Comments: 44 pages
Abstract: In the analytic theory of the linear differential equations
one distinguishes regular and irregular singular points. In a neighbourhood
of the irregular singular point any solution may be sought as the product
of the three factors: of the appropriate exponential function, of a power
function (with a non-integer exponent, in general) and of a Taylor series.
The exponent of this power function and the Taylor series coefficients
may be found algorithmically, by substitution of the product into the equation.
The convergence radius of the Taylor series equals zero, so the series
is a formal one. Despite of this the series coefficients save
in principle all information on the solution. This information may
be recovered by means of a summation method applied to the series.
The Borel summation method provides a basis for such summation methods
adjusted to the considered differential equation. In this paper this program
is realised for the Bessel differential equation. The Hankel functions
were chosen as a natural basis of solutions related to the infinity which
is an irregular singular point for the equation. Both the Hankel functions
are the Laplace transforms of the different branches of the same multivalued
analytic function - the Gauss Hypergeometric function, which serves as
the Borel transform
for each of these Hankel functions. Using this, we express the Stokes
matrix for the Bessel Equation from an entry of a connection matrix
for the Hypergeometric Equation. The branching properties of Hypergeometric
Functions play the crucial role here. The paper may be considered as an
Introduction into the Introduction into the so-called Resurgent Analysis.
This is the first manuscript in the presumed serie of manuscripts dedicated
to the Laplace-Borel-like transforms adjusted to the differential equations
in the complex domain.