# Usefulness of Watson's theorem

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.

• 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 $0$. Mar 24, 2020 at 23:37