  This paper is devoted to the calculation by Mellin-Barnes transform of a
especial class of integrals. It contains double integrals in the position space
in d = 4-2e dimensions, where e is parameter of dimensional regularization.
These integrals contribute to the effective action of the N = 4 supersymmetric
Yang-Mills theory. The integrand is a fraction in which the numerator is a
logarithm of ratio of spacetime intervals, and the denominator is the product
of powers of spacetime intervals. According to the method developed in the
previous papers, in order to make use of the uniqueness technique for one of
two integrations, we shift exponents in powers in the denominator of integrands
by some multiples of e. As the next step, the second integration in the
position space is done by Mellin-Barnes transform. For normalizing procedure,
we reproduce first the known result obtained earlier by Gegenbauer polynomial
technique. Then, we make another shift of exponents in powers in the
denominator to create the logarithm in the numerator as the derivative with
respect to the shift parameter delta. We show that the technique of work with
the contour of the integral modified in this way by using Mellin-Barnes
transform repeats the technique of work with the contour of the integral
without such a modification. In particular, all the operations with a shift of
contour of integration over complex variables of two-fold Mellin-Barnes
transform are the same as before the delta modification of indices, and even
the poles of residues coincide. This confirms the observation made in the
previous papers that in the position space all the Green function of N = 4
supersymmetric Yang-Mills theory can be expressed in terms of UD functions.
