Mooc Work

Analysis of a Complex Kind

September 1st, 2016

This coursera mooc is offered by Wesleyan University, taught by Petra Bonfert-Taylor.

I've never actually formally studied complex analysis before. Before taking calculus in school, I taught it to myself first and I tried learning complex variables, but all I saw were the same results as with real variable calculus. Having taken honors analysis as my undergraduate degree, afterward I found a graduate text giving a rigorous treatment of the topic. I spent several months with that book, going over every proof. It helped a lot, but I had spent too much time seeing the trees instead of the forest, as they say. I understood the proofs and their techniques, but I never got to see the application of the theory.

To be frank, the biggest issue I've had with the complex number system is not that it was overwhelming or too complicated, rather because I saw the main calculus results as being the same as for the real number system, I didn't see the point really. As a lifelong learner I've decided it's worth overcoming this mental block, and getting my hands dirty and putting in the effort to really understand just how the reals and the complex numbers are different.

For example, I have a greater appreciation of the real numbers themselves because I know just how they're different from the rational numbers. From a set-theoretic lens, the most basic set-based structure (after a set itself) is a relation. The three main relations are functions, equivalence relations, and ordering relations. As there is no redundancy in these number systems there's no relevance in looking for equivalence. That just leaves functions (monoids), and orderings. The thing is, both the rationals and reals are linearly ordered. Both are densely ordered. Both are algebraic fields given their respective monoid operators (addition, multiplication).

So how are they different? The real number system is complete. This is to say it has the completion property: For every non-empty subset (of the reals) with an upper-bound, such a set also has a supremum (a least upper bound). This single fundamental difference is why you have unique limits, convergence, continuity within the real number line while you don't with the rationals. That's it. Reader beware: I intend to offer both a summary and a light critique of this mooc for anyone thinking of taking it themselves, but I'm also exploring this alternative motivation of mine: Looking for narrative differences between the complex numbers and their predecessor the reals.

It is for all these reasons I'm glad to be taking this mooc: I found it gives a nice tour around the application side of complex analysis. Not too soft, not too hard, just right.

• Week 1
• Week 2
• Week 3
• Week 4
• Week 5
• Week 6
• Week 7

Week 1

September 4th, 2016

This is a math mooc that takes itself seriously. Not to say it's not fun or somehow uninviting, but right in week one we're immediately thrown into the basic arithmetic and algebraic properties of the complex numbers.

Definitions worth taking away: modulus, conjugate, polar coordinates, principal argument (Arg z), nth roots of unity. Topological concepts are introduced, but I've been looking ahead, and it seems like they're not used too much, which makes sense: This isn't a proof heavy class, it's a 7-week mooc after all.

Identities, formulas, equations:

• $z\; conj(z)=|z|$²
• $|z|=0$ iff $z=0$
• The triangle inequality
• $e$^{conj(i theta)} = e^{-i theta}
• $1/e$^{i theta} = e^{-i theta}
• DeMoivre's formula

The Fundamental Theorem of Algebra is mentioned of course, this is not unexpected. As I've studied a proof-oriented narrative of complex numbers before, I especially enjoyed how exponential notation was introduced this early, and the clever way it side-steps the complications of introducing full exponentiation which would otherwise require heavier analytic tools, namely power-series.

Week 2

September 12th, 2016

My favorite part of this week by far was Julia sets. Touching on chaos theory, fractals, and the Mandelbrot set! I've been exposed to these ideas, but just barely, I didn't even know the natural way to access them was through complex analysis.

Otherwise what stood out for me, a connection I hadn't made myself, was visualizing the hard to visualize 4D space when graphing complex valued functions. In particular mapping circles from one plane to another. In any case, this week also introduced limits, sequences, continuous functions.

Week 3

September 17th, 2016

This week we finally got to the basics of complex variable calculus, namely differentiation, analytic functions, the Cauchy-Riemann equations, the complex exponential function, complex trigonometric functions. Similarly to last week, I was pleasantly surprised as to the practical techniques I'd never considered before in developing visual intuition for complex functions. In particular, the effectiveness of mapping horizontal and vertical lines from plane to plane.

This week I also finally started making some intuitive connections in how to distinguish the complex number system from the real number system. Even back in the day, having studied complex numbers a little previously, the non-linear ordering of the complex numbers (by means of the modulus) differ from the linearly ordered real numbers. A difference such as that is obvious, so when I say I've been looking to distinguish, it's more like: Because I've studied "the story" of the real numbers, how would "the story" of the complex numbers differ? More importantly, why would it differ?

Over these past 3 weeks (of the course), I've been sticking to the basics, and I've come to realize a main narrative of complex numbers that distinguish them from reals:

• multiplication

  Think about origins: The very motivation that creates a need for complex numbers in the first place is is the identity $i*i\; =\; -1$. This implies a definition of complex multiplication which is an extension of real multiplication. If you look more closely at this extended definition, it embeds rotation, and as a result it intertwines the Re z, Im z components. The real and imaginary components oscillate based on the rotation factor introduced by any given multiplication, and those oscillations are connected. Although this is a simple observation of the basic nature of complex multiplication, notice how this shows itself in the definition of $e$^z as well as the Cauchy-Riemann equations. It's like, those analytic (continuous) complex valued functions are only continuous because regardless of how they map a complex variable, they still preserve this rotational behaviour overall. For example $f(x)=x$² is a parabola for the real numbers, it's a very well studied continuous function. But when we extend it to $f(z)=z$² it involves multiplication and thus rotation. If it weren't continuous, it would have to tear some kind of hole in its rotational behaviour.

