Math 4555 (Complex Variables), Fall 2022
| Course Information | |||||
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| Instructor | Class Times | Office Hours | |||
| Evan Dummit edummit at northeastern dot edu |
MWR 1:35pm- 2:40pm Kariotis Hall 104 |
MW noon-1:00pm MWR 3:00pm-4:00pm Online, via Zoom |
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| We will use Piazza for any course-related discussion: here is the Piazza page. | |||||
| For detailed information about the course, please consult the 4555 Course Syllabus. Note: any information given in class or on this webpage supersedes the written syllabus. | |||||
| All homework assignments will be posted on this webpage (see below). Homework assignments will be collected via Canvas. Please submit scans of your homework pages by 11:59pm Eastern on the due date. Late assignments may be penalized at the grader's discretion. |
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| The instructor will write lecture notes for the course (see below) in lieu of an official textbook as the semester progresses. The course will generally follow chapters I-VIII of Lang's "Complex Analysis" (2nd edition) and chapters 1-7 of Saff and Snider's "Fundamentals of Complex Analysis" (3rd edition), but it is not necessary to purchase either book for the course. | |||||
| Homework Assignments + Exams | |||||||
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| Some Tips on Problem Solving are available as suggestions for the homework assignments. | |||||||
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Homework #1, due Wed Sep 14th. (solutions)
Homework #2, due Wed Sep 21st. (solutions) Homework #3, due Wed Sep 28th. (solutions) Homework #4, due Wed Oct 5th. (solutions) Homework #5, due Wed Oct 12th. (solutions) Homework #6, due Thu Oct 21st. (solutions) Homework #7, due Fri Oct 28th. (solutions) Homework #8, due Fri Nov 4th. (solutions) Homework #9, due Sun Nov 13th. (solutions) Homework #10, due Sun Nov 20th. (solutions) Homework #11, due Fri Dec 2nd. Homework #12, due Fri Dec 9th. The final exam is optional and is only available for students who are below an A grade after Homework #12 in case they want to increase their grade. You will be notified via Canvas by Sunday Dec 11th whether you already have an A (and are thus exempt from the final exam) or whether you are below an A (and may thus wish to take the final exam to raise your grade). | |||||||
| Handouts / Lecture Notes | |||
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| Handout | Topics | ||
| Chapter 1: Complex Numbers and Derivatives (19pp, v1.00, posted 9/6) | 1.1 ~ Complex Arithmetic (Complex Numbers, Polar and Exponential Forms, Topology of C) 1.2 ~ Complex Derivatives (Limits, Complex Derivatives, Partial Derivatives and the Cauchy-Riemann Equations, Holomorphic Functions and Angles) |
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| Chapter 2: Complex Power Series (24pp, v1.00, updated 10/5) (updates in v1.00: fixed errors in proofs, added examples to 2.3.1, added 2.3.2) (updates in v0.60: added examples at end of 2.2.2, added 2.2.3-2.3.1) |
2.1 ~ Formal Power Series (Formal Power Series, Formal Laurent Series) 2.2 ~ Convergence of Power Series (Sequences and Series, Convergence of Power Series, Continuity and Differentiability, Analytic Functions) 2.3 ~ Elementary Functions as Power Series (Complex Exponentials, Trigonometric Functions, Logarithms and Powers) |
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| Chapter 3: Complex Integration (23pp, v1.00, updated 10/24) (updates in v1.00: added 3.2.4-3.2.6) (updates in v0.40: added 3.2.1-3.2.3) |
3.1 ~ Integrals on Complex Curves (Integrals via Riemann Sums, Evaluating Line Integrals, Antiderivatives and the Fundamental Theorem of Calculus) 3.2 ~ Cauchy's Integral Theorem (via Green's Theorem, via Deformation of Contours, with Laurent Series, Winding Numbers, Cauchy's Integral Formula, Differentiability Properties) |
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| Chapter 4: Applications of Cauchy's Integral Formula (29pp, v1.00, updated 11/14) (updates in 1.00: added simple pole residue formula and examples at end of 4.3.1, added 4.3.2-4.3.4) (updates in 0.50: added 4.2.1-4.3.1) |
4.1 ~ Estimates for Holomorphic Functions (Cauchy Estimates, Entire Functions, Liouville's Theorem, the Maximum Modulus Principle) 4.2 ~ Laurent Series and Singularities (Laurent Series, Zeroes, Removable Singularities, Poles, Essential Singularities) 4.3 ~ Residues and Residue Calculus (Residues, the Residue Theorem, Calculating Definite Integrals via Residue Calculus) |
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| Chapter 5: Local Behavior of Holomorphic Functions (21pp, v1.00, updated 12/3) (updates in v1.00: added 3-transitivity to end of 5.2.2, added 5.2.3-5.2.4) |
5.1 ~ Locations of Zeroes and Poles (Counting Zeroes and Poles, Rouché's Theorem, Open Mapping, Local Invertibility and Preimages) 5.2 ~ Conformal Mapping (The Point at Infinity, The Extended Complex Plane, Fractional Linear Transformations, Conformal Maps and Analytic Isomorphisms, Functions on the Unit Disc, Harmonic Functions) |
