MATH 185 Introduction to Complex Analysis (Spring 2004):

Instructor: Alexander Yong, 1035 Evans Hall, ayong@math.berkeley.edu

Lectures: Tuesdays and Thursdays 12:30-2:00PM, 9 Evans Hall

Office hours: Tuesdays 10-11 AM and Thursdays 2-3 PM. **Aaron Greicius will be the pool GSI for this course. His office hours are Monday 9-11 AM and 2-5 PM; Tuesday 11-1 AM and 2-5 PM in room 891 Evans.** Tony Chiang, our grader will also hold an office hour on Wednesdays 11-12 AM in 935 Evans Hall. He asks that you do NOT email him, but rather see him during that time if you feel you need a regrade. See also Tony Chiang's grading guide.

Required Text: J. Brown and R. Churchill "Complex Variables and Applications", 6th edition .

Other recommended texts:

1. D. Sarason, "Notes on Complex Function theory"

2. E. B. Saff and A. D. Snider, "Fundamentals of Complex analysis"

Grading: homework (25%), Midterm (25%), final exam (50%) OR homework (25%) and final exam (75%) [whichever is better]

Homework: Assignments will be posted weekly on the course webpage and are due in class the following week. The lowest two assignments will be dropped. Except for the first assignment, all assignments will be due on Tuesdays. This is so as to coordinate with the office hours of our pool GSI.

Homework 0: Send me an email about you and your (mathematical) background so I can get to better know you.

Exams: One midterm **now officially** Thursday March 18th, 2004 during class hours ******BUT IN 0170 BARROWS HALL****** . The final exam will be cumulative, ** we are exam group 14 (Wednesday May 19, 5-8PM) in 101 LSA (Life Sciences Building Addition)**.

Prerequisites: Elementary real analysis (Math 104).

Syllabus: Complex analysis is a fundamental tool that is used throughout mathematics. This will be an introductory course on this subject. Basic properties of complex numbers, the complex plane. Differentiation, (contour) integration, Cauchy's theorem, Liouville's theorem, proof of the Fundamental theorem of algebra, Maximum modulus principle, Taylor and Laurent series, residues and introduction to conformal mappings.

Outline: I will spend ~4 weeks on "preliminaries" (chapters 1-3 of the text), ~6 weeks on integration, series, residues and poles (chapters 4-7), and ~4 weeks on (conformal) mappings (chapters 8-10). Of course, we will not have time to cover **every aspect** of these chapters!


Part I: Preliminaries:

Lecture 1: We talked a little about the things that will come up in the course. Five things that remind me of Complex Analysis, and in particular its usefulness: (1) the Riemann sphere is the complex numbers union the point at infinity, (2) the fundamental theorem of algebra, (3) a **real** integral (that we will later deduce easily from complex analysis methods), (4) the classical sum of 1/n^2 = (pi^2)/6 formula can be deduced from complex analysis, (5) the Jacobian conjecture is an example of a famous open problem in the subject.

Looking at (2) we should note that it is a statement purely about algebra. Yet it can be naturally derived from complex analysis! It follows from the following claim: the bounded entire functions from complexes to complexes are just the constant functions! This is surprising since this is definitely not true for real valued functions. In turn, this claim follows from the *central* theorem of the course: the Cauchy integral formula, which is an equality of f(z_0) with a contour integral. This theorem says that if you want to calculate the contour integral ("hard"), you simply have to evaluate at a point ("easy"). On the other hand, it gives a geometric calculation associated to f(z_0), providing a chance to apply "analysis" to deduce properties of f(z). We didn't really talk much about (3) and (4), the point being that Complex analysis is also useful as a tool to deduce (real valued) problems that might be harder if without this tool.

Here is the first day survey (PDF file) ,

Lecture 2: We will talk about basic properties of complex numbers, complex conjugation, the modulus (norm) of a complex number, the triangle and Cauchy-Schwarz inequalities. Here is Assignment 1 , due next Thursday.

Note 1: Aaron Greicius will be the pool GSI for this course. His office hours are Monday 9-11 AM and 2-5 PM; Thursday 11-1 AM and 2-5 PM in room 891 Evans. You may see him outside (or even during) my office hours for this class.

