# asking qualitative questions about how the geometric structure of functions on the complex plane relates to the construction of their power series representations

This is the power series for one of the Airy functions:
$\mathrm{Bi}(z) = \frac{1}{\pi\sqrt[6]{3}} \sum_{n=0}^{\infty} \frac{3^{n/3} \Gamma((1+n)/3) \sin(2(1+n)\pi/3)z^{n}}{n!}$

And this is a domain colored picture of $Bi(z)$ plotted with mpmath.

The general question here is sort of higher order to the “where are the zeros and poles of this function?” — on seeing a picture of the function you have a visceral appreciation of its geometry. I don’t need to talk about residues and whatnot: you apprehend the function (on the subset of the complex plane that has been domain colored) all at once. If I had plotted a polynomial, one could read off the zeroes very easily. But in the case of more complicated functions, it is not always clear that Weierstrass factorizations are effective, or easy to accomplish. How much of the qualitative geometric structure of complex functions can be directly read off from the general form of the power series coefficients?

# not “Misiurewicz points” but “Misiurewicz stalks”.

It’s known that at the Misiurewicz points, there’s asymptotic self similarity. But this got me thinking: when we do magnifications of fractals, we’re not really exploring the points that the set is composed of so much as their asymptotic structure, and that asymptote is not zero dimensional information, but a open subset of the plane.

So instead of a Misiurewicz point $M_{n,k}$, we’re really examining the Misiurewicz stalk $\mathcal{F}_{M_{n,k}}$ of the Mandelbrot sheaf $\mathcal{F}$.

# un abus de langage

The following is a list of words that start with “eigen”. some of these are legitimate mathematical concepts, and some may be fruitful things to look at.

eigentorsor
eigentensor
eigenvector
eigengerbe
eigenvariety
eigentwistor
eigenfractal
eigensheaf
eigenstack
eigenvector
eigenvalue
eigenintegral
eigenderivative
eigenbuilding
eigengroup
eigenfunctor
eigenshtuka
eigencohomology
eigenbundle
eigenmotive
eigenalgebra
eigenmodule
eigenring
eigenspectrum
eigencategory
eigenmorphism
eigencover
eigendiagram
eigenlemma
eigentheoerem
eigenyoga
eigenhomology
eigenresolution
eigencoherent
eigenfield
eigenlogarithm
eigenfactorial
eigenproof
eigenconjecture
eigentheorem
eigenpacket
eigendistribution

# weekend lark

$\prod_{n=1}^{\infty} \log\left(e+\frac{1}{2^{n}}\right)\approx 1.390436264416635011574732\ldots$

The obvious generalization is to consider
$H(z) = \prod_{n=1}^{\infty} \log\left(e+z^{n}\right)$ for $|z|<1$, which looks, after domain coloring, like this:

I think, that besides all the singularities at rational multiples of $\pi$, that the thing is aggressively one on most of the unit disk

# funny morning story

$e^{iz} = i^{ez}$ when $z=\frac{4\pi n}{e\pi-2}$ and $n\in\mathbb{Z}$

# weird theta function product

Consider that $\forall z\in\mathbb{C}$, $\theta_{3}(z,0)=1.0$. This should instantly cue us into the the potential existence of interesting infinite products. Therefore define

$g(z)=\prod_{n=1}^{\infty} \theta_{3}(z,e^{-n})$