reexpanding

Reexpanding is the process of expressing a function, expression, or object that has already been expanded as a series in terms of a new center or a different set of variables. In mathematics, it most often means reexpressing a Taylor or Laurent series around a different expansion point, rather than keeping the original center fixed.

Method and example: If a function f(x) has a series expansion about x0, such as f(x) = sum_{n≥0} a_n (x − x0)^n, reexpanding about a new point x1 involves substituting x = x1 + h and re expanding in powers of (x − x1). A simple illustration is e^x, which has the expansion about 0 as sum_{n≥0} x^n/n!. Reexpanding about x = 1 gives e^x = e · sum_{n≥0} (x − 1)^n / n!.

Applications and purpose: Reexpansion helps improve convergence in a target region, facilitates the combination of expansions from different centers, and supports piecewise or multivariable approximations. In numerical analysis, reexpansion can yield more stable or accurate representations when the original center lies far from the region of interest.

Limitations: The radius of convergence around the original center bounds the validity of the reexpanded series, and reexpansion cannot fix intrinsic divergence of the function’s representation. In practice, multiple centers or hybrid methods (such as piecewise expansions or rational approximants) are used to cover larger domains.

Relation to other concepts: Reexpansion differs from resummation, which aims to assign finite values to divergent series. It is also related to analytic continuation, change of variables, and the use of alternative bases for representing a function.