Thus no single choice of z₀ will give a power series expansion of f(z) valid for all z∈ℂ⧵-1,1 even though f is analytic on this set.

In this we may use as our starting-point a general discussion of the theory of power series with complex variables and complex coefficients. The construction of such a theory of power series offers no difficulty once we define the concept of limit in the domain of complex numbers; in fact, it follows the theory of real power series almost exactly.

1899, Oskar Bolza, The Theory of Functions, 2013, Reprint, Books on Demand, page 69,
From this theorem (for which in many cases Cauchy's theorem on double sums may be substituted) follow easily the rules for the multiplication and division of power series, Taylor's theorem for power series along with the rules for differentiation of power series and series of power series.

Power series

Meaning

Noun