Watson's 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.

For $g(s)=-(1/a)e^{ias}{\int }_0^{\infty }{x}^{s+\sigma -1}{e}^{-tx}/(ia+\log x)dx$ with positive $a,t,\sigma$ and $s\ge 0$, differentiation cancels the logarithmic denominator and gives $g_s=-(1/a)e^{ias}\Gamma (s+\sigma )t^{-s-\sigma }$. Integrating this derivative expresses $g(0)$ as a finite parameter-Laplace integral plus $g(s)$. The elementary lower bound $|ia+\log x|\ge a$ gives the displayed remainder, smaller than every logarithmic term at fixed $s>0$. Watson's lemma then determines all coefficients from the finite integral's endpoint Taylor series.

A cut-off endpoint integral with amplitude $(\log u{)}^{-2}$ has a logarithmic rather than a Taylor endpoint expansion. Set $v=xu$ and expand $(\log x-\log v{)}^{-2}$ against $e^{-v}dv/x$. The derivatives of the gamma function at one supply the logarithmic moments: ${\int }_0^{\infty }{e}^{-v}\log vdv=-{\gamma }_E$ and ${\int }_0^{\infty }{e}^{-v}(\log v{)}^{2}dv={\gamma }_E^2+{\pi }^2/6$. Other peaks or singularities must be treated separately before applying this endpoint result to a whole integral.

For real $x\to +\infty$, deformation into the upper half-plane separates a logarithmically singular endpoint at zero from the smooth endpoint at one. The first contributes the exact logarithmic term displayed. The second yields its factorial asymptotic expansion by Watson's lemma. The two endpoints must both be retained; repeated integration by parts without treating the logarithmic singularity loses its leading contribution.

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