Note 2: The current edition of the textbook is the 7th edition. However, you may still use the 6th edition.

Lecture 3: We finished up talking about basic properties of complex numbers, with the n roots of a complex number (computed as a consequence of DeMoivre's formula). Then we moved to a discussion of "elementary functions": e^z, log(z), sin(z), cos(z). The definitions, and all of the slightly annoying differences from their real valued analogues flow from the fact that e^z is **not** one to one. It is instead periodic. Another question that went unanswered: what justified our definitions of these functions anyway (other than the fact that they agree with the real-valued analogue when z is real).

Lecture 4: We continue to talk about elementary complex valued functions. We proved that log(z) is the inverse of e^z **when you restrict to a branch**. We proved some usual properties of log (with slight twists of course). Knowing how log(z) works allows us to deal with a^b for complex numbers a and b. This leads to three possibilities of how many different values a^b can take depending on the type of numbers a and b are. Then we began to talk about the geometry of complex functions. Assignment 1 was due today; here is Assignment 2 , due **Tuesday February 10, 2004**. Here are the partial solutions to Assignment 1.

Lecture 5: We talked about the "point set topology" of the complex numbers: what is an open set, what is a closed set? What does continuity of a function mean? What does it mean for a function to have a limit? Uniqueness of limits? We also talked about sequences and limits. The main point that we will get to (next class) is that beyond all the delta-epsilon stuff (which we need to _prove_ various things), the main point is that a "good" function is one that is continuous, and "good" function are defined in terms of open sets: a function is continuous if the inverse image of any open set is open.

Lecture 6: The old 1-1+1-1+1-1+... example illustrating which our definitions of convergence of sequences makes sense. Proof that a set is closed if and only if every convergent sequence converges to a point inside the set. We used this sequence definition of closed to get a nice characterization of continuous functions f: the inverse image of any open/closed set is respectively open/closed. Relative continuity, open and closed sets. Connected sets: two definitions here: path connected and ("not") connected. Path connectivity implies connectivity but not the other way around ("topologist's since curve"; exercise: explain this). Began talking about compact sets. There are three definitions: (easy one) closed and bounded; (property we want) every sequence has a convergent subsequence; (technical one useful for proofs) every open cover has a finite subcover. Assignment 1 was handed back.

Lecture 7: Finished off the discussion about compact sets: showed the image of a compact set is compact, the extreme value theorem and the distance lemma. Began talking about basic properties of analytic functions. Assignment 2 was collected and here is Assignment 3 , due Tuesday February 17, 2004. Here are the partial solutions to Assignment 2. ***The new (and official) date for the midterm is Thursday March 18th.***

Lecture 8: Proved the Cauchy-Riemann theorem which gives a simple necessary and sufficient condition for a function f to be analytic. We applied the Cauchy-Riemann theorem to prove some basic differentiation facts for elementary functions such as e^z, sin(z) and cos(z). Started on contour integrals, which means we talked about paths gamma:[a,b]->complexes.

Lecture 9: Defined the contour integral of a function f:C->C along a piecewise C^{1} curve gamma:[a,b]->C; i.e., gamma is continuous on [a,b], and there is a subdivision of [a,b] such that the derivative of gamma exists on each open subinterval (a_i, b_i), and the derivative is continuous on each closed subinterval [a_i,b_i]. We did some examples of contour integrals, and stated some routine, natural properties of countour integration. We asked and gave an answer to the question of whether how you describe a curve changes the value of the integral (answer: basically no --where "basically" has a technical meaning: reparameterization). We collected Assignment 3 and here is Assignment 4 , due Tuesday February 24, 2004. Here are the partial solutions to Assignment 3.

