189-235A: Algebra 1

Blog

Week 1 (January 7 and 9). The first week was devoted to a brief motivational overview of the subject, giving a discussion of complex numbers, their cartesian and polar representations, a proof of the Euler identity $e^{i\theta}= \cos(\theta) + i \sin(\theta)$, and three different proofs of the fundamental theorem of algebra asserting that every complex polynomial has a complex root. The discussion of the latter topic is taken from Section 1.2. of the book of Romik, and a delightful account of the Euler identity can be found in this mathologer video.

From now on we will be following the book of Stein very closely. (The other texts are for the enjoyment and edification of the more hardy complex analysis enthusiasts, and while they might be occasionally be referred to in the blog they will not be strictly required for the course.)

Week 2 (Jan 14 and 16). This week's lectures were devoted to Chapter 1 in the book of Stein, giving a careful discussion of convergence of complex valued functions of a complex variable, power series, and the notion of integration along curves. We will prove a few special cases of Cauchy's important theorem that the integral of a complex valued function along a closed curve vanishes when the function is holomorphic on a region in the complex plane containing the curve and its interior.

Week 3 (Jan 21 and 23). This week's lecture focussed on the end of Chapter 1 and the beginning of Chapter 2 of Stein, and were largely devoted to a discussion of integrals of holomorphic functions along paths. Chapter 1 gave some basic properties of this path integral. Notably, if $f(z)$ is the derivative of a holomorphic function $F(z)$ on a region $\Omega$, then the integral of $f(z)$ along a path $\gamma$ is equal to $F(z_1)-F(z_0)$, where $z_0 = \gamma(0)$ and $z_1=\gamma(1)$ are the endpoints of the path. This is essentially the fundamental theorem of calculus. In particular, the integral of such an $f$ along a closed path (where $\gamma(0)=\gamma(1)$) is $0$. This conclusion therefore holds in particular for a power series in its open disc of convergence, since power series have antiderivatives.

Cauchy's Theorem is a basic foundational result that asserts that the integral of $f(z)$ along a closed curve $\gamma$ is likewise zero whenever $f$ is holomorphic in a region $\Omega$ that contains $\gamma$ along with its interior. The fact that a smooth closed non self intersecting curve breaks up the complex plane into two two pieces -- the interior and `the exterior -- is one of those ostensibly self-evident things that turn out to be a bit tricky to prove rigorously (it is known as the Jordan Curve Theorem) and this accounts for some of the drama in the proof of Cauchy's theorem. We will get by, initially, by working only with curves $\gamma$ that the book refers to as ``toy contours", like circles or triangles, where the interior region really is explicit and evident.

The proof of Cauchy's theorem breaks up into two steps. In the first step, one proves it for a very special class of curves, namely, triangles in the complex plane. This special case is enough to show that a holomorphic function on a disc $\Omega$ has a primitive, and from this, to deduce Cauchy's theorem in the special case that $\Omega $ is a disc.

We then discussed a number of applications of Cauchy's theorem, notably, to proving somewhat surprising regularity properties of holomorphic functions, which imply in particular that they are analytic, and thus infinitely differentiable in particular, a rather striking fact to obtain from the mere assumption that $f$ posseses a first derivative.

Week 4 (Jan 28 and 29).

Week 4 (Jan 28 and 30).

Week 5 (Feb 4 and 6).