Function element

A pair (analytic function, domain) -- one piece used to build up an analytic continuation.

A function element is a pair consisting of an analytic function together with the domain (typically a disk) on which it is given by a convergent power series. Analytic continuation proceeds element by element, chaining together overlapping function elements that agree on their common region, so a full continuation is really a sequence, or a whole family, of function elements linked this way rather than one formula valid everywhere at once.

Examples

The pair consisting of $\sum z^n$ and the disk $\{|z|<1\}$ is a function element; continuing it along the negative real axis produces new function elements centered elsewhere.
Two function elements are direct continuations of each other exactly when their domains overlap and they agree on the overlap.