Lecture 10: We proved the Fundamental theorem of Calculus for contour integrals: if F is a function that is analytic on an open neighbourhood containing some curve gamma, then there is an easy way to compute the contour integral for the _derivative_ F' along gamma, just like in ordinary calculus, as a difference of the value of F at the endpoints of gamma. The fundamental theorem naturally suggests the question of to what extent the choice of curve (even more than just parameterization) matters. The answer, given in our "path independence" theorem essentially says that the assumption of the existence of a "global" (on a given open set A) antiderivative is what is demanded. The hard part of the proof was to show that path independence implies the existence of a global antiderivative, which we achieved via a delta-epsilon argument. We concluded with another example of a contour integral calculation that demonstrated some of the concepts of the day.

Lecture 11: NEW! See The grader Tony Chiang's grading guide for the assignments (#4 on). We reviewed the path independence theorem. When does the path independence theorem apply? When does the fundamental theorem apply? We moved on to the main theorem of the course: Cauchy's theorem, which states that the contour integral of a function f that is analytic on and inside a closed curve is zero. We proved some weaker versions of this. The first was for simple closed curves, with the extra assumption that the derivative of f is continuous (later in the course we'll see this assumption is unnecessary). This follows easily from Green's theorem. From this we proved an early version of the deformation theorem; question to test understanding: why isn't the theorem pointless? We assume f is analytic on the region containing the two curves, doesn't that mean the contour integrals are all zero?? We finished by proving another weak version of Cauchy. This time where we assume that the contour is a rectangle, but where we remove the "f' is continuous" assumption. Here is Assignment 5 , due next Tuesday. Here are the partial solutions to Assignment 4.

Lecture 12: Note: assignment 5, Q 3 should say "gamma is the upper half of the unit circle". Today we finished up the proof of Cauchy's theorem for a rectangle. We then used it to prove Cauchy's theorem for a disk (i.e, where a closed curve is contained in a disk).

Lecture 13: We defined what it means for two curves (with fixed endpoints z_0 and z_1, or that are both closed) to be homotopic, in terms of a continuous function H(s,t):[0,1]x[0,1]--> G (a given set). This was the rigorous definition of what it means to "continuously deform" two curves anyway. We also gave a totally not rigorous explanation in terms of stretching rubber bands when pins are in the way ---you had to be there. With this we could define simply connected sets, and we showed convex sets are a wide class of examples. Then we stated and proved the (strong) deformation theorem. This is a powerful tool that allows us to replace integrals along difficult to handle contours with easy ones. Here is Assignment 6 , due next Tuesday. Here are the partial solutions to Assignment 5.

Lecture 14: IMPORTANT!!: Our midterm is **now officially** Thursday March 18th, 2004 during class hours ******BUT IN 0170 BARROWS HALL****** . We talked about the index I(gamma,z_0) or "winding number" of a closed curve about a given point z_0. Our definition is motivated by the observation that how many times a curve goes around a point is obvious for circles e^{it} for 0<=t<=2n*pi (the number is n here). On the other hand, the integral of 1/z for that circle is 2i*pi*n. So dividing by 2*pi*i suggests a natural general definition of winding number. We proved this definition always gives an integer, and it naturally explains what it means for a point to be "inside" a closed curve. With winding numbers, we could state an all important result: the Cauchy integral formula. Why is this formula important?

Lecture 15: We finished the proof of the Cauchy integral theorem by proving two lemmas that allow us to obtain a technically stronger version of Cauchy's theorem. From this all kinds of things flowed: we proved the Cauchy integral theorem for derivatives, the Cauchy inequality (bounding the absolute value of f'(z_0)), Liouville's theorem (every entire bounded function is a constant), and almost finished the proof of the fundamental theorem of algebra. Here is Assignment 7 , due next Tuesday. Here are the partial solutions to Assignment 6. The midterm next Thursday will cover all the material covered up to Lecture 16.

Lecture 16: Finished the proof of the fundmamental theorem of algebra. We stated and proved Morera's theorem (yet another easy consequence of Cauchy's integral formula): this says that if a function integrates to zero on all closed curves and is continuous, then it is in fact analytic. We also went into the maximum modulus principle (both a "local" and a "global" version): roughly, a function that has a relative maximum is constant "near" the max. The local (easier) version followed from the mean value property (also easy from Cauchy's integral formula). The global version needed a little technical language about boundaries of regions etc. Also the proof was a little more involved. I emphasize once more how the theorems from this lecture and the last follow very much from Cauchy's integral theorem.

