Important: Give complete explanations or proofs for all homework questions, show all work. Please write up your solutions neatly, be sure to put your name on the first page and staple all pages. Illegible homework will not be graded. If collaborating with other students or using any other resources, your solutions should be written up in your own words, and all your collaborators and sources should be acknowledged (see academic integrity policy below). No late homework is accepted.

Week 1. Sections 1-10: definition and algebraic properties of complex numbers, complex numbers and vectors, complex conjugate, exponential form, argument and modulus, roots of complex numbers. You should be able to add, subtract, multiply, and divide complex numbers, show that a given expression defines a complex number by reducing it to the form a + bi, plot complex numbers on the complex plane, express them in exponential form, find argument and modulus, find and plot the complex conjugate of a given number, be able to work with basic properties of complex numbers including the vector interpretation, use exponential form to find powers of complex numbers and to solve equations of the form z^n=a. You should also be able to use geometry of vectors, know interpretation of |z-a| as distance between z and a and use it to plot circles, and use the triangle inequality to make estimates and geometric conclusions. Read the textbook!

Homework 1, due Wednesday, Sept 4 at 5pm: Homework 1 (pdf)

Week 2. Sections 11, 13-18: you should be able to find and plot solutions of equations of the form z^n=a, know what "roots of unity" means and be able to find them; interpret functions as transformations of the complex plane (very simple examples only), make plots like section 14 example 2; understand intuitive definitions of limits and continuity (epsilon-delta encouraged but not required), compute simple limits, and determine that the limit doesn't exist is the function approaches different values when z approaches z0 from different directions (simple examples only).

There will be a quiz on Monday, Sept 9. The quiz will test the skills from Week 1 in simple examples (finding roots of complex numbers or using the triangle inequality in complicated situations will not be required).

Homework 2, due Wednesday, Sept 11 at 5pm: Homework 2 (pdf)

Week 3. Sections 19-23: you should be able to determine whether the function is differentiable at a given point and compute the derivative if it exists, in three ways: 1) analyzing the limit directly from definition; show that f'(z) does not exist if different limiting values are obtained by approaching z from different directions; 2) using limit laws; 3) using Cauchy-Riemann equations. You should know what the Cauchy-Riemann equations are and why they must be satisfied if f'(z) exists. Proof that the C-R equations are sufficient for differentiability (Sec 23) is optional, but you should know and be able to use this fact.

Homework 3, due Wednesday, Sept 18 at 5pm: Homework 3 (pdf)

Week 4. Sections 25, 26, 30-38: you should know the definitions of analytic and entire functions and basic properties (such as: sums, products, compositions of analytic functions are analytic). You should know how to define and be able to work with the elementary functions of a complex variable: the exponential functions, sine, cosine, log, and power functions. You should be able to compute their values (in simple examples) and establish simple identities. You should know that the logarithm and power functions (such as square root) are multivalued, understand the notion of a branch and the principal value, and be able to compute the values of specific branches in simple examples.

There will be a quiz on Monday, Sept 23. The quiz will test the skills from Week 3 as well as definitions of exponential and trig functions of complex numbers. You should remember the Cauchy-Riemann equations and be able to use them, and to show that the derivative does not exist by finding different limiting values along different directions. You should remember the formulas for exp(z), sin(z), cos(z) and be able to use them. You will not be asked to work with complicated trig identities. Multivalued functions and their branches are not on the test.

Homework 4, due Wednesday, Sept 25 at 5pm: Homework 4 (pdf)

Week 5. Sections 41-46, 48, part of 49 (a implies b implies c only): you should know the definitions of a parameterized contour in the complex plane and of the integral of a function of a complex variable over a contour, work with basic properties of integrals, and be able to compute integrals directly from parameterization. You should know the notion of antiderivative, be able to find antiderivatives in simple situations, and know and be able to use the VERY IMPORTANT THEOREM in section 48. Caution: the contour must lie in the domain where the antiderivative is defined for the theorem to apply. If the antiderivative involves multi-valued functions such as log, you need to be very careful with branches. See Example 2 in section 48. You should understand this phenomenon and be able to compute integrals in similar situations (when the integral over a closed contour will not be zero).

Homework 5, due Wednesday, Oct 2 at 5pm: Homework 5 (pdf)

Week 6. Sections 47, 49, 50, 52, 53: you should know bounds for the modulus of the integral in terms of a bound on the modulus of the integrand and the length of the contour. You should know that a complex function has an antiderivative if and only if its integral over any closed contour is zero, and be able to use this fact. You should also that for any analytic function in a domain D, its integral over a contour C is zero if the domain contains the region enclosed by the curve C. (This is another VERY IMPORTANT THEOREM, sec. 50). You should be able to use this fact to compute some integrals (the answer is zero if you can check that the function is analytic in the appropriate domain) and to relate integrals over different contours (as in section 53).

The midterm will be on October 23, in class.

