Weierstrass Analytic Functions

Weierstrass introduced a complete analytic function as a set consisting of a power series and all its analytic continuations. This definition — a fundamental expansion of Lagrange’s introduction of a function as a power series — was formany decades the only alternative to the classical introduction of functions as laws or rules associating or pairing numbers with numbers or à la Dirichlet — as those associations themselves. These traditional introductions are not entirely transparent; for, while there exist definite procedures for operations with sets (e.g., with sets of power series or of numbers or of pairs of numbers) traditional mathematics lacks definitions of operations with laws or rules or associationes. Weierstrass’ alternative was confined to the realm of analytic functions but it was perfectly clear.