====== Math 185: Complex Analysis ====== //UC Berkeley, Spring 2023, Section 5. Course Number: 23829// Lecture: MWF 10:00A-10:59A at Etcheverry 3107 Final Exam: Tue, May 9 • 3:00P - 6:00P • Etcheverry 3107 **Instructor: ** Peng Zhou, pzhou.math@berkeley.edu\\ Office hour: 753 Evans Hall, MWF 11-12, \\ ** GSI ** Magda Hlavacek, mhlava@math.berkeley.edu \\ Office hour:1061 Evans Hall, M-F: 1-3pm ==== Syllabus ==== * Stein-Shakarchi's first 3 chapters + additional topics if time allows * Grading Policy: 40% homework, two midterms 20%, final 40%. (You can drop one midterm and weighing final by 50%). * two lowest homework will be dropped === References === * [S] Stein-Shakarchi, //Complex Analayis// * [A] Ahlfors: //Complex analysis. An introduction to the theory of analytic functions of one complex variable// * [G] Gamelin: //Complex Analysis (Undergraduate Text in Math) [[https://link.springer.com/book/10.1007/978-0-387-21607-2 | springer-link]]// ==== Links ==== * [[https://discord.gg/gfwBQy6NPa | Discord Discussion]], informal place to ask questions. * [[https://www.gradescope.com | Gradescope]]: Entry Code 8NZ5WW * [[https://berkeley.zoom.us/j/96924942825?pwd=V3ZOMStGdm5sSWRSc0kweDJaaEtRdz09 | Zoom Link ]] Lectures will be recorded and posted * [[https://forms.gle/EHKHRQahW35j6Zbv5 | Anonymous Google Feedback form]] * [[./s/ | Students Homepages]] A place where you can share your lectures notes, and homework solutions (after the due date), so that other people can benefit. ==== Resources ==== * Richard Borcherds's [[https://www.youtube.com/playlist?list=PL8yHsr3EFj537_iYA5QrvwhvMlpkJ1yGN | youtube lectures]] {{ :halmos.jpg?nolink&400 |}} ===== Journal ===== | Lecture # | Date | Reading | Content | Video | | Lecture 1 | Jan 18 | [S] 1.1.1 [A] 1.2 | Overview, and review of complex number | [[https://berkeley.zoom.us/rec/share/IH3zP6F7r5ROd-7ZQL7DeneJzxyVelMSSe3DUgoKzC1jDKscFCgUP8vgHknB3LFa.EcGOcx9lBh2eSaOP | video]] | | Lecture 2 | Jan 20 | [S] 1.1.3, 1.2.2 | Review of topology | [[ https://berkeley.zoom.us/rec/share/YazNl7Enc21CejfjX6vdIH9GZjINEoNh2-ssNGi31HhYH7Wl5X20hTZO0XUPSOr1.njEFgftJXEFwsAq9 | video]] | | Lecture 3 | Jan 23 | [S] 1.2.2 | Holomorphic Functions and Examples | {{ :math185-s23:note_jan_22_2023_lecture_3_definition_of_holomorphic_functions.pdf |note}}, [[https://berkeley.zoom.us/rec/share/gl1Ts78JYrgNaOgRFJ77kYSQXCAisLNbG3iRELwHjgSpZXhsxi7N4iIfLc6nfpHD.8fpxAfvVxJa3kqaH|video]] | | Lecture 