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A simple model for QCD dynamics in which the DGLAP integro-differentialequation may be solved analytically has been considered in our previous papersarXiv:1611.08787 [hep-ph] and arXiv:1906.07924 [hep-ph]. When such a modelcontains only one term in the splitting function of the dominant partondistribution, then Bessel function appears to be the solution to thissimplified DGLAP equation. To our knowledge, this model with only one term inthe splitting function for the first time has been proposed by Blumlein inhep-ph/9506403. In arXiv:1906.07924 [hep-ph] we have shown that a dualintegro-differential equation obtained from the DGLAP equation by a complex mapin the plane of the Mellin moment in this model may be considered as the BFKLequation. Then, in arXiv:1906.07924 we have applied a complex diffeomorphism toobtain a standard integral from Gradshteyn and Ryzhik tables starting from thecontour integral for parton distribution functions that is usually taken bycalculus of residues. This standard integral from these tables appears to bethe Laplace transformation of Jacobian for this complex diffeomorphism. Here wewrite up all the formulae behind this trick in detail and find out certainimportant points for further development of this strategy. We verify that theinverse Laplace transformation of the Laplace image of the Bessel function maybe represented in a form of Barnes contour integral.

Analytical solution to DGLAP integro-differential equation via complex maps in domains of contour integrals