kernel singular integral, Fourier series, Poisson singular integral, Fatou theorem, Poisson-Abel method, harmonic function, meromorphic function, the Mittag-Leffler theorem, the formula Schwartz formula Hilbert integral of Cauchy type, etc.

With the help of K alpha − summation formula functional series [1] obtained a new singular integral having numerous applications. Of singular integral is obtained as a consequence of the singular Poisson integral and thus – Schwartz formula, enabling express an analytic function in the disc through its real part on circle up to a constant imaginary term. In addition, from new singular integral is obtained as a consequence of the singular integral with Hilbert associated with the boundary value problem of Hilbert.

Kuliev, Valekh Dzhafarovich Dr. Sci. Phys. & Math., Professor, Head of the Department of the Higher Mathematics, Moscow State Machine-Building University (MAMI), Moscow