INFORMATION ON READING BEFORE CLASS

Below are some comments to help you prepare for each class' lecture. For each section in the book (Complex Analysis by Stein and Shakarchi, ISBN13: 978-0-691-11385-2, click here for the introduction, click here for chapter 2), I'll mention what you should have read for class. In other words, what are the key points. When you come to class, you should have already read the section and have some sense of the definitions of the terms we'll study and the results we'll prove. This does not mean you should know the material well enough to give the lecture; it does mean that you should have a familiarity with the material so that when I lecture on the math, it won't be your first exposure to the terminology or results. Everyone processes and learns material in different ways; for me, I find it very hard to go to a lecture on a subject I'm unfamiliar with and get much out of it. I need to have some sense of what will happen, as otherwise I spend too much time absorbing the definitions, and then I fall behind. I'm hoping the bullet points below will help you in preparing for each lecture. If there is anything else I can do to assist, as always let me know (either email directly, or anonymously through mathephs@gmail.com, passsword 11235813).

CHAPTER 1: Preliminaries to Complex Analysis: Click here for my lecture notes.

• Section 1: Complex numbers and the complex plane
  • This is a quick summary of basic properties of complex numbers and reviews the terminology we'll need from real analysis / topology. You should make sure you are comfortable with these terms and how to use them. I strongly urge you to attempt some of the practice problems in my lecture notes.
• Section 2: Functions on the complex plane
  • We see the definition of the complex derivative formally looks the same as in one variable, but as the limit must exist for a paths in the plane and not just on a line, there will be enormous consequences. The key word is holomorphic; a function is holomorphic if it is complex differentiable. Don't worry about the analysis on page 11 of complex-valued functions as mappings. The most important part starts at the bottom of page 11 and continues to page 12, where the book proves the Cauchy-Riemann equations. We'll see an interpretation of this in terms of Green or Stokes' theorem. The last part of the section deals with sequences and series, especially convergence. We won't delve too deeply into this in the course, but we will quickly review some (but not all) of this. The most important point is Theorem 2.5, determining the radius of convergence of a series. We will frequently use an immediate consequence, Theorem 2.6, which tells us that if we have a power series then we can differentiate term by term inside its radius of convergence.
• Section 3: Integration along curves
  • The section ends with a discussion of path integrals. If you haven't seen path integrals in multivariable calculus, please see me. We'll discuss some of this material now, and again when we do Cauchy's Residue Theorem. The two most important results are Theorem 3.2 (which implies the integral of a holomorphic function along a closed curve is zero) and the exercise right after Corollary 3.3 (which says the integral of 1/z on a circle centered at the origin is 2πi).

CHAPTER 2: Cauchy's Theorem and its Applications: Click here for my lecture notes.

• Section 1: Goursat's theorem
  • Obviously, Goursat's theorem is the highpoint! Think about dividing a triangle into smaller triangles versus a square into smaller squares -- which do you prefer and why? Note how we used real analysis / topology to show that there was a unique point in all the triangles, and then used local properties of f at that point.
• Section 2: Local existence of primitives and Cauchy's theorem in a disc
  • Learn the different key contours -- depending on the symmetries of your problem, different contours are more useful than others. The key fact is Theorem 2.2, Cauchy's theorem for a disk. Similar to the generalizations of Green and Stokes' theorem, this too will be massively expanded to include more contours.
• Section 3: Evaluation of some integrals
  • Example 1 on page 42 is very important; if you take ξ = 0 this is essentially the normalization constant of a Gaussian distribution from probability! You might remember doing this integral in Calc III by squaring it and switching to polar coordinates. Unfortunately, to evaluate this integral for arbitrary ξ we need to know its value when ξ = 0. I thus find this example a bit unsatisfying, though it does give the critical result that the Fourier transform of a Gaussian is a Gaussian. This is used in proving the Central Limit Theorem, one of the gems of probability. I thus find Example 2 more satisfying.
• Section 4: Cauchy's integral formulas
  • Cauchy's integral formula is a generalization of Cauchy's theorem for the disk (Theorem 2.2), but now we allow our function to be holomorphic everywhere save finitely many points where it has poles. It is hard to understate the value of this result -- we convert integration (hard!) to finding a specific coefficient in a Taylor expansion (this is algebra, and easy!). It takes a few exercises to get a hand on all the different tricks to find expansions / how to set contours; we'll do more as the semester progresses. Corollary 4.3 (the Cauchy inequalities) give wonderful bounds on how large the derivatives can be. This has many applications, including a holomorphic function is infinitely differentiable and equals its power series expansion (thus the words holomorphic and analytic are the same!). Another is Liouville's theorem: a bounded entire function (differentiable on all of the complex plane) must be constant. A paragraph later we have a proof of the Fundamental Theorem of Algebra! Click here for another proof; this is essentially using Stokes' theorem.
• Section 5: We will not cover this section

