What exactly is a local expansion? From time to time, I come across authors writing about performing a local expansion of a complex function $f$ at some point. This usually means performing a singular expansion (such as Laurent or Puiseux) or something more subtler &ndash; e.g., performing a singular expansion for one argument of a multivariate analytic function.

I have wondered if this term indeed means the same as "singular expansion" or possibly arbitrary "series expansion". I have not been able to find any text, in which the notion of local expansion is actually defined: it just usually strikes as bolt out of the blue. And in fact, I just cannot see any motivation for such term, as (at least in my understanding) almost all series expansions, possibly except some with infinite radius of convergence, are inherently valid only locally.

For these reasons, I would like to ask what exactly is meant by a local expansion and in case it is a synonym of some other term, what is the usual motivation of using this term instead of the other one. Thanks a lot.
Defining things locally is an extremely useful and ubiquitous modus operandi in modern mathematics. A function having what you call a 'local expansion' is one example. This appears in the definition of an analytic function:


  an analytic function is a function that is locally given by a convergent power series 
  (Wikipedia)


This means that although the function itself may not be given by a convergent power series, we can always find a small set around every point in its domain such that it is equal to a convergent power series when restricted to this set.

I think you might call this a 'local expansion'.
Defining things locally is an extremely useful and ubiquitous modus operandi in modern mathematics. A function having what you call a 'local expansion' is one example. This appears in the definition of an analytic function:


  an analytic function is a function that is locally given by a convergent power series 
  (Wikipedia)


This means that although the function itself may not be given by a convergent power series, we can always find a small set around every point in its domain such that it is equal to a convergent power series when restricted to this set.

I think you might call this a 'local expansion'.