This is the power series for one of the Airy functions:

And this is a domain colored picture of 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?*