  In any case, although this is all just intuition on my part (and although I'm jumping ahead because I have some knowledge of what's to come), I'm willing to say this idea of rotation (and of circles) shows up again and again as a major theme as a result of this fundamental property of complex numbers.

It's good to be doing math again :)

Week 4

September 24th, 2016

A lot was new for me this week. When I had studied complex analysis previously, I never touched on the theory of conformal mappings which then specialize to Mobius transformations then leading up to the Riemann Mapping Theorem. The fact that you can translate the half-plane into an open unit disk is amazing! And I actually enjoyed the treatment of decomposing and classifying Mobius transformations into atomic variants, furthermore using circles and lines to circles or lines.

In any case, as far as the rest of the week went, we finally get to see some topology, not to mention inverse functions as well as the very important $Log\; z$.

As an aside, I was giving it some thought and what I find interesting is when you have isolated singularities such as with $f(z)=1/z$ at $z=0$, and then you create a primitive (anti-derivative) of this function seems to not only "preserve" the discontinuity, but the act of reshaping this otherwise continuous mapping seems to create an even more pronounced tear which is why you have an entire slit (ray) discontinuity within the domain of $Log\; z$. That's my current intuition about it anyway.

Week 5

September 27th, 2016

This week was more familiar to me. Having previously taught myself the honours version of this complex analysis, the book I read spent a lot of time and effort detailing the full Cauchy Theorem, and as a result so did I. Everything from complex integration, to the partial Cauchy theorems to the Cauchy integral formula, all of it leads up to the general Cauchy theorem, but they're each just beautiful in their own right. Not only are they elegant as asthetic, they're very powerful tools of proof. It's the first time I really started to see the beauty in the complex number system.

In week 3 I discussed what I feel is a main narrative as to why the complex numbers and the reals differ. I've been thinking things over this week and I feel I've realized another:

• completeness

  The complex numbers form an algebraic field, but you can't quite say they have the completeness property because you can't say every non-empty set with an upper bound has a least upper bound (because that would require linear ordering).

  And yet you still have a calculus of sorts, so what kind of calculus do you have? First of all, as I've taken honours analysis as my undergrad, we studied N dimensional vector calculus (mapping to M dimensional vector spaces). Having that background, I can say even without the supremum property in full, you still have strong enough topological properties to define unique limits, and so you still have ideas of differentiation and even integration. Here, the complex plane mapped to the complex plane is just a special case of vector calculus.

  Beyond this though, we have a unique situation: As mentioned at the start, this vector space is also an algebraic field. With that said, you can say you end up with a complex calculus/analysis which is quite similar to real calculus/analysis. The subtle differences show up because it can be said the complex number system contains the completeness property, but it doesn't have the completeness property.

  This is to say, because of the algebraic field, and because you can trace paths (which are isomorphic to the real line, and thus have the completeness property) throughout the complex plane, you end up with invariance-of-path line integrals. The amazing thing is, once you have this equivalence class of loops, you can choose which closed paths to work with, and since we know circles are very well-behaved in this system of numbers which privilege rotation, we return to this pattern of circles yet again. It becomes a natural extension that Cauchy's Theorem as well as the Cauchy Integral Formula behave so consistently throughout this space.

It's a lot of narrative conjecture to take in I admit, but maybe something to think about?

Week 6

October 2nd, 2016

This week we got into infinite series, their types, as well as tests for convergence. Aside from the concept of radius of convergence, the treatment of series is otherwise pretty well identical to that of infinite series in real analysis. As mentioned before, having seen this sort of identical-ness when studying complex calculus the first time around, this is why I didn't know how to take complex variables seriously.

The most fascinating part of the week for me was the Riemann zeta function along with the infamous Riemann hypothesis. I've looked into these ideas before, but I never had much insight into them. I'm grateful to Petra Bonfert-Taylor for the way she introduced this topic, as she has enough background to navigate it in a way the rest of us can appreciate.

Week 7

October 4th, 2016

The residue theorem with the application of evaluating improper integrals was one of my original motivations in studying complex variables when I was young and looking into them on my own. I admit, with all my efforts at the time, I never really understand the ideas in the sense of being able to prove them, or even using the technique to solve any improper integrals that came my way.

After having sat with my thoughts and reflections on the complex number system, both refreshing and relearning as well as simply just learning humbly, I no longer take the Laurent series or singularities for granted. I no longer wish to rush past them to get to the residue theorem, but rather am happy to see them for their own importance.

I'll leave this course with these final thoughts: These last two weeks on series tie in with the Cauchy integral formula due to the fact that it says all the information required to determine the image of an analytic function is embedded in a closed curve surrounding the point to be mapped. In terms of power series of analytic functions: The information within the coefficients are sufficient to completely (locally) describe its function. In a way you have both a local and global perspective of analytic functions—power series being local representations, the Cauchy integral formula being global. analytic functions seem to stabilize—to the extent that they have infinite derivatives. Furthermore, regarding power series, notice how circles show up yet again as such series are described by their annuli as radii of convergence? Maybe all of this is also a property of the interconnectedness of the component parts due to the multiplication definition? Analytic functions are forced by their rotational nature to be exceptionally smooth.