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Watson lemma expands a Laplace integral by integrating the small-$t$ expansion term by term. More precisely, if $f(t)\sim {\sum }_{j\ge 0}{a}_j{t}^{\lambda +j-1}$ with $\mathrm{Re}\lambda >0$, and the tail satisfies bounds permitting this localization, thenFor ordinary powers $f(t)\sim \sum c_j{t}^j$, the coefficients are $c_{j}j!$. For the Gaussian weight, put $s=t^2$ separately on the positive and negative half-lines. The integral becomes a Laplace integral with amplitude $[f(\sqrt{s})+f(-\sqrt{s})]/(2\sqrt{s})$. Odd terms cancel and Watson lemma givesThese assertions concern asymptotic expansions, not necessarily convergent series.

For the complex integral in the lower half-plane, the identity $(z-t{)}^{-1}=i{\int }_0^{\infty }{e}^{-i(z-t)s}ds$ and the Gaussian Fourier transform giveThe last expression, initially obtained by completing the square, is entire and therefore gives the requested analytic continuation. Expanding the denominator formally, with odd Gaussian moments zero, gives the algebraic expansionHere $(-1)!!=1$. The complementary error function expansion shows $I(z)\sim \mathcal{A}(z)$ on closed subsectors of $-5\pi /4<\arg z<\pi /4$, using an unwrapped argument centered on the lower half-plane.

In the upper half-plane, the directly evaluated real-contour integral is $J(z)=-i\pi e^{-z^2}\mathrm{erfc}(-iz)$. Since $\mathrm{erfc}(w)+\mathrm{erfc}(-w)=2$, continuation from below givesThe upper expansion for $J$ holds on closed subsectors of $-\pi /4<\arg z<5\pi /4$. The added term is the residue acquired by the continued contour passing the pole; it is exponentially dominant in the upper sector $\pi /4<\arg z<3\pi /4$ and exponentially small near either real ray. On the real axis the continued value is the principal-value integral plus $i\pi e^{-x^2}$.

To make the Stokes phenomenon convention explicit, the exponential switches its asymptotic multiplier across the real rays $\arg z=0,\pi$; these are the Stokes switching lines. The diagonal rays $\arg z=\pi /4+j\pi /2$ separate exponential growth from decay and are often called anti-Stokes lines, or critical boundaries when regions are classified by dominance. Thus both sets of critical directions and the exact jump are specified, independently of which terminology is used. This is the Gaussian Cauchy transform and its Stokes jump.