Abstract
The re-quantization method—one of the resurgent analysis methods of current importance—is developed in this study. It is widely used in the analytical theory of linear differential equations. With the help of the re-quantization method, the problem of constructing the asymptotics of the inverse Laplace–Borel transform is solved for a particular type of functions with holomorphic coefficients that exponentially grow at zero. Two examples of constructing the uniform asymptotics at infinity for the second- and forth-order differential equations with the help of the re-quantization method and the result obtained in this study are considered.
Highlights
This study is devoted to the application of the re-quantization method to constructing asymptotics of solutions to ordinary differential equations with holomorphic coefficients in the neighborhood of infinity
The infinite extensibility theorem is proven, the Laplace–Borel transform is applied one more time, and asymptotics are constructed for the resulting equation; they make it possible to find
We have developed the re-quantization method
Summary
This study is devoted to the application of the re-quantization method to constructing asymptotics of solutions to ordinary differential equations with holomorphic coefficients in the neighborhood of infinity. This result is generalized to the case of the linear partial differential equations with additional constraint conditions imposed on the differentiation operator symbol This result makes it possible to apply the resurgent analysis methods to construct asymptotics of solutions to both ordinary and partial linear differential equations with holomorphic coefficients. This method is used in the case where the integro-differential equation in the dual space cannot be solved by the method of successive approximations and is reduced, in turn, to an equation with beak-type degeneracies In this case, the infinite extensibility theorem is proven (it was already proven for ordinary differential equations), the Laplace–Borel transform is applied one more time, and asymptotics are constructed for the resulting equation; they make it possible to find. It should be noted that for the Laplace–Borel k-transform, the following formulas are valid: Bk
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