An exact method for direct calculation of the Jost function and Jost solutions for a repulsive singular potential is presented. Within this method the Schrödinger equation is replaced by an equivalent system of linear first-order differential equations, which after complex rotation, can easily be solved numerically. The Jost function can be obtained to any desired accuracy for all complex momenta of physical interest, including the spectral points corresponding to bound and resonant states. The method can also be used in the complex angular-momentum plane to calculate the Regge trajectories. The effectiveness of the method is demonstrated using the Lennard-Jones (12, 6) potential. The spectral properties of the realistic interatomic 4 He− 4 He potentials HFDHE2 and HFD-B of Aziz et al.[Mol. Phys. 61, 1487 (1987)] are also investigated.