CHAPTER 3: Meromorphic Functions and the Logarithm: Click here for my lecture notes.

• Section 1: Zeros and poles
  • A function f has a zero of order n at z₀ if f(z) = (z-z₀)ⁿ g(z) and g(z₀) is not zero, and f has a pole of order n at z₀ if 1/f(z) has a zero of order n at z₀. Thus f(z) = z² (z-1)³ / (z-4)⁷ has a zero of order 2 at z=0, a zero of order 3 at z=1, and a pole of order 7 at z=4. A Laurent series is a Taylor series with finitely many negative terms, such as g(z) = Sum_{n = -6 to ∞} (z-4)ⁿ / (n²!+1). The principal part of f are the terms corresponding to negative n in the Laurent expansion, which here would be Sum_{n = -6 to -1} (z-4)ⁿ / (n²!+1). The residue of g at z₀ is the a_-1 term in its Laurent expansion. For the example above, the residue at z=4 is 1/(-1)²! + 1) = 1/2, and the residue at any other point is zero as the function is well-defined for all other z. For the function  f(z) = z² (z-1)³ / (z-4)⁷ the only non-zero residue is when z=4: what is its residue? The main theoretical result is that if P(z) is the principal part of f(z) at z₀ then (1/2πi) Integral_gamma f(z) dz =  (1/2πi) Integral_gamma P(z) dz = a_-1. To highlight where the action takes place, we sometimes write a_-1(z₀).
• Section 2: The residue formula
  • The residue formula is perhaps the most important in all of complex analysis, at least from an applications standpoint. It says that for a function f holomorphic on an open set containing a circle C except at finitely many points (say z₁, ..., z_N), then (1/2πi) Integral_gamma f(z) dz = Sum_{k = 1 to N} a_-1(z_k). This converts integration (hard) to evaluting the -1 term in Laurent expansions (algebra -- much easier!). The contour need not be a circle; in general, the hard part of the problem is choosing a contour that avoids the poles and has the function experiencing rapid decay.
• Section 3: Singularities and meromorphic functions
  • First is Riemann's theorem on removable singularities (basically if the function is holomorphic everywhere save at one point, and f is bounded on the complement of that point, then we can extend f to be holomorphic everywhere). There are three types of singularities: removable (f bounded near z₀), pole (|f(z)| tends to infinity as z tends to z₀), and essential (anything else). If f has an essential singularity, the Casorati-Weierstrass theorem says that in small neighborhoods of that point, the values the function takes on is dense in C.
• Section 4: The argument principle and applications
  • There are many Pavlovian responses in mathematics; one of the central ones in complex analysis is that you should integrate the logarithmic derivative of a function. The argument principle states that (1/2πi) Integral_gamma f'(z) / f(z) dz = #(zeros of f inside gamma) - #(poles of f inside gamma); this formula is sometimes replaced with a weighted version: (1/2πi) Integral_gamma f'(z) /f(z) * g(z) dz with g holomorphic. The argument principle yields many important results, such as Roche's theorem, the Open Mapping Theorem, and the maximum Modulus Principle. Roche's theorem discusses when two holomorphic functions have the same number of zeros in a region. The Open Mapping Theorem says that if f is holomorphic and U is open then f(U) is open. Is a similar result true for real functions? Namely if f: R to R is differentiable and I is an open interval, must f(I) be open? The Maximum Modulus Principle says that the maximum value of a non-constant holomorphic function must be attained on the boundary (again, try and see if the corresponding statement is true for functions of a real variable).
• Section 6: The complex logarithm
  • This section constructs the complex logarithm. We'd like to say that if z = r exp(i θ) then log z = log r + iθ; unfortunately, z' = r exp(i (θ + 2nπ)) for any integer n, and thus we cannot define the logarithm for the entire complex plane. This leads to the notion of a principal branch, which frequently is just a ray emanating from the origin and going to infinity.

