---
tag: BW830VCE98
kind: statement
role: lemma
name: Watson's lemma
msc: [41A60, 44A10, "30E15"]
uses: [B53THC8F79, 05XT5AVJBK, 0V3FZMVB4K]
about: [BJQTEPEZ9G, 0V49JEZW06]
sources:
  - url: https://en.wikipedia.org/wiki/Watson%27s_lemma
same_as:
  wikidata: Q7974787
  wikipedia: Watson's lemma
---

## Statement

Let $0 < T \le \infty$, let $\lambda > -1$, and let $f \colon (0, T) \to \C$ be measurable.
Assume that:

1. $f$ has the [[asymptotic expansion|B53THC8F79]] $f(t) \sim \sum_{n \ge 0} a_n t^{n + \lambda}$ as $t \to 0^+$, with complex $a_n$.
2. There is a real number $x_0$ such that $\int_0^T e^{-x_0 t} \abs{f(t)} \, dt < \infty$.

Then the integral $F(x) = \int_0^T e^{-xt} f(t) \, dt$ converges absolutely for every $x \ge x_0$, and
$$F(x) \sim \sum_{n \ge 0} a_n \frac{\Gamma(n + \lambda + 1)}{x^{n + \lambda + 1}} \quad \text{as } x \to +\infty,$$
where $\Gamma$ is the [[gamma function|0V3FZMVB4K]].

## Remarks

Idea of the proof: for $x$ large, the factor $e^{-xt}$ makes the integral over $[\delta, T)$ exponentially small in $x$, for any fixed $\delta > 0$.
On $(0, \delta)$, replace $f$ by its first $N$ terms plus an error bounded by $C t^{N + \lambda}$.
Each term gives $\int_0^\infty e^{-xt} t^{n + \lambda} \, dt = \Gamma(n + \lambda + 1) x^{-n-\lambda-1}$, up to an exponentially small change, and the error contributes $O(x^{-N-\lambda-1})$.
Example: $\int_0^\infty \frac{e^{-xt}}{1 + t} \, dt \sim \sum_{n \ge 0} (-1)^n \frac{n!}{x^{n+1}}$, a divergent series obtained from $\frac{1}{1+t} = \sum (-1)^n t^n$ near $0$.
The condition $\lambda > -1$ is needed: if $\lambda \le -1$ and $a_0 \ne 0$, then $\abs{f(t)}$ is of order $t^{\lambda}$ near $0$ and the integral diverges at $t = 0$.

## Explanations

### L1 human

For a large parameter $x$, the weight $e^{-xt}$ is tiny except for $t$ very close to $0$.
So the integral only sees the behaviour of $f$ near $0$.
Watson's lemma makes this precise: expand $f$ in powers of $t$ near $0$, integrate term by term, and the result is the correct asymptotic expansion of the integral.

### L2 human

The lemma gives the large-$x$ behaviour of the [[Laplace transform|05XT5AVJBK]] $F(x)$ of $f$ from the small-$t$ behaviour of $f$.
Together with a change of variables it yields [[Laplace's method|9G7K8DBT9Y]] and the method of steepest descent for integrals $\int e^{x \phi(t)} g(t) \, dt$.
The resulting series often diverge, and [[Borel summation|8DJPSYSHGJ]] can be seen as the inverse process that recovers $F$ from its series.