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Where can I learn about Watson's theorem. The linked Wikipedia article states:

If $f$ is holomorphic in some region |z| < R, |arg(z)| < π/2 + ε for some positive R and ε, and, in this region, $f$ has an asymptotic series $a_0 + a_1z + ...$ with the property that the error

$f(z)-a_{0}-a_{1}z-\cdots -a_{n-1}z^{n-1}$

is bounded by $C^{n+1}n!|z|^{n}$ for all $z$ in the region (for some positive constant C); then $f$ is given by the Borel sum of its asymptotic series (in that region).

My main question is about the usefulness of the theorem. Since $f$ being holomorphic is among the hypothesis, then one must prove $f$ is holomorphic before using the theorem. But, if I only have the asymptotic series, say

$f(z) = \sum_{n = 0}^∞ \frac{(-1)^n a^{-2 (1 + n)} (2 n - 1)!}{(n - 1)!} z^n$

then how can I infer $f$ is holomorphic?

Also, any pointers to books or articles will be appreciated.

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  • $\begingroup$ If you can prove that series is absolutely convergent, then you can do a simple "swap the summation and integral" argument to prove any closed loop line integral evaluates to $0$. $\endgroup$ Mar 24, 2020 at 23:37

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