The generalized Borel transform and Stokes multipliers

We propose a new approach to the complex WKB method for matrix linear differential equations with meromorphic coefficients. The essence of our method which is the further development of the well known Borel summation method (see [1, 2] and also [3, 4]) is the following. It is known that the WKB (formal) solution of equation is presented as a product of some exponential factor with factorially divergent power series. One can construct the one-to-one correspondence between this formal series and some analytical function on Riemann surface. This function is analogous to the classical Borel transform. Many of the properties of the actual solutions can be studied using this transform. In particular one can evaluate the so-called connection coefficients or Stokes multipliers. We shall present the explicit formulae for such generalization of the Borel transform and for evaluation of the Stokes multipliers in the general case. There is a crucial difference between the second order and the third or higher order differential equations which probably was not known so far. Our method could find applications in the scattering and spectral theories. We shall illustrate the details of this approach in our second paper.