Geodesic
Legendre uniformization of multi-valued analytic functions
Sbornik. Mathematics, Tome 41 (1982) no. 2, pp. 217-234

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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. 
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     author = {B. Yu. Sternin and V. E. Shatalov},
     title = {Legendre uniformization of multi-valued analytic functions},
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B. Yu. Sternin; V. E. Shatalov. Legendre uniformization of multi-valued analytic functions. Sbornik. Mathematics, Tome 41 (1982) no. 2, pp. 217-234. http://geodesic.mathdoc.fr/item/SM_1982_41_2_a3/