This semester has been a bit of a chore. I have had to try and be consistent with my working habits, instead of sprint forward, pausing, and repeating. This change has been both invigorating and exhausting and has left me lost in left field mathematically. On the one hand, I have always been a so called “algebra-boy” down to the way I eat my corn. On the other hand, after soldiering through enough of the boring jargon, over-technical ideas and cumbersome proof of things in analysis I am starting to enjoy it.
Now hear me out, I am not a flip-flopper as much as it just also has its merits. The early analysis everyone takes part in is still the bane of my existence. It feels like a pep-rally where the cheerleaders yell, “Epsilon,” and the students yell back, “Delta.” There really isn’t much more to early analysis than this, which makes it useful for lots of things. Analysis is the framework behind calculus though one would never know it. Analysis thus support all the outlandish estimations made by engineers, and provides easy rules for physicists to solve integrals. Sadly though this low level analysis isn’t much food for a brain.
The real kicker is higher level analysis, the Lebesgue integral and the Cauchy-Riemann equations for complex integration. The Lebesgue integral is almost impossible to compute by hand in but is provably integrable and allows us to talk about integrals of crazy functions such as the Dirichlet function which the regular (Riemann) integral just can’t handle. Thus the Lebesgue integral is this powerful tool which allows us to examine some particularly interesting functions. The Cauchy-Riemann equations are another wild ride which tell us what exactly complex derivates must look like and what they depend on. The real kicker though is that they can be used to show that some functions aren’t integrable at all. That is some functions we normally would expect to have antiderivates don’t have them. This is an absolute shocker.
Between these two facts my mind has been in a haze of analysis. I have been waking up in the middle of the night to write down little facts about a proof. It’s gotten so bad that I have actually woken up and solved proofs at 4 am because an idea came to me in my sleep that was so good I just got up and had to write it down. All in all, analysis has becoming interesting.