For the removable singularity result, it is easier to consider g(z) = z^2 f(z) which is differentiable everywhere including at the origin and has continuous derivative so g is holomorphic. As it has power expansion starting at z^2 we conclude that f(z) = g(z) /z^2 is holomorphic everywhere. Nice video as always !
Thanks Prof. Borcherds, this was a eally enjoyable lecture. Finally we're moving beyond what I remember from my complex analysis course in 1998-99. Could we see a proof of Picard's theorem? Or maybe a reference to a readable version of it? And towards the end of this course, could you give a quick survey of "several complex variables" in comparison/contrast to "1 complex variable".
There's a lovely stochastic proof in Koerner, as follows: The image of a random walk under a holomorphic map is again a random walk, b.c. conformal. The fundamental group of a plane minus a point is abelian, while the fundamental group of a plane minus two points is not, it's a free group on two generators, which is an infnitely branching tree. Because the walk is random, the image of a random walk which passes through an essential singularity will tangle in a nonabelian way around the two missing values (if there were two missing values) once you apply the function, and this gives a random walk over a nonabelian group. A random walk on a tree can't returns to the origin, it has a probability bias toward going further and further out on the branches, so as the walk progresses, it will produce more and more complicated tangling, only coming back to no tangling finitely many times. But the random walk can't do that, because the original space is only missing one point, and you can define the homotopy classes on the original space, which has an abelian fundamental group, so the walk will untangle infinitely many times on the original space. That means it must untangle when lifted using the function, which means the image of the function must have an abelian fundmanetal group, which means it can't be missing two points or more. This proof is incredibly intuitive and immediate to understand, but it is probabilistic, and a continuous Brownian random walk is extrardinarily annoying to embed into ordinary set theory.
Many of the figures in this video came from a book Is that book just called table of functions by jankhe and edme? I can't exactly tell what the title was but this book came up after a search and I'm wondering if someone could verify
