I have a real number $I_h$ depending on a small parameter $h>0$. I want to show that it has an asymptotic expansion in integer powers $h$, i.e. there exists a sequence $(J_k)_{k}$ such that
$ I_h \sim \sum_{k=0}^\infty \ h^{k} \ J_{k} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*)$
Assume I am not able to show this directly, but that I can construct for every
$\alpha $ say in $[\frac 12,1) $ a double sequence $(J_{k,m}^{(\alpha)})_{k,m}$ such that asymptotically
$I_h \sim \sum_{k,m=0}^\infty \ h^{k + \alpha m} \ J^{(\alpha)}_{k,m}$
This should imply what I want (I consider for example $\alpha_1 =\frac 12$ and $\alpha_2=\frac{\sqrt 2}{ 2}$, so only the coefficients of integer powers can be nonzero, otherwise I have a contradiciton. )
My question is (given that what I wrote is correct): is this somehow a standard trick in asymptotic analysis? Can you give me examples of situations where this argument is used?
EDIT 1: I changed $\alpha\in[\frac 12,1]$ to $\alpha\in [\frac 12,1)$. If I had also $\alpha=1$ I would be done of course.
EDIT 2: The coefficients $(J_{k,m}^{(\alpha)})_{k,m}$ are very complicated and I would say it is hopeless to write them all down in a closed formula (they arise from a combiantion of Laplacae asymtptotics and several other expansions). And even if one manages to write down a formula, there appear quantities derived from a WKB expansion, for which it seems hard to me to get directly much more information then just existence (to show directly $(*)$ I would need to know that some complicated combinations of arbitrary high derivatives vanish at some point... ).
In brief: even if there is a direct argument to prove $(*)$, the undirect argument is much shorter and painless.