Homework 6, due Wednesday, Oct 9 at 5pm: Homework 6 (pdf)

There was a quiz on Wednesday, Oct 9. The quiz focused on integration (the skills from Weeks 5 and 6, which include knowing and being able to use VERY IMPORTANT THEOREMS from sections 48 and 50). Specifically:

-- know the definition of the integral over given contour from point z1 to point z2, via paramaterization; be able to compute from parameterizations; know that the value of this integral may depend on the actual choice of the curve (not only its endpoints z1 and z2)
-- know that if the function has an antiderivative in the given domain OR the function is analytic and the domain has no holes or punctures, then the integral over a contour from z1 and z2 depends ONLY on the endpoints z1 and z2 (not the actual curve); this fact can be used to simplify calculations by using a simpler contour connecting the same points
-- be able to find antiderivatives in simple situations and use them to compute integrals; know that the integral over a closed contour is zero if the function has an antiderivative defined on the contour
-- know that the integral of a function over a closed contour C is zero if the function is analytic on C and inside the contour C (must be analytic on all of the region enclosed by the contour, without any holes or bad points); use this to compute integrals simply by referring to the analytic property
-- be able to relate the integral over a closed contour C to the sum of integrals over other contours C_1, .., C_k (usually encircling "bad" points), if the function is analytic between the contours C and C_1,..,C_k; use this to simplify calculations (in particular, if there's only one bad point a inside C, the integral equals to the integral over a small circle around a)
-- know (with justification) the value of the integral of 1/(z-a) over a small circle enclosing a
-- the Cauchy integral formula can be used on the quiz but is not required

Comments on the Quiz : the questions had quick solutions if you used the above skills to pass to an easier contour or to notice that the integral must be zero (in some parts) because the function is analytic inside the given closed contour. For the integrals on non-closed contours, you needed to find the antiderivatives. The last question was the only one with a bit of challenge: you could either get the answer immediately from the Cauchy integral formula, or solve it as above by decomposing into partial fractions first. I recommend that you try to redo the quiz (not for grade) if you feel that you didn't do very well.

Week 7. Sections 54, 55, 57 section 56 optional): you should know the Cauchy integral formula and be able to use it to compute integrals. Similarly, you should know the extensions (Cauchy formulas for the derivatives) and be able to use them in claculations. You should know the corollaries (section 57).

Homework 7, due Wednesday, Oct 16 at 5pm: question 3 p.159; questions 1ac, 2ab, 4, 7 p.170-171. Optional: question 9 p.171 (note that you need to prove this fact for a complete non-circular proof that a function that has one complex derivative has derivatives of all orders).

Week 8. No class on Monday 10/14. Sections 58 (including Theorem 3 from section 57), 59 (we'll finish the maximum modulus principle on Monday, but you should know the statement and use it in the homework). You should know the statements and understand the proofs of Liouville's theorem, the fundamental theorem of algebra, and of the maximum modulus principle. (You can be asked to state these theorems on the test; you will not be asked to reproduce the proofs). You should be able to use these theorems in simple situations, for example, apply them to auxiliary functions to derive similar results (see Homework 8).

Homework 8, due Monday, Oct 21, at the beginning of class: questions 1, 2, 3, 5 p.177-178. Ignore "harmonic" in question 1, this simply means that you are working with u = Re f. Optional: question 10 p.172, question 7 p.178.

We will discuss Homework 8 in class on Monday after the homework is collected (so no late homework will be accepted under any circumstances).

The midterm will be on Wednesday, October 23, in class. The test will cover the material of Chapters 1-5 (refer to the week-by-week schedule on this page; the sections we omitted are not on the test). For practice, you can do extra questions from the textbook: you should definitely be able to do the questions at the beginning of each exercise set in the book.

Most of Monday 10/21 class will be spent on reviewing the previous material (bring your questions).

Special office hours on Monday, 10/21, 2pm-3:40pm. There will be no office hours until Homework 8 is due, but I will do my best to answer questions by email (excluding last minute emails).

Week 9. Review and exam.

Week 10. Sections 61--65 (we didn't discuss section 65 yet but it's very easy material and we'll do some examples on Monday). Sections 60-61 discuss the notions of convergence of sequences and series. If you have the background from MAT 319, you should understand the rigorous definitions of convergence. If not, you can stick to a calculus-like definition of the limit. In any case, you should know that the sum of the series is defined as the limit of partial sums, when this limit exists. [You should know what partial sums are and be able to write them for a given series, even if they can be rarely computed directly.] You should know what the Taylor series are, and the heuristic explanation of why the coefficients are given by the formulas with derivatives (via differentiation of the series). You should be able to compute the Taylor series from definition (in simple cases). You should know the Taylor series for the functions e^z, sin z, cos z, as well as the geometric series, and be able to use these and algebraic manipulations to find Taylor series for related functions. You should know that if the function is analytic in a disk centered at z0, then the Taylor series about z0 converges in that disk (be able to use this fact to say where the series for a given function converges).

