Abstract.
Until now, the area of symbolic computation has mainly focused on the manipulation of algebraic expressions. It would be interesting to apply a similar spirit of “exact computations” to the field of mathematical analysis.
One important step for such a project is the ability to compute with computable complex numbers and computable analytic functions. Such computations include effective analytic continuation, the exploration of Riemann surfaces and the study of singularities. This paper aims at providing some first contributions in this direction, both from a theoretical point of view (such as precise definitions of computable Riemann surfaces and computable analytic functions) and a practical one (how to compute bounds and analytic continuations in a reasonably efficient way).
We started to implement some of the algorithms in the MMXLIB library. However, during the implementation, it became apparent that further study was necessary, giving rise to the present paper.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by the ANR Gecko project.
The paper was originally written using GNU \(\hbox{\TeX}_{\rm MACS}{}\) (see www.texmacs.org). Unfortunately, Birkhäuser insists on the use of \(\hbox{\LaTeX}{}\) with its own style files. Any insufficiencies in the typesetting quality should therefore be imputed to Birkhäuser.
Rights and permissions
About this article
Cite this article
van der Hoeven, J. On Effective Analytic Continuation. Math.comput.sci. 1, 111–175 (2007). https://doi.org/10.1007/s11786-007-0006-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11786-007-0006-6