The reason that the notion of analytic continuation is interesting is the rigidity theorem for complex functions, which implies that analytic continuation is unique. This is in marked contrast to what would happen if, instead of analyticity, we only required a weaker condition such as continuity. In that case, there would exist a great number of continuations of a given function to a larger domain. In fact there are so many possibilities that the choice of continuation is largely arbitrary — one can find a continuation which agrees with any continuous function on a subset of which is separated from .
One can generalize this notion to continuation along a chain of open sets. Suppose that one has a sequence of open sets such that is not empty for between and and functions such that when for all between and . Then we say that is the indirect analytic continuation of along the chain .
A related notion is that of analytic continuation along an arc. Given an arc with endpoints and , a function defined in open neighborhood of , and a function defined in open neighborhood of we say that is the analytic continuation of along the arc if there exists an open set containing and and an analytic function defined on this open set such that agrees with at those points of the complex plane where both and are defined and agrees with at those points of the complex plane where both and are defined. By the rigidity theorem, the analytic continuation of a function along an arc is unique. This concerns especially extending a real function to an analytic function of a complex variable.
It is worth noting that it is possible to obtain different analytic continuations of the same function along two arcs with the same endpoints. For instance, if we let be the square root function, let and , and let and be the two halves of the unit circle joining to , then the result of analytically continuing from to along differs from the result of analytically continuing from to along .
This seems to contradict the uniqueness of analytic continuation. This contradiction, however, is only apparent because there does not exist a single open set which contains both and to which can be analytically continued. In fact, in order to accommodate both the analytic continuation along and the analytic continuation along , one needs to define on a Riemann surface. Then there exists an open subset of the Riemann surface which contains the lifts of both and . Since the lift of the endpoint of the lift of lies on a different sheet of the Riemann surface than the endpoint of the lift of , there is nothing contradictory about the fact that the analytic continuation of assumes different values at these two distinct points on the Riemann surface.
The notion of analytic continuation along an arc can be extended to analytic continuation along a path. Any path can be written as the union of overlapping arcs. To continue a function along a path, one successively continues it along the various arcs into which the path has been decomposed. It is a simple consequence of the rigidity theorem that the result of analytically continuing a function along a path is independent of the decomposition of the path into arcs.
|Date of creation||2014-03-13 16:38:30|
|Last modified on||2014-03-13 16:38:30|
|Last modified by||pahio (2872)|
|Defines||analytic continuation along an arc|
|Defines||analytic continuation along a path|