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?


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