Next week, we'll be covering Laurent series (sections 65-68). It is useful to read ahead.

Homework 9, due Wednesday, Nov 6 at 5pm: Homework 9 (pdf)

Week 11. Sections 66--68; brief discussion of sections 69-73 (focus on applications) You should be able to know Taylor series are and what Laurent series are, find them using algebraic manipulations and known series or by taking derivatives, when possible. You should be able to determine the domain of convergence (a disk or an annulus) by checking where the given function is analytic. Often, there are different Laurent/Taylor series for the same function is different domains. You should also be able to do operations with power series, such as differentiation or integration or algebraic operations, and use those to find series for related functions. Discussion of convergence (such as sections 69, 70 and justification of operations in 71-73) is optional but recommended for students familiar with the MAT 320 material.

Homework 10, due Wednesday, Nov 13 at 5pm: questions 3, 4, and 6 p.205-206; questions 1, 2, 3, 4 p.218-219. In question 6 p.206, please also find the series representing the finction in the other two domains (as in question 5).

Week 12. Sections 74-76, 78-81. Section 77 is optional. You should be know what the residue of the function at a singular point is, and be able to compute residues by using the Laurent series decomposition of the function. (Typically you need to find the Laurent series first.) You should know the Cauchy residue theorem and be able to use it together with the residue calculations to compute integrals. You should be able to identify the type of an isolated singular point (removable, essential, or a pole). You should be able to identify poles using the theorem from section 80, and compute the residue at a pole via derivatives of the related analytic function (same theorem).

Homework 11, due Wednesday, Nov 20 at 5pm: Homework 11 (pdf)

Quiz on Monday 11/18 The quiz covered Taylor and Laurent series and computing residues from a Laurent series decomposition. (Integration will not be on this quiz.) You should memorize the formulas for the series for sine, cosine, the exponential function and the geometric series, and use these formulas and simple algebra to find related series. You will not be asked to do very hard algebra, but basic manipulations with the geometric series expression will be required. In simple examples, you should also be able to find the Taylor series directly by differentiating the function and using the formula for the coefficients of the Taylor series.

Week 13. Sections 82-84 From the Taylor series of f at z0, you should be able to tell if f has a zero at z0, and what is the order of that zero. Know and be able to use an important fact: f has a zero of order m at z0 if and only if f=(z-z0)g, where g is analytic and nonzero at z0. For a function of the form f=p/q, you should understand what the interplay of zeros and poles of p and q means for the function f, and use this information to compute residues of f (section 83). You should also know how the function has different behavior near singular points of different type (section 84), and that an analytic function can only have isolated zeros (Thms 2, 3 section 82). An important consequence: if two analytic functions f, g take the same values along a line segment or on a convergent sequence of points, then f(z)=g(z) for all z.

Homework 12, due Wednesday, Dec 4 at 5pm: Homework 12 (pdf)

Classes on 11/25, 12/2, 12/4 Sections 85-87, Sections 88-91 optional, Sections 93-94, review as time permits.

Some questions on HW 12 are for the material of sections 85-87 (computing real-valued integrals via complex analysis), to be covered on 11/25 and 12/2. Sections 93-94 (argument principle and applications) will be covered on 12/4. There will be a short homework that week on the argument principle (in the "do not submit" or "extra credit" format).

Homework 13 (complete by Monday, Dec 9; do not submit): textbook section 94, p.293: questions 2, 3, 7, 8, 9.

The final exam will be December 11, 11:15am-1:45pm in Earth and Space 131 (our usual classroom). This reference page will be included as the second page of your exam. You do not have to memorize these formulas (but you still need to know what they mean and how to use them).

You might want to use past quizzes for the exam review: Quiz 1, Quiz 2, Quiz 3, Quiz 4. (Note that the quizzes do not cover all of the exam topics, even if they include some of the important ones.)

What's on the test? The final exam covers everything we studied (refer to the week-by-week schedule for specific textbook sections and the description of the material). For

To pass the exam, you must be able to work with complex integrals. This includes the Cauchy integral formula, the residue formula, being able to compute residues by various methods (including by working with series), using parameterizations and antiderivatives, knowing important results about integration (eg integral of a function over a closed contour is zero if the function is analytic inside the contour), and also requires working confidently with all the basics such as the arithmetic of complex numbers and properties of the standard functions (including writing Taylor series and Laurent series for a given function).

All of the other topics we covered are also on the test: working with Taylor and Laurent series, different types of singular points, special properties of functions such as branches of logarithm, Cauchy-Riemann equations, computing improper real integrals via complex analysis methods, theorems about analytic functions such as Liouville's theorem, maximum modulus principle, argument principle, etc. The more you know, the better. For a good grade, you have to show that you understand and can work with a number of course topics (but you don't have to complete 100% of everything.)

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