This is more of a generic question. Method of matched asymptotic expansion is a well-established method to approximate a function that is a solution of a given differential equation. However, what about the functions given by some integral representation $I(x)$ defined for $x\in(0,\infty)$, for which one can obtain an asymptotic expansion for small $x$,
$$ I(x) \sim f_1(x) + f_2(x) + \ldots, ~~~ x \to 0, $$
as well as for large $x$
$$ I(x) \sim g_1(x) + g_2(x) + \ldots, ~~~ x \to \infty. $$
Knowing these $f_n$ and $g_n$ functions how does one construct an approximate function for $I(x)$ valid on whole domain $(0,\infty)$? How does one match these two expansions in some in-between region? Is there even a well-defined procedure?
As a simple example, let's consider a Bessel function $J_0(x)$. Let me suppose I know only an integral form of this function, e.g.
$$ J_0(x) = \frac{1}{2\pi} \int_{-\pi}^\pi e^{i x \sin t} dt, $$
and that I was able to compute the asymptotics from this
$$J_0(x) = 1 - \frac{x^2}{4} + \ldots, ~~{\rm as}~~ x \to 0, $$
and
$$J_0(x) =\sqrt{\frac{2}{\pi x}}\cos\left(\frac{\pi}{4} - x\right) + \ldots, ~~{\rm as}~~ x \to \infty. $$
Just by looking at the plot (see attached), there should be a way to do the matching somehow! But how to proceed?
