189-249B: Honors Complex Variables

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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.

A power series centered at $z_0$ is a formal expression of the form $$ f(z) = \sum_{n=0}^\infty a_n (z-z_0)^n,\qquad a_n\in \mathbb C.$$ It converges pointwise for all $z$ in the open disc $D_R(z_0)$, where the radius of convergence $R$ is the inverse of the lim sup of the quantities $|a_n|^{1/n}$. The convergence is even uniform on any disc of radius $r \lt R$, and the same is true for the power series derivative $\sum_{n=0}^\infty n a_n z^{n-1}$. This implies that $f(z)$ is differentiable on $D_R(z_0)$, and is even, in fact, infinitely differentiable. The proof is given in the textbook from first principles and can be generalised to the statement that if a sequence of differentiable functions $f_n(z)$ converges uniformly to a function $f(z)$ and its derivatives also converge uniformly, then $f(z)$ is itself differentiable and $f'(z)$ is the limit of the $f_n'(z)$.

A complex-valued function $f(z)$ on an open subset $\Omega$ of $\mathbb C$ that admits a power series expansion at every $z_0\in \Omega$ is called analytic. It follows from our discussion that every analytic function is holomorphic, and ostensibly satisfies much stronger regularity properties, since analytic functions possess derivatives of all orders. It is all the more striking that we will eventually show that holomorphic functions are in fact analytic, a fact which is not at all apparent at this stage of our development of the theory.

Week 3 (Jan 21 and 23). This week's lectures focussed on the end of Chapter 1 and roughly the first half 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 assertions that turn out to be a bit tricky to prove rigorously. It is known as the Jordan Curve Theorem and 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, which is known as Goursat's Lemma, is then used 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, which already suffices for many applications.

We gave one application of Cauchy's theorem to the evaluation of explicit integrals; notably, a proof that the standard Gaussian $f(x) = e^{-\pi x^2}$ is its own Fourier transorm: $$ {\hat f}(t) := \int_{-\infty}^\infty e^{-\pi x^2} e^{-2\pi i x t} dt = e^{-\pi t^2}.$$

Week 4 (Jan 28 and 30). In this week's lectures we wrapped up Chapter 2 of Stein's book by discussing Cauchy's integral representation, which asserts that if $f(z)$ is a holomorphic function in a region $\Omega$ and $D=D_r(z_0)$ is a disc in $\Omega$ centered at $z_0$ with $C_r(z_0)$ the associated bounding circle, oriented in the counter-clockwise direction, then $$ f(z_0) = \frac{1}{2\pi i} \int_{C_r(z_0)} \frac{f(z)}{z-z_0} dz.$$
(Note that the function $f(z)/(z-z_0)$ is holomorphic on $C_r(z_0)$ but fails to be holomorphic at $z_0$ which lies in the interior of this curve. This illustrates the necessity of the hypothesis in the statement of Cauchy's theorem on the vanishing of integrals along closed contours.)

Cauchy's integral representation of holomorphic functions is a fundamental property of which has two far-reaching consequences:

1. Regularity. We will use it to show that holomorphic functions are infinitely differentiable, and even admit power series expansions. More precisely, if $f(z)$ is holomorphic on $\Omega$ which contains an open disc $D_r(z_0)$, then $f(z)$ has a power series expansion $$ f(z) = \sum_{n=0}^\infty a_n (z-z_0)^n, $$ whose radius of convergence is $\ge r$. This is a striking assertion: from the ostensibly far weaker definition that a holomophic function is once differentiable,we have shown that such a function is in fact infinitely differentiable, and analytic on its domain of definition.

2. Rigidity. This adjective refers to the fact that a holomorphic function is often completely determined by its restriction to smaller subsets (something which is rarely true for functions of a real variable). For instance, any two holomorphic functions that agree on a non-empty subset of $\mathbb C$, however small, must agree everywhere that they are defined. The same is true for two holomorphic functions that agree on $\mathbb R$. In particular, natural functions that you might first have encountered as functions of a real variable, like the exponential, sine or cosine functions, admit at most one extension to a holomorphic function on $\mathbb C$. So these extensions are anything but arbitrary!

