Recently, a circuits-based approach for modeling the mutual coupling of reconfigurable surfaces, which comprise sub-wavelength spaced passive scattering elements coupled with electronic circuits for enabling the reconfiguration of the surface, has been introduced. The approach is based on a finite-length discrete dipole representation of a reconfigurable surface, and on the assumption that the current distribution on each thin wire dipole is a sinusoidal function. Under these assumptions, the voltages at the ports of a multi-antenna receiver can be formulated in terms of the voltage generators at a multi-antenna transmitter through a transfer function matrix that explicitly depends on the mutual coupling and the tuning circuits through the mutual impedances between every pair of thin wire dipoles. In currently available works, the mutual impedances are formulated in an integral form. In this paper, we show that they can be formulated in a closed-form expression in terms of exponential integral functions.