### A class of generalized uniform asymptotic expansions.

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The paper studies the relation between asymptotically developable functions in several complex variables and their extensions as functions of real variables. A new Taylor type formula with integral remainder in several variables is an essential tool. We prove that strongly asymptotically developable functions defined on polysectors have ${C}^{\infty}$ extensions from any subpolysector; the Gevrey case is included.

Small perturbations of an equilibrium plasma satisfy the linearized magnetohydrodynamics equations. These form a mixed elliptic-hyperbolic system that in a straight-field geometry and for a fixed time frequency may be reduced to a single scalar equation div$\left({A}_{1}{\Delta}_{u}\right)+{A}_{2}u=0$, where ${A}_{1}$ may have singularities in the domaind $U$ of definition. We study the case when $U$ is a half-plane and $u$ possesses high Fourier components, analyzing the changes brought about by the singularity ${A}_{1}=\infty $. We show that absorptions of energy takes...

The maximal operator S⁎ for the spherical summation operator (or disc multiplier) ${S}_{R}$ associated with the Jacobi transform through the defining relation $\widehat{{S}_{R}f}\left(\lambda \right)={1}_{\left|\lambda \right|\le R}f\u0302\left(t\right)$ for a function f on ℝ is shown to be bounded from ${L}^{p}(\mathbb{R}\u208a,d\mu )$ into ${L}^{p}(\mathbb{R},d\mu )+L\xb2(\mathbb{R},d\mu )$ for (4α + 4)/(2α + 3) < p ≤ 2. Moreover S⁎ is bounded from ${L}^{p\u2080,1}(\mathbb{R}\u208a,d\mu )$ into ${L}^{p\u2080,\infty}(\mathbb{R},d\mu )+L\xb2(\mathbb{R},d\mu )$. In particular ${{S}_{R}f\left(t\right)}_{R>0}$ converges almost everywhere towards f, for $f\in {L}^{p}(\mathbb{R}\u208a,d\mu )$, whenever (4α + 4)/(2α + 3) < p ≤ 2.

We give a proof of the fact that any holomorphic Pfaffian form in two variables has a convergent integral curve. The proof gives an effective method to construct the solution, and we extend it to get a Gevrey type solution for a Gevrey form.

Here we present basic ideas and algorithms of Power Geometry and give a survey of some of its applications. In Section 2, we consider one generic ordinary differential equation and demonstrate how to find asymptotic forms and asymptotic expansions of its solutions. In Section 3, we demonstrate how to find expansions of solutions to Painlevé equations by this method, and we analyze singularities of plane oscillations of a satellite on an elliptic orbit. In Section 4, we consider the problem of local...

We analyze the Charlier polynomials C n(χ) and their zeros asymptotically as n → ∞. We obtain asymptotic approximations, using the limit relation between the Krawtchouk and Charlier polynomials, involving some special functions. We give numerical examples showing the accuracy of our formulas.