This paper reports on a comprehensive study of the long-range interaction of the K-39(4s) + K-39(4p) asymptotic system. We present a detailed discussion of the R-dependent angular momentum couplings and correlation between the Hund’s case (a) and case (c) molecular states. Analytical expressions for the 16 adiabatic Hund’s case (c) long-range potential curves are derived including the higher order dispersion forces and the effects of retardation. Experimentally, six Hund’s case (c) long-range molecular states (0(u)(+), 1(g), and 0(g)(-) dissociating to the 4 S-2(1/2) + 4 P-2(3/2) asymptote and 0(u)(+), 1 g and 0(g)(-) to the 4 S-2(1/2) + 4 P-2(1/2) limit) are observed with rovibrational resolution by photoassociative spectroscopy of ultracold K-39 atoms in a high density magneto-optical trap (MOT). Among the six observed long-range states, the upper 0(g)(-) "pure long-range" state has negligible short-range chemical exchange contributions and the measured molecular binding energies (v = 0-26) are used to precisely determine the long-range potential constants of the 4s + 4p interaction. We determine : C-3(Pi) = 8.436(14) a.u., C-3(Sigma) = 16.872(28) a.u., C-6(Pi) = 6272(94) a.u., and C-6(Sigma) = 9365(141) a.u. Molecular constants for the three special pure long-range states, the 0(g)(-) and 1(u) (dissociating to the 4 P-2(3/2) limit and with potential minimum) and the 1(u) (dissociating to the 4 P-2(1/2) and with potential maximum), are reported. The internal consistency of the theoretical model used in this work is confirmed by the excellent agreement between the long-range potential curve of the 1(g) state obtained in present work (from the 0(g)(-) srate) and the long-range portion of the RKR potential curve of the 1 (1) Pi(g) state previously determined by conventional molecular spectroscopy. The radiative lifetime of the K 4p state derived from the n is also in excellent agreement with a recent fast-beam dipole-dipole interaction constant C-3(Pi) is also in excellent agreement with a recent fast-beam measurement.