Generalized Eisenstein functions

In the previous works, analytical exact and approximate formulas for the effective conductivity of two-dimensional random composites were deduced. These formulas contain the discrete convolutions of the Eisenstein series (e-sums, generalized Rayleigh–Eisenstein–Mityushev sums). The considered objects become the classic lattice sums for regular doubly periodic locations of the centers of inclusions. In the present paper, we modify the above approach and extend it to Natanzon's series having applications in the elasticity theory. The conditionally convergent series are investigated by the Eisenstein summation method. We derive a computationally effective formula for conditionally convergent series. Numerical simulations suggest new formulas in which some lattice sums are equal to π.