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On an integral transform of complex analytic functions. (English. Russian original) Zbl 0634.44002
Math. USSR, Izv. 29, 407-427 (1987); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50, No. 5, 1054-1076 (1986).
The authors introduce an invertible integral transform in the space of multivalent complex analytic functions (analogous to the real d/d$$\tau$$- integral Fourier transform of V. P. Maslov: Perturbation theory and asymptotic methods (1965; French translation 1972; Zbl 0247.47010), in order to solve complex differential equations and to give an explicit method of investigation of the singularities of the obtained solution. For an integral transform of Fourier type for holomorphic functions see: J. Sjöstrand, Astérisque 95, 1-166 (1982; Zbl 0524.35007)].
In the first two sections the Legendre transform of an analytic set is defined and the integration contours (homology classes) are studied. The third section is devoted to the study of the space of the convenient multivalent complex analytic functions. In the central section (four) the integral transform is defined, its invertibility is proved and some basic commutation formulae are given. In the fifth section the solution of the Cauchy problem for differential equations with constant coefficients is given, and in the final section some concrete examples illustrate the method of investigation of the singularities of the solutions.

##### MSC:
 44A15 Special integral transforms (Legendre, Hilbert, etc.) 35A22 Transform methods (e.g., integral transforms) applied to PDEs 35C15 Integral representations of solutions to PDEs 58J40 Pseudodifferential and Fourier integral operators on manifolds
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