Cauchy’s integral formula in complex analysis

I took a graduate course in complex analysis a while ago as an undergraduate. However, I did not actually understand it well at all, to which is a testament that much of the knowledge vanished very quickly. It pleases me though now following some intellectual maturation, after relearning certain theorems, they seem to stick more permanently, with the main ideas behind the proof more easily understandably clear than mind-disorienting, the latter of which was experienced by me too much in my early days. Shall I say it that before I must have been on drugs of something, because the way about which I approached certain things was frankly quite weird, and in retrospect, I was in many ways an animal-like creature trapped within the confines of an addled consciousness oblivious and uninhibited. Almost certainly never again will I experience anything like that. Now, I can only mentally rationalize the conscious experience of a mentally inferior creature but such cannot be experienced for real. It is almost like how an evangelical cannot imagine what it is like not to believe in God, and even goes as far as to contempt the pagan. Exaltation, exhilaration was concomitant with the leap of consciousness till it not long after established its normalcy.

Now, the last of non-mathematical writing in this post will be on the following excerpt from Grothendieck’s Récoltes et Semailles:

In those critical years I learned how to be alone. [But even] this formulation doesn’t really capture my meaning. I didn’t, in any literal sense learn to be alone, for the simple reason that this knowledge had never been unlearned during my childhood. It is a basic capacity in all of us from the day of our birth. However these three years of work in isolation [1945–1948], when I was thrown onto my own resources, following guidelines which I myself had spontaneously invented, instilled in me a strong degree of confidence, unassuming yet enduring, in my ability to do mathematics, which owes nothing to any consensus or to the fashions which pass as law….By this I mean to say: to reach out in my own way to the things I wished to learn, rather than relying on the notions of the consensus, overt or tacit, coming from a more or less extended clan of which I found myself a member, or which for any other reason laid claim to be taken as an authority. This silent consensus had informed me, both at the lycée and at the university, that one shouldn’t bother worrying about what was really meant when using a term like “volume,” which was “obviously self-evident,” “generally known,” “unproblematic,” etc….It is in this gesture of “going beyond,” to be something in oneself rather than the pawn of a consensus, the refusal to stay within a rigid circle that others have drawn around one—it is in this solitary act that one finds true creativity. All others things follow as a matter of course.

Since then I’ve had the chance, in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group, who were much more brilliant, much more “gifted” than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle—while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things that I had to learn (so I was assured), things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates, almost by sleight of hand, the most forbidding subjects.

In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still, from the perspective of thirty or thirty-five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have had to rediscover in themselves that capability which was their birthright, as it was mine: the capacity to be alone.

Grothendieck was first known to me the dimwit in a later stage of high school. At that time, I was still culturally under the idiotic and shallow social constraints of an American high school, though already visibly different, unable to detach too much from it both intellectually and psychologically. There is quite an element of what I now in recollection with benefit of hindsight can characterize as a harbinger of unusual aesthetic discernment, one exercised and already vaguely sensed back then though lacking in reinforcement in social support and confidence, and most of all, in ability. For at that time, I was still much of a species in mental bondage, more often than not driven by awe as opposed to reason. In particular, I awed and despaired at many a contemporary of very fine range of myself who on the surface appeared to me so much more endowed and quick to grasp and compute, in an environment where judgment of an individual’s capability is dominated so much more so by scores and metrics, as opposed to substance, not that I had any of the latter either.

Vaguely, I recall seeing the above passage once in high school articulated with so much of verbal richness of a height that would have overwhelmed and intimidated me at the time. It could not be understood by me how Grothendieck, this guy considered by many as greatest mathematician of the 20th century, could have actually felt dumb. Though I felt very dumb myself, I never fully lost confidence, sensing a spirit in me that saw quite differently from others, that was far less inclined to lose himself in “those invisible and despotic circles” than most around me. Now, for the first time, I can at least subjectively feel identification with Grothendieck, and perhaps I am still misinterpreting his message to some extent, though I surely feel far less at sea with respect to that now than before.

Later I had the fortune to know personally one who gave a name to this implicit phenomenon, aesthetic discernment. It has been met with ridicule as self-congratulatory artificialized by one of lesser formal achievement, a concoction of a failure in self-denial. Yet on the other hand, I have witnessed that most people are too carried away in today’s excessively artificially institutionally credentialist society that they lose sight of what is fundamentally meaningful, and sadly, those unperturbed by this ill are few and fewer. Finally, I have reflected on the question of what good is knowledge if too few can rightly perceive it. Science is always there and much of it of value remains unknown to any who has inhabited this planet, and I will conclude at that.

So, one of the theorems in that class was of course Cauchy’s integral formula, one of the most central tools in complex analysis. Formally,

Let $D$ be a bounded domain with piecewise smooth boundary. If $f(z)$ is analytic on $D$, and $f(z)$ extends smoothly to the boundary of $D$, then

$f(z) = \frac{1}{2\pi i}\int_{\partial D} \frac{f(w)}{w-z}dw,\qquad z \in D. \ \ \ \ (1)$

This theorem was actually somewhat elusive to me. I would learn it, find it deceptively obvious, and then forget it eventually, having to repeat this cycle. I now ask how one would conceive of this theorem. On that, we first observe that by continuity, we can show that the average on a circle will go to its value at the center as the radius goes to zero. With $dw = i\epsilon e^{i\theta}d\theta$, we can with the $w - z$ in the denominator, vanish out any factor of $f(z + \epsilon e^{i\theta})$ in the integrand. From this, we have the result if $D$ sufficiently small circle. Even with this, there is implicit Cauchy’s integral theorem, the one which states that integral of holomorphic function inside on closed curve is zero. Speaking of which, we can extend to any bounded domain with piecewise smooth boundary along the same principle.

Cauchy’s integral formula is powerful when the integrand is bounded. We have already seen this in Montel’s theorem. In another even simpler case, in Riemann’s theorem on removable singularities, we can with our upper bound on the integrand $M$, establish with $M / r^n$ establish that for $n < 0$, the coefficient in the Laurent series about the point is $a_n = 0$.

This integral formula extends to all derivatives by differentiating. Inductively, with uniform convergence of the integrand, one can show that

$f^{(m)}(z) = \frac{m!}{2\pi i}\int_{\partial D} \frac{f(w)}{(w-z)^{m+1}}dw, \qquad z \in D, m \geq 0$.

An application of this for a bounded entire function would be to contour integrate along an arbitrarily large circle to derive an $n!M / R^n$ upper bound (which goes to $0$ as $R \to \infty$) on the derivatives. This gives us Liouville’s theorem, which states that bounded entire functions are constant, by Taylor series.