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Remarks about asymptotic expansions. (Remarques sur les développements asymptotiques.) (French) Zbl 1157.30322
The authors consider analytic functions $$f(z)$$ bounded on a sector $$\alpha\leq \arg\,z\leq \beta, 0<|z|<\rho$$, with vertex at the origin of the complex $$z$$-plane. They show that if $$f(z)$$ has an asymptotic expansion as $$|z|\rightarrow 0$$ along the ray $$\arg\,z=\theta_0$$ of the sector, then the asymptotics remain valid on the whole sector. The type of expansion considered is a Gevrey expansion of order $$k$$. This is defined as follows: if $$f(z)$$ has an asymptotic expansion $$\widehat{f}_N(z)$$ along the ray $$\arg\,z=\theta_0$$, where $$N$$ is a positive integer, then $$|f(z)-\widehat{f}_N(z)|<C_N|z|^N (|z|\rightarrow 0)$$, where $$C_N$$ is a constant independent of $$z$$. If $$C_N$$ has the form $$C_N=CA^N\Gamma(1+Nk^{-1})$$, with $$A$$ and $$C$$ positive constants independent of $$N$$, then $$f(z)$$ possesses $$\widehat{f}_N(z)$$ as an asymptotic expansion of Gevrey type of order $$k$$ in the direction $$\theta_0$$. In the particular case $$k=1$$, this corresponds to the familiar “factorial divided by a power” (when $$z$$ is replaced by $$1/z$$ as the asymptotic variable) employed in Dingle’s 1973 book. When $$k>1$$ the growth in the coefficients is slower than $$N!$$.
If, in addition, the constant $$A$$ in the above bound has the form $$A=1/R(\theta)+\delta$$ for some $$\delta>0$$, then it is established how the quantity $$R(\theta)$$ varies across the sector $$\alpha\leq\arg\,z\leq\beta$$ for different $$k$$. Exponentially small expansions are also considered and the behaviour of the exponent $$R(\theta)$$ in the order estimate $$f(z)=O(\exp {-R(\theta)|z|})$$, valid along the ray $$\theta_0$$ in the sector, is obtained throughout the sector $$\alpha\leq\arg\,z\leq\beta$$.
A $$q$$-analog of these results is also discussed.

MSC:
 30E15 Asymptotic representations in the complex plane 34M30 Asymptotics and summation methods for ordinary differential equations in the complex domain 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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References:
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