Watson lemma

Suppose that, as $t\downarrow 0$,and that the corresponding Laplace integral has suitable integrability and growth control away from zero. Watson lemma gives the endpoint expansionas $x\to +\infty$. Each finite truncation of the local amplitude expansion produces the corresponding finite truncation and remainder of the integral expansion.

Under the corresponding holomorphic and uniform remainder hypotheses, the expansion in Watson lemma remains valid for a complex large parameter $z$, uniformly in every closed subsectorwith $0<\delta <\pi /2$. A rotation of an endpoint contour reduces oscillatory factors to this decaying Laplace form.

Expanding $(1+t{)}^{-1}$ at the endpoint and applying Watson's lemma gives