Analytic Continuation - Formal Definition of A Germ

Formal Definition of A Germ

Let

be a power series converging in the disc D_r(z₀) defined by

.

Note that without loss of generality, here and below, we will always assume that a maximal such r was chosen, even if that r is ∞. Also note that it would be equivalent to begin with an analytic function defined on some small open set. We say that the vector

g = (z₀, α₀, α₁, α₂, ...)

is a germ of f. The base g₀ of g is z₀, the stem of g is (α₀, α₁, α₂, ...) and the top g₁ of g is α₀. The top of g is the value of f at z₀, the bottom of g.

Any vector g = (z₀, α₀, α₁, ...) is a germ if it represents a power series of an analytic function around z₀ with some radius of convergence r > 0. Therefore, we can safely speak of the set of germs .