= Watson lemma
{c}
{wiki=Watson's_lemma}
= Watson's lemma
{c}
{synonym}
Suppose that, as $t\downarrow0$,
$$
f(t)\sim\sum_{n=0}^{\infty}a_nt^{\lambda_n-1},
\qquad
0<\lambda_0<\lambda_1<\cdots,
$$
and that the corresponding Laplace integral has suitable integrability and growth control away from zero. Watson lemma gives the endpoint expansion
$$
\int_0^b e^{-xt}f(t)\,dt
\sim\sum_{n=0}^{\infty}a_n\Gamma(\lambda_n)x^{-\lambda_n}
$$
as $x\to+\infty$. Each finite truncation of the local amplitude expansion produces the corresponding finite truncation and remainder of the integral expansion.
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