4 | Jan 25 | [G]1.4 - 1.8, [S] 1.2.3 | More examples and Series | [[https://www.youtube.com/watch?v=Ed1NpQAztbE&list=PL8yHsr3EFj537_iYA5QrvwhvMlpkJ1yGN&index=4 | Borcherds]],[[https://berkeley.zoom.us/rec/share/yM_6Uy4otEE8PRPcNB5TU06VbbOSLA9ind9ZsG_9-UVmX24hqCu6D7_f1Crldcd9.nv0Ux5_cQBCBqF6q|video]],{{ :math185-s23:note_jan_24_2023_lecture_4.pdf |note}} | | Lecture 5 | Jan 27 | [S] 1.3 | Contour Integral | [[ https://berkeley.zoom.us/rec/share/FZkv6Ym3tM5IuMl8nMaGl-5DQWvpojvk4ynF0MOzqou1J6h2N7MnrPQw2H3L2rRq.hnmBuhVSqlbfnL3W | video]] {{ :math185-s23:note_jan_26_2023_lecture_5.pdf |note}} | | Lecture 6 | Jan 30 | [S] 2.1 | Goursat's Theorem |[[https://berkeley.zoom.us/rec/share/N7IUrVClNQ7vw3zbDrAZ31Ef_sqnh3NieXqXOFZSlGzxwNDdivghR6ytlmsJq478.AlblG9xMVly_K--k|video]] | | Lecture 7 | Feb 1 | [S] 2.2, 2.3 | Cauchy’s theorem in a disc | [[https://berkeley.zoom.us/rec/share/behmjjM19MP05bO5lUYleW8e5-wZOKy9-Br3RxHmrvuIhiWOoNYzKehVpjB1ABda.WUDehb5ah_8cuCfL | video]] | | Lecture 8 | Feb 3 | [S] 2.3, 2.4 | Some Integrals Examples | [[https://berkeley.zoom.us/rec/share/pyyiqS9RgHPlBCnZYVy-C3Z4m4th7xh5nSTKYwMorgVPuVo47dnAuevJ9IwkoIyk.Nw9zYS_uRzzWOiHK | video]] | | Lecture 9 | Feb 6 | [S] 2.4, | Existence of Taylor expansion. Liouville Theorem | [[https://berkeley.zoom.us/rec/share/GUbtyN6GyakbW_kPkQnrT1SPKfcS9Aepr5Dv_-BOHa4BFifh4M6UtjckF_jZVSZ5.G9PboYwPIXf0OVZ3 | video]] | | Lecture 10 | Feb 8 | [S] 2.5 | Uniqueness of extension, Morera Theorem | [[https://berkeley.zoom.us/rec/share/gUl1rwlnUrQQWwaUzRJApfjJQkPmAMrIncaNyKOjfsIgwh53y7fDt6vUQDO37P76.a5nYpP_vwuQUM--p | video]] | | Lecture 11 | Feb 10 | [S] 2.5, 3.1 | Schwarz reflection, Runge Approximation. Zero and Poles | [[https://berkeley.zoom.us/rec/share/ct89u2tz3p7X4wVuIruNWQn4gPMBb3OiBZCYMGfEYBgAFJ7zOobjKkgmBZ10S6qE.AmW-D2_iD5Ymbvsw | video]] | | Lecture 12 | Feb 13 | [S] 3.2 | more examples on Residue Formula | [[https://berkeley.zoom.us/rec/share/-Xg-393pwpn-35DGzwfc3Z4dnmmVMY_97I17gE_tiBYGdEoTq5_yRx6MOMimi8FD.M3Gg8vJl2ZMDLNKR | video]] | | Lecture 13 | Feb 15 | [S] 3.3 | Classification of Singularities, Riemann sphere | [[https://berkeley.zoom.us/rec/share/Vqgs4NRwugOmuZYyT2kJjV9eLqh6GEtR29PpQTZPTIjqyF4cSYjKY5eclJrVvEL2.8Y-6n00x9evpOubN | video]] | | Lecture 14 | Feb 17 | midterm 1 | [[mt1-review]], {{ :math185-s23:note_feb_16_2023_midterm_1_sol.pdf |solution}}| {{ :math185-s23:math_185-2020-midterm-1.pdf |past version}}, {{ :math185-s23:2020-midterm1-solution.pdf |sol'n}} | | no class | Feb 20 | | | | | Lecture 15 | Feb 22 | [S] 3.4 | Argument Principle | [[https://berkeley.zoom.us/rec/share/TtzeU56kjxdRqFOp712VZRponpgzRXD6mcvtVLXYmOmz1d1E9Wja4fK9RlUsU9Ly.bTGjKbYAFoDyG738 | video]] | | Lecture 16 | Feb 24 | [S] 3.4 | Rouché Thm and Open mapping thm | [[https://berkeley.zoom.us/rec/share/yA9OlBmr5SqBhwQ6djGEvA8zo_ua3QP4Wv4B9WvHCpaVqV1ocoXUA56aPLse0zxc.V3yP-2HPhQiK-NBM | video]] | | Lecture 17 | Feb 27 | [S] 3.7, 4.1 | Mean Value Thm. Fourier Transform | [[https://berkeley.zoom.us/rec/share/vKFBc0ZNn1YXHh1lGdGdN-_oLu5pAOI6rEgfSaqFS99v-8eoKhznmhKN8Wje_Sn-.RFX5lSyJJpD-LHgb | video]] | | Lecture 18 | Mar 1 | [S] 4.2 | Fourier Transform | [[https://berkeley.zoom.us/rec/share/IRv--V54KRETWI4kxEDIehnmChKMTXf8KTq-486uzjZu54WbInbaeGV5FEW0oAo1.uG3QU9_OsS2uf2XJ | video]] | | Lecture 19 | Mar 3 | [S] 4.3 | Poisson summation, Thm 4.3.1 | [[https://berkeley.zoom.us/rec/share/zUkX13siE9y3ZOIHzxUAquEWwrl3wuEbGBWgM38HWGzpxs7drUXFPvBCyYnHd0my.HOqadbesjt7xilbi | video]] | | Lecture 20 | Mar 6 | [S] 4.3 | Paley-Wiener | [[https://berkeley.zoom.us/rec/share/8XSlUpPcxjTf2VL0ucEeG3N-v3DlEs7UtInBwcPRZfJRcqL4kLGtglZdSj4SkoeY.W_hxZjCcclAEfQ3C | video]] | | Lecture 21 | Mar 8 | [S] 5.1, 5.2 | Jensen's Formula | [[ https://berkeley.zoom.us/rec/share/NSkHn7WM0nvjqQfF_FjTOEU_rpPHTb5cODwYpsqO8ZZCtkoUmibK4yewXe0C7Gx9.q5PLY8SdV41BfIwA | video]] | | Lecture 22 | Mar 10 | [S] 5.2, | Growth Order | [[https://berkeley.zoom.us/rec/share/DrEuM8OcorN6cyakgMsoKsuwSHYTt_Xxa9GpmgbW4M-rtuh8Z5jV60oNGuSNjeHD.FT8BFNtXPYr4WEJA | video]] | | Lecture 23 | Mar 13 | [S] 5.3 | Infinite Product | [[https://berkeley.zoom.us/rec/share/hYZEpD_OreMX-QP2sVH_Xxb_wuv3u4fVkJNscfkTfB0d0lj86lB7sp-TlvmNCuCA.8IzLoy9eHO8yZj-t | video]] | | Lecture 24 | Mar 15 | [S] 6.1 | Gamma function | [[https://berkeley.zoom.us/rec/share/z9YDza8UAEiWClRCP0ls1qSm3kC-VqqcNPayS6aG4VOdQ1uOVofDpZNmGXMVtcg.nMks95A3kvYHLX-S | video]] | | Lecture 25 | Mar 17 | [S] 6.1 | Gamma function | [[https://berkeley.zoom.us/rec/share/KjXWOA8DFdFZk3AoE3B47agCZTYG1_IjGgRCne7OeXZnbKKitQ-M2zS1IpEEpjro.R3EAU6nJ6ITFiVcv | video]] | | Lecture 26 | Mar 20 | [S] 6.1 | Finishing up Gamma function | [[https://berkeley.zoom.us/rec/share/NQJkP4IJHe9rmOHeuUjIY016EcXm9mqZaN-ukq2KkEfgop5GuBjNXuGdOmsTQmQa.dNxU1ylz4iKbmuJ5 | video]] | | Lecture 27 | Mar 22 | | Review | [[ https://berkeley.zoom.us/rec/share/QFXBpOgX7fX7Ap5pCSuRq9AgOpYMv4uVTVC96FfLfEXnFeyRD-zE4EANaMtRflTz._LII5bnYcIjSuPmj | video ]] | | Lecture 28 | Mar 24 | midterm 2 | {{ :math185-s23:midterm_2_review.pdf | review and sample problems}} | {{ :math185-s23:note_mar_22_2023_midterm_2_solution.pdf |solution}} | | no class | Mar 27 | | | | | no class | Mar 29 | | | | | no class | Mar 31 | | | | | Lecture 29 | Apr 3 |[G] 9.1 | Schwarz Lemma. $Aut(\mathbb{D})$. | [[https://berkeley.zoom.us/rec/share/JmxSqw-JvMUgYEVBku9CdodRadhgSQ1IRma3qLMyDbB2BB6CPa9mIKiwQKxoI_T9.UoPCJe32OBjnw0iv | video]] | | Lecture 30 | Apr 5 | [G] 9.2, 9.3 | Pick's Lemma, Hyperbolic Geom | [[https://berkeley.zoom.us/rec/share/qGErNjqc64L5-1rrwji3gSWc0gPHkUQ9UUiWCFpWzc-gqFYGlERpk4L2Pxx0U0x2.nqpsEIufXU0DmvV_ | video]] | | Lecture 31 | Apr 7 | [G] 9.3 | Hyperbolic Geom | [[https://berkeley.zoom.us/rec/share/phU9eGrc-L-Np4us1zwM3U72xZ49o0wMdtNelQQdqxNlbxrYd-ZQE8smZuzBdK5g.rYT1kRejLWMHEIiM | video]] | | Lecture 32 | Apr 10 | [G] 11.1 | mappings to unit disk and upper half-plane |[[https://berkeley.zoom.us/rec/share/_J-mHAkF-BB1LJ7gw6ocAzmfLoQsM_F7o8Btfy6cXLYE3coWb6-qJlokbBqth4wr.npFWu5Q4onfFwdnn|video]] | | Lecture 33 | Apr 12 | [G] 11.2, 11.3 | Riemann mapping | [[https://berkeley.zoom.us/rec/share/TBC3haNH2Gl3WHmJ9Ge9d2yhBcR3hJTkJ-pLWjvWeJt8x8ugZtrowOonelEfZxlm.zCqneuFF6B-5b8qo | video]] | | Lecture 34 | Apr 14 | [G] 11.3 | The Schwarz-Christoffel Formula | [[https://berkeley.zoom.us/rec/share/Sv7C6bnuXnMdapvjjPMK9C6Sdmu9f2F0Lm0k0f91u7Gkqtb_LH2NRWTNK6fmIfR9.SaotB68BuZGikNWj | video]] | | Lecture 35 | Apr 17 | [S] Ch 8.4, Ch9.1 | Elliptic Integral and Doubly periodic function | [[https://berkeley.zoom.us/rec/share/L1U3Y2YbDh1_OGfgdFdIobwLd203j0EMPznV50iL3JobsJbWXz2z_vrJjktDYkKu.zpt-7OQESP1_Xo6k | video]] | | Lecture 36 | Apr 19 | [S] 9.1.2 | Weierstrass P-function | [[https://berkeley.zoom.us/rec/share/c13mtVXuTfb5ZxrypK40PEWhvqBfYhTlR4gxu1KifNpI8YqQ8ntg6RlMMaTriyPQ.7WDeNYSdYq_M2c3- | video ]] | | Lecture 37 | Apr 21 | [S] 9.1 | more on Weierstrass P-function | [[https://berkeley.zoom.us/rec/share/8rOCDfC3hdBbRVOwqQgGHkvgHiWbQGxdrVoxsPFSvEkQSrvMyRq-Y3rGEDNg5FzD.r5ymh7kJwYMADKdZ |video]] | | Lecture 38 | Apr 24 | [S] 9.2 | Eisenstein Series | [[ https://berkeley.zoom.us/rec/share/GvwCy4wcX6J2_7UYOpDbCsBTHeicrV3uEFyDUSvSCW6Jc9-RkSxPUy31QzrQVcTH.qvsbRfkNie45sWno | video]] | | Lecture 39 | Apr 26 | | Review 1 | [[ https://berkeley.zoom.us/rec/share/NKBw-aBkHRYkSgrA2By0SIYNkgnWuomZRiq2_2IDFglCAr4xncRq_jKcI05ReOnY.w93ilDjkd9tNCnuJ | video ]] | | Lecture 40 | Apr 28 | | Review 2 | [[https://berkeley.zoom.us/rec/share/ldSi7iV4jh7dSCW54qy2Ypj3i7qv7-LBRxBsX_yHVaErFJ987kSZfZLZuwE3y5RK.p-Ah9fXoBvxPCQ4x|video]] | | Final | May 9th | {{ :math185-s23:final_with_solution.pdf |solution}} | | Final Zoom Office Hour: May 6th, 9:30PM. ==== Homeworks ==== | HW # | Due Date | Problems | Solutions | | HW 1 | Due Jan 23 8pm | {{ :math185-s23:ch_1_exercise.pdf |Stein Ch 1 Ex}}: 1,2,3,7,8 | {{ :math185-s23:note_jan_23_2023_hw1.pdf |solution}} | | HW 2 | Due Jan 30 8pm | {{ :math185-s23:ch_1_exercise.pdf |Stein Ch 1 Ex}}: 10, 11, 13, 15, 16(a,c,e) | {{:math185-s23:hw2.pdf |solution}} | | HW 3 | Due Feb 07 8pm | {{ :math185-s23:ch_1_exercise.pdf |Ch 1}}: 25, {{ :math185-s23:ch2_exercises.pdf | Ch2}}: 1,2,3,4 | {{ :math185-s23:hw3.pdf |solution}}, {{ :math185-s23:hw3-1.pdf |errata}} | | HW 4 | Due Feb 13 8pm | Stein {{ :math185-s23:ch2_exercises.pdf | Ch2}}, 6,7,8,10,12 |{{ :math185-s23:math185_hw4.pdf | solution}} | | HW 5 | Due Feb 22 8pm | Stein {{ :math185-s23:ch3-exercises.pdf | Ch3}}, 1,2,3,7, [[hw5-extra]] | {{ :math185-s23:hw5_solution.pdf |solution}} | | HW 6 | Due Mar 1 8pm | Stein {{ :math185-s23:ch3-exercises.pdf | Ch3}}, 9, 12, 16, [[hw6-extra]] | {{ :math185-s23:hw6_sol.pdf |solution}} | | HW 7 | Due Mar 8 8pm | Stein {{ :math185-s23:ch4-exercises.pdf | Ch4}}, 3, 4, 6, 8, 9(a) | {{ :math185-s23:hw7-solution.pdf |solution}} | | HW 8 | Due Mar 17 8pm |{{ :math185-s23:gamelin-p356.pdf |[G] p356}} Ex 1,10,14 {{ :math185-s23:stein_ch5.pdf |[S] Ch5}} Ex 3, 4(a) | {{ :math185-s23:hw8-sol.pdf |solution}} | | HW 9 | Due Mar 27 8pm |{{ :math185-s23:gamelin-p356.pdf |[G] p356}} Ex 15, 16(a),(b) {{ :math185-s23:gamelin_p360.pdf |[G]p360}} 2, 5 | {{ :math185-s23:hw9-sol.pdf |solution}} | | HW 10 | Due Apr 14 8pm | [G] Ch IX.1 #2,#3, IX.2 #2, #3, #4 | {{ :math185-s23:hw10-sol.pdf |solution}} | | HW 11 | Due Apr 19 8pm | {{ :math185-s23:gamelin_xi.1_ex.pdf |[G] Ch XI.1 #1,3,5,7,8 }} | {{ :math185-s23:hw11-solution.pdf |solution}} | | HW 12 | Due Apr 28 8pm | [S] p278 #2, #4 [[hw12|and one more]] | {{ :math185-s23:hw12-solution.pdf |solution}} |