | HW 6 | Due 10/30 10:00pm | [[draft:math185-3:hw6]] | | {{ math185-2020:math185-hw6-sol.pdf |solution}}| | HW 7 | Due 11/6 10:00pm | [[draft:math185-3:hw7]] | | {{ math185-2020:math185-hw7.pdf |solution}}| | HW 8 | Due 11/16(Mon) 10:00pm | 9,10,14,15,16 [[draft:math185-3:hw8| hint]]| {{ math185-2020:ch3-exercises.pdf | Ch 3}} |{{ math185-2020:hw8_sol.pdf | solution}} | | HW 9 | Due 11/30(Mon) 10:00pm | [[draft:math185-3:hw9]] | | {{ math185-2020:hw9_sol.pdf |solution}} |