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This study is concerned with the Green function of the theory of potential flow about a body in regular (time-harmonic) water waves in deep water, that is with the linearized velocity potential of the flow due to a source of pulsating strength at a fixed position below the free surface (or a pulsating flux across the free surface) of a quiescent infinitely deep sea. An asymptotic expansion and a convergent ascending-series expansion for the Green function are obtained from two alternative complementary 'near-field' and 'far-field' single-integral representations in terms of the exponential integral. The asymptotic expansion and the ascending series allow efficient numerical evaluation of the Green function for large and small distances, respectively, from the mirror image of the singularity (submerged source or free-surface flux) with respect to the mean sea surface.

Three alternative single-integral representations for the Green function of the theory of ship wave resistance are derived in a unified manner from a basic double-integral representation. These alternative single-integral representations, which essentially are modifications of well-known double-integral representations due to MicheU, Havelock, and Peters, are compared and discussed. Another object of this study is to examine the field equation and the boundary condition satisfied by the Green function in the limiting case when the singular point is exactly at the free surface.

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