Legendre uniformization of multi-valued analytic functions

In the first part of this article, the notion of Legendre uniformization of a multi-valued analytic function is introduced. This is a global generalization of the Leray uniformization, and consists in representing the function as an integral of Feynman type naturally associated with a certain Legendre manifold.
In the second part of the article, a transformation of Fourier type is defined for a certain class of uniformized functions, and an inversion formula is proved. Also, some natural commutation relations with differentiations and multiplications by the independent variables are established.
Bibliography: 11 titles.