Exact Evaluation of the Maliuzhinets Half-Plane Function

The Maliuzhinets half-plane function psi
_pi
(z) has been used for many applications, but its closed-form has not been known. In this Communication, we exactly evaluate a known integral representation of psi
_pi
(z) to formulate an exact closed-form of the function, which is expressed in terms of the dilogarithm function. The exact formulation is numerically verified by comparing it with the results obtained by numerical integration of a integral representation of general Maliuzhinets function. A closed-form of psi
_pi/2
(z) is also obtained since it can be expressed in terms of psi
_pi
(z)