We introduce and study the sheaf of Deligne to describe singular points of a linear differential operator $D$ and we develop a technique based on homological algebra to prove index theorems for $D$.As particular cases, we obtain index theorems for $D$ acting in spaces of multisummable series and a new proof of the index theorem of Malgrange in the space of convergent power series and of the index theorems of Ramis in the spaces of Gevrey series.We compute the values of these indices in terms of the formal...

This paper is concerned with square integrable quasi-derivatives for any solution of a general quasi-differential equation of $n$th order with complex coefficients $M\left[y\right]-\lambda wy=wf(t,y^{\left[0\right]},...,y^{[n-1]})$, $t\in [a,b)$ provided that all $r$th quasi-derivatives of solutions of $M\left[y\right]-\lambda wy=0$ and all solutions of its normal adjoint $M^{+}\left[z\right]-\overline{\lambda }wz=0$ are in $L_w^2(a,b)$ and under suitable conditions on the function $f$.

We investigate the growth and Borel exceptional values of meromorphic solutions of the Riccati differential equation w' = a(z) + b(z)w + w², where a(z) and b(z) are meromorphic functions. In particular, we correct a result of E. Hille [Ordinary Differential Equations in the Complex Domain, 1976] and get a precise estimate on the growth order of the transcendental meromorphic solution w(z); and if at least one of a(z) and b(z) is non-constant, then we show that w(z)...

In this paper we analyze movable singularities of the solutions of the equation for self-similar profiles resulting from semilinear wave equation. We study local analytic solutions around two fixed singularity points of this equation- ρ = 0 and ρ = 1. The movable singularities of local analytic solutions at the origin will be connected with those of the Lane-Emden equation. The function describing approximately their position on the complex plane will be derived. For ρ > 1 some topological considerations...

This paper is divided in two parts. In the first part we study a convergent $q$-analog of the divergent Euler series, with $q\in (0,1)$, and we show how the Borel sum of a generic Gevrey formal solution to a differential equation can be uniformly approximated on a convenient sector by a meromorphic solution of a corresponding $q$-difference equation. In the second part, we work under the assumption $q\in (1,+\infty )$. In this case, at least four different $q$-Borel sums of a divergent power series solution of an irregular singular...

This work is devoted to the study of a Cauchy problem for a certain family of q-difference-differential equations having Fuchsian and irregular singularities. For given formal initial conditions, we first prove the existence of a unique formal power series X̂(t,z) solving the problem. Under appropriate conditions, q-Borel and q-Laplace techniques (firstly developed by J.-P. Ramis and C. Zhang) help us in order to construct actual holomorphic solutions of the Cauchy problem whose q-asymptotic expansion...

We study conditions involving the critical set of a regular polynomial endomorphism f∶ℂ2↦ℂ2 under which all complete external rays from infinity for f have well defined endpoints.

The zeros $c_k\left(\nu \right)$ of the solution $z(t,\nu )$ of the differential equation $z^{\text{'}\text{'}}+q(t,\nu )z=0$ are investigated when $\underset{t\to \infty }{lim}q(t,\nu )=1$, ${\int }^{\infty }|q(t,\nu )-1|dt<\infty$ and $q(t,\nu )$ has some monotonicity properties as $t\to \infty$. The notion $c_{\kappa }\left(\nu \right)$ is introduced also for $\kappa$ real, too. We are particularly interested in solutions $z(t,\nu )$ which are “close" to the functions $sint$, $cost$ when $t$ is large. We derive a formula for $d{c}_{\kappa }\left(\nu \right)/d\nu$ and apply the result to Bessel differential equation, where we introduce new pair of linearly independent solutions replacing the usual pair $J_{\nu }\left(t\right)$, $Y_{\nu }\left(t\right)$. We show the concavity of $c_{\kappa }\left(\nu \right)$ for $\left|\nu \right|\ge \frac{1}{2}$ and also...

We first introduce the notion of microdifferential operators of WKB type and then develop their exact WKB analysis using microlocal analysis; a recursive way of constructing a WKB solution for such an operator is given through the symbol calculus of microdifferential operators, and their local structure near their turning points is discussed by a Weierstrass-type division theorem for such operators. A detailed study of the Berk-Book equation is given in Appendix.