Editing Extrapolation (section)

==In the complex plane==

In [[complex analysis]], a problem of extrapolation may be converted into an [[interpolation]] problem by the change of variable <math>\hat{z} = 1/z</math>.  This transform exchanges the part of the [[complex plane]] inside the [[unit circle]] with the part of the complex plane outside of the unit circle.  In particular, the [[compactification (mathematics)|compactification]] [[point at infinity]] is mapped to the origin and vice versa.  Care must be taken with this transform however, since the original function may have had "features", for example [[pole (complex analysis)|poles]] and other [[mathematical singularity|singularities]], at infinity that were not evident from the sampled data.

Another problem of extrapolation is loosely related to the problem of [[analytic continuation]], where (typically) a [[power series]] representation of a [[function (mathematics)|function]] is expanded at one of its points of [[limit of a function|convergence]] to produce a [[power series]] with a larger [[radius of convergence]].  In effect, a set of data from a small region is used to extrapolate a function onto a larger region.

Again, [[analytic continuation]] can be thwarted by [[function (mathematics)|function]] features that were not evident from the initial data.

Also, one may use   [[sequence transformation]]s like [[Padé approximant]]s and [[Levin-type sequence transformation]]s as extrapolation methods that lead to a [[summation]] of  [[power series]] that are divergent outside the original [[radius of convergence]]. In this case, one often obtains 
[[rational approximant]]s.