I was trying to figure out how this works algebraically, and I think I understand for polynomials (or power series) at least. Basically, for a really small num

I was trying to figure out how this works algebraically, and I think I understand for polynomials (or power series) at least. Basically, for a really small number a, ai has the property that (ai)^2 is effectively zero, so one can think of this as working with the dual numbers themselves (i.e., throwing out the x^2 term and onward in a power series). The focus on analytic functions is then because they are representable by power series at each point. I think this is what lloda is referring to.