This week should roughly conclude our discussion of the more ``foundational" parts of the theory and from here on we will focus somewhat more on examples and applications of the general theory.

Week 5 (Feb 4 and 6).

This week was devoted to the material in Chapter 3 of Stein's book, discussing meromorphic functions and the logarithm. We covered mainly sections 3.1 and 3.2. We discussed the three different types of singularity that one can encounter: removable singularities, poles,and essential singularities, and we showed that in a neighbourhood of a pole a function admits a Laurent series expansion.

This allowed us to define the notion of a residue of a meromorphic function at a point and to state and prove the residue theorem, asserting that the sum of the residues of a meromorphic function in a region bounded by a closed curve $C$ is equal to $2\pi i$ times the integral of $f$ along $C$ (oriented in the positive, counterclockwise direction).

We were able to use the residue theorem to give closed form evaluations of certain definite integrals.

Week 6 (Feb 11 and 13).

This week was devoted to the material in the latter half of Chapter 3 of Stein's book.

We first continued on our discussion of singularities (removable, poles, and essential) of analytic functions on a punctured disk, and extended the notion to the setting where $z_0= \infty$. This allows us to make sense of what it means for a complex function to be analytic at infinity, meromorphic at infinity, to have a pole of order $m$ at infinity: the latter just meaning that $f(z)/z^m$ is bounded as $z\rightarrow \infty$. One of the striking things we showed is that an entire function which is meromorphic at $\infty$ is a polynomial, and a function which is meromorphic everywhere including at $\infty$ is a rational function (a ratio of polynomials). This leads to some simple instances of the basic principle that a meromorphic function is often determined, up to scaling, by its set of zeroes and poles, counted with multiplicity.

After a further elaboration of the principle that a meromorphic function is determined by its singularities (section 3.3) we derived the argument principle of Section 3.4. which asserts that one can count the zeroes and poles with multiplicities of a meromorphic function $f$ inside a closed contour $C$ by integrating the logarithmic derivative of $f$ along $C$.

Week 7 (Feb 18 and 20).
This week, we will finally get to a discussion, taken from the end of Chapter 3 of Stein, of the logarithm on a simply connected domain in $\mathbb C-\{ 0\}$.

It would be natural to define $$ \log(z) = \int_{\gamma_z} \frac{dw}{w},$$ where $\gamma_z$ is a path joining $1$ to $z$ in the complex plane, but this integral depends on the choice of $\gamma_z$. For instance, for a closed path $\gamma$ that goes from $1$ to $1$ while winding once around the origin, one has $$ \int_\gamma \frac{dw}{w}= 2\pi i.$$ The fact that $1/z$ therefore does not have a well-defined continuous antiderivative on the entire punctured complex plane where it is defined is what explains some striking phenomena like Cauchy's integral formula and the residue theorem which we discussed last week.

This led us to the notion in Section 5 of Chapter 3 of homomotopy equivalence of paths and to the notion of simply connected domains, the key result being that the function $1/z$ admits a holomorphic primitive on any simply connected region $\Omega$ that does not contain $0$. Finally, Section 6 described the complex logarithm. The subtlety of the logarithm is that is is of necessity a ''multi-valued function'' when considered as a function on $\mathbb C-\{0\}$. It can only be consistently defined on a connected region that does not contain $0$ and is also simply connected, like the complement in $\mathbb C$ of the set of negative real numbers $x\le 0$. After defining the argument of a complex number in this region to be a number in the open interval $(-\pi, \pi)$, one obtains a well defined branch of the complex logarithm which is defined on all of $\mathbb C$ except the negative real numbers. It is sometimes called the <\i>standard branch of the complex logarithm.

Week 8 (Feb 25 and 27).
This week is devoted to the material in chapter 5 on entire functions. The main theme is to explore the interactions between the set of zeroes of an entire function, and other properties of the function. To what extent do the zeroes of a function determine the function? Affect its growth rate as the variable approaches infinity? Can any discrete subset of the complex plane arise as the zero set of an entire function?