# zbMATH — the first resource for mathematics

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.)
Full Text:
##### References:
 [1] Fruchard ( A. ) et Schäfke ( R. ) .- On the Borel transform , C. R. Acad. Sci. Paris , Série I , 323 ( 1996 ), pp. 999 - 1004 . MR 1423209 | Zbl 0861.44001 · Zbl 0861.44001 [2] Malgrange ( B. ) .- Sommation des séries divergentes , Exp. Math. 13 , n^\circ 2 -3 ( 1995 ), pp. 163 - 222 . MR 1346201 | Zbl 0836.40004 · Zbl 0836.40004 [3] Ramis ( J.-P. ) . - Les séries k-sommables et leurs applications , Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory , Lecture Notes in Physics 126 ( 1980 ), pp. 178 - 199 . MR 579749 · Zbl 1251.32008 [4] Sibuya ( Y. ) . - Linear differential equations in the complexe domain, problems of analytical continuation , A.M.S. , Providence (RI) , 1990 . MR 1084379 | Zbl 00048899 · Zbl 1145.34378 [5] Titchmarsh ( E.C. ) .- The Theory of Functions , Second edition, Oxford Science Publications ( 1939 ). MR 882550 | JFM 65.0302.01 · JFM 65.0302.01 [6] Tougeron ( J.- Cl. ) . - An introduction to the theory of Gevrey expansions and to the Borel-Laplace transform with some applications , Preprint University of Toronto , Canada ( 1990 ). [7] Wasow ( W. ) . - Asymptotic expansions for ordinary differential equations , Interscience , New-York ( 1965 ). MR 203188 | Zbl 0133.35301 · Zbl 0133.35301 [8] Zhang ( C. ) .- Les développements asymptotiques q-Gevrey, les séries Gq-sommables et leurs applications , Ann. Inst. Fourier 49 , n^\circ 1 ( 1999 ) à paraître. Numdam | MR 1688144 | Zbl 0974.39009 · Zbl 0974.39009 · doi:10.5802/aif.1672 · numdam:AIF_1999__49_1_227_0 · eudml:75334
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.