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Can a power series or a Laurent series always be analytically continued into a domain strictly larger than its convergent annulus of finite radii? Is there a way to find its maximal domain that it can analytically continue into?

As an example, what are the answers to the above questions for the following two summations from this answer?

$$p(x)=\sum_{n=0}^\infty \frac{x^n}{e^{\sqrt n}},\quad l(x)=\sum_{n=-\infty}^\infty \frac{x^n}{e^{\sqrt{|n|}}}.$$

Hans
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  • There are certainly power series that cannot be extended beyond the radius of convergence, but are you assuming the power series converges everywhere at the radius of convergence, as happens for the series in the question you link to? – Gerry Myerson Dec 28 '18 at 03:02
  • @GerryMyerson: Could you please give an example of power series that cannot be analytically continued beyond the radius of convergence? What happens if the power series 1) converges everywhere, 2) converges at some points on the circumference of the convergence disk? For case 2) the power series is continuous there by Abel's theorem. – Hans Dec 28 '18 at 03:31
  • @Story123: That is the wrong example. Your series analytically continues to the meromorphic function $\frac z{1-z}$ on the whole complex plane with a simple pole at $1$. – Hans Dec 28 '18 at 05:41
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    I've wondered about something similar in Are the convergence radii circles of a Laurent-series always caused by isolated singularities?. The questions How to switch to a Laurent series' next convergence ring? and How to properly translate the coefficients of a Taylor series? are also related. – Tobias Kienzler Dec 28 '18 at 11:17
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    The concept you're looking for, Hans, is "natural boundary", and a search for that will bring you examples. I think the standard example of a power series with natural boundary the unit circle (that is, converges inside, and can't be analytically continued at all outside) is $\sum_1^{\infty}z^{2^n}$. – Gerry Myerson Dec 28 '18 at 18:39
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    See also https://math.stackexchange.com/questions/1347085/find-the-natural-boundary-of-sum-n-1-infty-fraczn1-zn and https://math.stackexchange.com/questions/1962354/given-complex-series-the-natural-boundary-is-the-unit-circle and https://math.stackexchange.com/questions/1320321/connection-between-convergence-on-natural-boundary-and-weierstrass-functions and https://math.stackexchange.com/questions/610862/solution-verification-the-series-sum-zn-has-the-unit-circle-as-a-natura and probably several others. – Gerry Myerson Dec 28 '18 at 18:43
  • Had a chance to follow up on my leads, Hans? – Gerry Myerson Dec 30 '18 at 02:40
  • @GerryMyerson: Yes, I have. Thank you, Gerry. These, good, examples all treat the natural boundary on the unit circle. But they do not answer the more general question of how to obtain the natural boundary of any given complex function when the natural boundary is not necessarily the unit circle. The monodromy theorem https://en.wikipedia.org/wiki/Monodromy_theorem is related. Do you have any references on the more general question? – Hans Dec 30 '18 at 10:03
  • Sorry, Hans, I don't know whether there is a general answer to the general question. – Gerry Myerson Dec 30 '18 at 14:02

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