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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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