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| Tips For Success In This Course | |||
|---|---|---|---|
| Attend Lecture | Missing lecture is a bad idea! If for any reason you cannot make it to a class, you should review the notes and watch the lecture recording. You are responsible for all material covered in lecture. | ||
| Read the Lecture Notes | The lecture notes are a comprehensive source of material for the course. The notes are intended as review material, although many students like to read them as preparation before attending the lecture on the corresponding topics. Please note that the electronic notes are not identical to the material covered in class: this is by design, so as to provide you a slightly different perspective on the material. | ||
| Solve Homework Problems | Much of the learning in this course will take place as you solve the homework problems. Like many other activities, problem-solving and proof-writing are things that are learned by doing them, not by hearing someone else tell you about them or reading about them in a book. As such, the homework assignments are an integral part of the course, and are fundamental to learning the material. It is highly recommended that you look over the homework assignments as soon as they are available, and work on them well in advance of the deadline: many problems will take substantial time and effort to solve, and you should expect to spend as much time as you need to finish the assignments. | ||
| Attend Office Hours | Office hours are specifically reserved for you to receive individual, one-on-one help from the instructor. Office hours will be the most effective when you have already put in effort to learn the material on your own (including trying to solve the homework problems), and when you come in with a list of specific questions or topics you are struggling with. | ||
| Course Schedule | |||
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| The schedule is subject to change! All sections refer to the course lecture notes. | |||
| Week | Schedule | ||
| Week of Sep 5 (class starts 9/7) |
§1.1.1: Complex Arithmetic §1.1.2: Polar and Exponential Forms No homework this week. |
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| Week of Sep 12 | §1.1.3: Topology of C §1.2.1: Limits §1.2.2: Complex Differentiation §1.2.3: The Cauchy-Riemann Equations Homework #1 due Wednesday 9/14 on Canvas. |
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| Week of Sep 19 | §1.2.3: The Cauchy-Riemann Equations §1.2.4: Holomorphic Functions and Angles §2.1.1: Formal Power Series §2.1.2: Formal Laurent Series Homework #2 due Wednesday 9/21 on Canvas. |
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| Week of Sep 26 |
§2.2.1: Convergent Sequences and Series §2.2.2: Convergent Power Series §2.2.3: Continuity and Differentiability of Power Series Homework #3 due Wednesday 9/28 on Canvas. |
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| Week of Oct 3 | §2.2.1: Exponentials and Trigononetric Functions §2.2.2: Complex Logarithms Homework #4 due Wednesday 10/5 on Canvas. |
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| Week of Oct 10 (no class 10/10) |
§3.1.1: Line Integrals in C §3.1.2: Evaluating Line Integrals §3.1.3: Path Independence Homework #5 due Wednesday 10/12 on Canvas. |
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| Week of Oct 17 | §3.2.1: Cauchy's Integral Theorem via Green §3.2.2: Deformation of Contours §3.2.3: Integration of Power Series Homework #6 due Thursday 10/20 on Canvas. |
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| Week of Oct 24 | §3.2.4: Winding Numbers §3.2.5: Cauchy's Integral Formula §3.2.6: Higher Derivatives and Series Expansions Homework #7 due Friday 10/28 on Canvas. |
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| Week of Oct 31 | §4.1.1: The Cauchy Estimates §4.1.2: Entire Functions, Liouville's Theorem §4.1.3: The Maximum Modulus Principle Homework #8 due Friday 11/4 on Canvas. |
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| Week of Nov 7 | §4.2.1: Laurent Series Expansions §4.2.2: Zeroes of Holomorphic Functions §4.2.3: Removable Singularities, Poles, and Essential Singularities Homework #9 due Sunday 11/13 on Canvas. |
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| Week of Nov 14 | §4.3.1: Residues, The Residue Theorem §4.3.2: Residue Integrals: Circular Contours §4.3.3: Residue Integrals: Circular Contours with Detours Homework #10 due Sunday 11/20 on Canvas. |
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| Week of Nov 21 (no class 11/23-11/27) | §4.3.4: Residue Integrals: Other Contours (Rectangles, Keyhole) No homework this week. |
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| Week of Nov 28 | §5.1.1: Counting Zeroes and Poles §5.1.2: Rouché's Theorem §5.1.3: The Open Mapping Theorem, Local Invertibility §5.2.1: The Point at Infinity and the Extended Complex Plane Homework #11 due Friday 12/2 on Canvas. | ||
| Week of Dec 5 (class ends 12/7) |
§5.2.2: Fractional Linear Transformations §5.2.3: Conformal Mapping §5.2.4: Analytic Continuation Homework #12 due Friday 12/9 on Canvas. |
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| Optional FINAL EXAM due Thu Dec 15th | |||