CHAPTER 5: Entire Functions: Click here for my lecture notes.

• Section 3: Infinite products
  • The section begins with some needed preliminaries on what infinite products mean. Essentially we can understand such products by taking logarithms and converting them to infinite sums, using the Taylor series expansion of log(1+x). In most applications x ranges of some sequence of terms tending to zero, and thus eventually each x is less than 1/2 in absolute value (which is a key component in understanding the convergence). This is codified in Proposition 3.1. Proposition 3.2 discusses convergence of holomorphic functions. Section 3.2 has the important example of a product expansion for sin(πz)/π. This expansion is useful in many problems. The proof is a bit long, but involves a nice appeal to Liouville's theorem (bounded entire functions are constant), and shows why this can be such a useful result.
• Section 4: Weierstrass infinite products
  • Theorem 4.1 is of immense importance in understanding representations of functions; it tells us that two entire functions with the same set of zeros have ratio equal to the exponential of an entire function. This quantifies to what extent knowing the zeros determine a function. We then turn to infinite products and introduce the canonical factors E_k(z). These contain exponential factors needed to ensure the resulting products converge; the argument of the exponential is related to the series expansion of the logarithm function.

CHAPTER 8: Conformal Mappings: Click here for my lecture notes.

• Section 1: Conformal equivalence and examples
  • The key fact is Proposition 1.1, where we see how important it is for the first derivative to be non-zero at a point. The proof uses Rouche's theorem (ah, yet another reason we learned it). We then move into lots and lots of examples; see in particular all the maps on page 213. We won't cover Section 1.3.
• Section 2: The Schwarz lemma; automorphisms of the disc and upper half plane
  • The main ingredient in our analysis is Lemma 2.1. Key inputs are the notion of a removable singularity and the maximum modulus principle. We then classify the automorphisms of the disk. One of the uses of the Riemann mapping theorem is to allow us to pass from the classification of automorphisms of the disk to any simply connected open set not all of C. As a particular example, we do the upper half plane. Another nice fact is that the set of automorphisms is actually a group.
• Section 3: The Riemann mapping theorem
  • Obviously, the statement of the Riemann mapping theorem is essential. The big condition is that our simply connected region is not all of C. I particularly like how we can see that C cannot be conformally equivalent to the unit disk by invoking Liouville's theorem. A good amount of the Riemann mapping theorem is just standardization in how to present the map. The proof is long and broken into many technical stages. The first is Montel's theorem. This is in the spirit of advanced real analysis -- we need bounds / convergence to hold uniformly. We also use some topology, writing our set as a union of compact sets (remember convergence is much nicer on compact sets, so this is a good trade-off). The proof of the Riemann mapping theorem then follows  in three steps.
    • Step 1: Our set is conformally equivalent to an open subset of the unit disk containing 0. We use properties of the logarithm to do this. I particularly like the step where we write F(z) = 1 / [f(z) - (f(w) + 2πi)]; we've seen this trick of looking at 1/(const + f(z)) a few times before (for example, if the real part of f is bounded this is a good way to show that f is constant, as it then gives a new function that is bounded).
    • Step 2: We look at the holomorphic, injective maps of our set into the disk sending 0 to 0 and find the one where |f'(0)| is maximized. We use a lot of the analysis results here.
    • Step 3: We show that our candidate, which arose from maximizing |f'(0)| among certain classes of functions, gives the desired map.
• Section 4: Conformal mappings onto polygons
  • We will just skim this section, stating some of the results. The miracle is the Scharz-Christoffel formula (integral), which gives an explicit solution to what the map is from certain polygons to the unit disk.