Algebraic-analytical Study of non-local pseudo-differential equations in complex domains

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The aims of this research were as follows :[1]To study algebraic-analytically non-local pseudo-differential equations in complex domains.[2]To construct holomorphic solutions to such equations and to establish the operational calculus.[3]To study algebraic-analytically differential-differnce eqations in complex domains.For [1], at first, we have defined cohomologically the classes of the non-local pseudo-differential operators operating to holomorphic functions defined at a fixed point, as well as their composition and the operation to functions. And secondly, using the Fourier-Sato transform, we re-defined the class of non-local pseudo-differential operators and proved functorially their composition. For [2], in the case of constant coefficient, we established a concrete integral representation formula for solutions to the non-local pseudo-differential equation, using the formal inverse operator. We also obtained concretely generators of the solution space of corresponding homogeneous equation. For [3], using the formula obtained in [2], we calculated special solutions to differential-difference equations with constant coefficient.