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PreprintCC BY-SA 4.0arXiv

When can a power series be analytically continued?

  • Kei Beauduin
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Abstract

A classical line of research characterizes analytic continuation of power series through analytic interpolation of their coefficients. Under suitable hypotheses, the growth of the interpolating function reflects the geometry of the continuation domain, and conversely. We survey the development of these results from Leau and Le Roy through Lindelöf, Carlson, Dufresnoyunicodex2013Pisot and Arakelian. We formulate the main results in a unified notation using compactifications of the complex plane. In particular, we give a comprehensive treatment of Carlson's second continuation theorem, which has received little attention, and prove a refinement based on the Laplace transform.

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Cite

BibTeXRIS
BibTeX
@misc{beauduin2026when,
  title = {{When can a power series be analytically continued?}},
  author = {Kei Beauduin},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.16676},
  eprint = {2609.16676},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.16676},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - When can a power series be analytically continued?
AU  - Kei Beauduin
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.16676
UR  - https://arxiv.org/abs/2609.16676
AB  - A classical line of research characterizes analytic continuation of power series through analytic interpolation of their coefficients. Under suitable hypotheses, the growth of the interpolating function reflects the geometry of the continuation domain, and conversely. We survey the development of these results from Leau and Le Roy through Lindelöf, Carlson, Dufresnoyunicodex2013Pisot and Arakelian. We formulate the main results in a unified notation using compactifications of the complex plane. In particular, we give a comprehensive treatment of Carlson's second continuation theorem, which has received little attention, and prove a refinement based on the Laplace transform.
ER  -