On a Watson-like Uniqueness Theorem and Gevrey Expansions

We present a maximal class of analytic functions, elements of which are in one-to-one correspondence with their asymptotic expansions. In recent decades it has been realized (B. Malgrange, J. Ecalle, J.-P. Ramis, Y. Sibuya et al.), that the formal power series solutions of a wide range of systems of ordinary (even non-linear) analytic differential equations are in fact the Gevrey expansions for the regular solutions. Watson's uniqueness theorem belongs to the foundations of this new theory. This paper contains a discussion of an extension of Watson's uniqueness theorem for classes of functions which admit a Gevrey expansion in angular regions of the complex plane with opening less than or equal to (\frac \pi k,) where (k) is the order of the Gevrey expansion. We present conditions which ensure uniqueness and which suggest an extension of Watson's representation theorem. These results may be applied for solutions of certain classes of differential equations to obtain the best accuracy estimate for the deviation of a solution from a finite sum of the corresponding Gevrey expansion.Comment: 18 pages, 4 figure

Error bounds, duality and Stokes phenomenon: I

We consider classes of functions uniquely determined by coefficients of their divergent expansions. Approximating a function from such a class by partial sums of its expansion, we study how the accuracy changes when we move within a given region of the complex plane. Analysis of these changes allows us to propose a theory of divergent expansions, which includes a duality theorem and the Stokes phenomenon as essential parts. In its turn, this enables us to formulate necessary and sufficient conditions for a particular divergent expansion to encounter the Stokes phenomenon. We derive explicit expressions for the exponentially small terms that appear upon crossing Stokes lines and lead to improvement in the accuracy of the expansion

A new insight into Bessel's equation

The principal idea of this paper is to apply Fourier transforms to the Stokes structure rather than to the original differential equation. It will be shown below how to derive the Stokes structure directly from differential equation without any previous knowledge of Bessel or hypergeometric functions. It will be shown further how to adjust a Fourier transform for the Stokes structure and how to answer questions on the interrelation between formal and regular solutions of Bessel's equation using a respective Fourier analysis. It will be shown finally how to evaluate coefficients of the Stokes structure which yields a new insight into the Stokes phenomenon. Although the main object here is Bessel's equation our approach can be extended to more general matrix equations with many applications in areas such as spectral and scattering theory and hydrodynamics