Explain, without proof, how to obtain an asymptotic expansion, as $x \rightarrow \infty$, of

$$I(x)=\int_{0}^{\infty} e^{-x t} f(t) d t$$

if it is known that $f(t)$ possesses an asymptotic power series as $t \rightarrow 0$.

Indicate the modification required to obtain an asymptotic expansion, under suitable conditions, of

$$\int_{-\infty}^{\infty} e^{-x t^{2}} f(t) d t$$

Find an asymptotic expansion as $z \rightarrow \infty$ of the function defined by

$$I(z)=\int_{-\infty}^{\infty} \frac{e^{-t^{2}}}{(z-t)} d t \quad(\operatorname{Im}(z)<0)$$

and its analytic continuation to $\operatorname{Im}(z) \geqslant 0$. Where are the Stokes lines, that is, the critical lines separating the Stokes regions?