Lecture 17: We did some review. The midterm will have some proof problems, some computations, one or two proofs from class. It will by in 0170 Barrows from 12:40-2pm. Here are the partial solutions to Assignment 7.

Lecture 18: No lecture today. Instead we had the following midterm.

Lecture 19: Started talking about taylor series expansions of analytic functions. We emphasized the differences from the real variable case. We also reviewed some notions and theorems about convergence of series (pointwise vs uniform converence, absolute convergence, tests for convergence, the Cauchy criterion, the Weierstrass M-test). No assignment handed out today. Instead wait until Thursday.

Lecture 20: We worked towards proving that a function f:C->C is analytic if and only if it is locally representable as a (uniformly) convergent power series. To prove this we needed tha analytic convergence theorem that states in f_n->f uniformly and each f_n is analytic then f is analytic. Throughout we repeated exploited the weaker fact that the limit of a uniformly convergent sequence of continuous function f_n is continuous. We also talked about _the_ radius of convergence of a power series, which is sort of surprising: why shouldn't it be possible for a sequence to converge in disconnected regions?? **I'll post the next assigment on Tuesday.**

Lecture 21: We took up the midterm. Here are the Here are the partial solutions to the midterm. Here is Assignment 8.

Lecture 22: We proved the main goal of showing that any analytic function can be represented locally by a uniformly convergent power series. This followed from the Abel-Weierstrass lemma, that says that if the numbers (a_n)(r^n) are bounded, then the corresponding series converges inside a circle of radius r (uniformly and absolutely).

Lecture 23: Continued to talk about series. Stated the ratio and root tests for power series. Stated Taylor's theorem (why is it that this is _another_ theorem? Didn't we prove it already....?). Handed back Assignment 7. Here is Assignment 9 , due next Tuesday. Here Here are the partial solutions to Assignment 8.

Lecture 24: We talked more about Taylor's theorem and did a bunch of examples, including computing the Taylor series for a product of functions. The main point is that Taylor's theorem allows us in many cases to avoid the definition to compute the power series, which can be slow and cumbersome. Taylor's theorem also leads us to more information about analytic functions. In particular, we showed that the zeroes of an analytic function are isolated. We started talking about Laurent series, which deals with the question of finding a convergent power series of "center" c where the function f is not analytic at c (i.e., c is a "pole").

Lecture 25: Statement and proof of the Laurent series theorem. Next class will be given by the guest lecturer Tony Chiang. Here is Assignment 10 , due next Tuesday.

Lecture 26: Guest lecturer Tony Chiang.

Lecture 27: Started talking about the residue calculus. The principle idea being that the coefficients of the Laurent series are in fact computations of contour integrals. On the other hand, any way to computing the Laurent series (via, e.g., power series expansion tricks) gives you the coefficients, and in particular the "residue", i.e., the coefficient b_{-1}. We proved that if a function has a finite number of isolated singularities inside a contour, then the contour integral is the sum of the residues (times 2\pi i). Notes for the final: it will include a homework problem, proof from class, problem from a list of "practice" problems and new problems. Here are the partial solutions to Assignment 9. Here are the partial solutions to Assignment 10.

Lecture 28: More on the residue calculus. Here is Assignment 11 , May 11, 2004.

Lecture 29: Used the residue calculus to compute some indefinite integrals. The basic idea is to choose a function f:Complexes->Complexes and a (closed) contour such that integrating along one piece of the contour gives you the integral that you want (at least when you make the contour grow to infinity) while the integration along the "annoying" piece goes off to zero. On the other hand, you can apply the residue theorem to calculate the integral over the entire closed contour. We also mentioned the Cauchy Principal value, and how it differs from the actual doubly infinite improper integrals.

Lecture 30: We used residue theory to give a general theorem to compute some infinite sums. Most notably, this proves the sum of 1/n^2 = pi^2/6 result of Euler. Here are practice problems for the final exam. Assignment 11 is available in a box outside my office.