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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.

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    @Leonid Kovalev. Yes, I admit it seems strange and I'm not very happy with it since of course it would be nicer to give a direct proof (but...see EDIT 2). This is why I look for consolation given by other similar cases.2012-08-18

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Here is a fact which might help:

Let $h\mapsto I_h$ denote a function defined at least in an interval $[0,h_0)$ and such that there exists two increasing nonnegative sequences $(i_n)_{n\geqslant0}$ and $(j_n)_{n\geqslant0}$ and some nonzero coefficients $(a_n)_{n\geqslant0}$ and $(b_n)_{n\geqslant0}$ such that $ I_h=\sum_{n=0}^{+\infty}a_nh^{i_n}=\sum_{n=0}^{+\infty}b_nh^{j_n}, $ in the sense that, for every $k\geqslant0$, when $h\to0^+$, $ I_h-\sum_{n=0}^{k}a_nh^{i_n}=o(h^{i_k}),\qquad I_h-\sum_{n=0}^{k}b_nh^{j_n}=o(h^{j_k}). $ Then, $i_n=j_n$ and $a_n=b_n$ for every $n\geqslant0$.

To prove this fact, assume the result is not true, consider $k=\min\{n\mid (i_n,a_n)\ne(j_n,b_n)\}$ and treat separately the cases when $i_k\ne j_k$ and when $i_k=j_k$ but $a_k\ne b_k$.

To apply this result to your case, write $ I_h=\sum_{k=0}^{+\infty}J_kh^k=\sum_{n=0}^{+\infty}a_nh^{i_n},\qquad I_h=\sum_{k,m=0}^{+\infty}J^{(\alpha)}_{k,m}h^{k + \alpha m}=\sum_{n=0}^{+\infty}b_nh^{j_n}, $ where $(i_n)_n$ enumerates the set of $k$ such that $J_k\ne0$ and $(j_n)_n$ enumerates the set of $j$ such that $ \sum_{k+\alpha m=j}J^{(\alpha)}_{k,m}\ne0. $ If $\alpha$ is irrational, for example $\alpha=\frac{\sqrt2}2$, this shows that $J^{(\alpha)}_{k,m}=0$ for every $m\ne0$. If $\alpha=\frac12$, this shows that, for every $i\geqslant0$, $ \sum_{k=0}^iJ^{(\alpha)}_{k,2i-2k+1}=0. $