Vagner Jiqia

 It is known, that the Euler integral of the second kind converges for each z of the complex plane under the condition: Rez>0

 According to recent studies, the above-mentioned integral defines a generalized function G(-α+iγ) in the singularity region (α≥0) In this abstract, the analytical form of the functional G(-α+iγ) is proposed at the points α=0.

Recent studies show that the Euler integral of the first kind for some pairs of parameters is integrable in the sense of generalized functions in the singularity domain. In this abstract, an analytical functional B(iγ-m, -iγ-n) is proposed, which is determined by the above integral.   

The new representation was proposed for the Fourier transform of the unit fun¬cti¬on 1(x), xÎR as well as for the Fourier transforms of the Heaviside unit functions of various types. This approach allows us to study new types of singular integrals. 

The transformation of the Euler type is found, which is determined by the fun-ction of the complex argument. From this representation, one can obtain the well-known Laplace, Fourier, and Mellin integrals by changing the domain of definition of the parameters.

    The new relation between Legendre functions of the second kind for special cases of the parameters is presented. The asymptotic relation is assumed for the function Qnit(z) in the vicinity of the point z=1.

 Let note the fact, that using the mentioned results with other nearest calculations, we ha-ve constructed the consistent, systematic mathematical theory for two charged particles of the continuous spectrum.

  We have calculated the Euler first kind integral type singular integral. The met¬hod developed in the mentioned investigation turned out useful for study singular integ¬rals of the new kind. The application of the mentioned method is supposed.

 Let us point out the fact that the results obtained also turned out to be very useful in order to avoid the well-known ambiguities of Coulomb scattering.

  We have studied the Euler-type singular integrals and have shown that the ab¬ove integrals can be expressed by the Dirac delta function. In this Ref. we propose some simple applications of the mentioned novelties.  

  Using the smooth cutoff parameter, in the framework of the complex analy¬sis, for the radial off shell (q¹k) Coulomb T matrix the exact analytical expression is derived. As it was expected, it is shown that the mentioned quantum-mechanical fun¬ction in the area under consideration is the entire analytic function.

  We have investigated the Euler-type singular integral and shown that it exists in the sense of singular generalized functions. Furthermore we studied analytical continu¬ation formulas for the Gaussian hyperge¬ometric function and expanded the domain of definition of the parameters in the formu¬las mentioned above.           

  Let note that the both above results were very useful in avoiding the well-known am-biguities con¬nected with Coulomb scatterings.

As it is known, up to now the mathematical formalism of quantum mechanical problem of two charged particles of continuous spectrum has not a perfect form. We have tried to correct partially that deficiency, which has required some investigations in the theory of special and generalized functions. 

The high-energy asymptotic Faddeev-type integral equations for the few-particle system are obtained. The iterative series corresponding to the above equation is considered. It is shown that the mentioned series converges and satisfies the corresponding many-particle unitarity condition.

Using the infrared cutoff parameter, in the framework of the complex an¬alysis, for the radial part of the Coulomb scattering matrix, as well as for the radial part of the Fourier image of the Coulomb wave function, the exact analytical expressions are obtained outside the energy shell (q¹k) It is shown that the mentioned quantum-me¬ch¬anical functions outside the energy shell represent the entire analytical functions.

Doctoral Thesis Referee

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Scientific editor of monographs in foreign languages

Scientific editor of a monograph in Georgian

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