The purpose of this paper is twofold. The first is to provide a fast and accurate method of approximating the J(θ,β) function for a single resonance. The second objective is to provide a rapid method of averaging unresolved levels by use of this approximate J function and a recently developed quadrature scheme of the Gaussian type. These approximations are well suited for use in day-to-day reactor design and evaluation and are substantially faster and more accurate than other approximations currently available in the literature. The approximate J function has been tested on that portion of the θ,β plane for which β ≥ 5.0 × 10−5 and θ ≥ 5.0 × 10−4. This portion of the plane encompasses almost every conceivable practical situation. On this domain, typical relative errors incurred in J (θ,β) are 0.25% or less and the maximum relative error for any (θ,β) pair is 2.2% which is encountered at an extreme value of β = 5.0 × 10−5. The technique for J-function averaging produces relative errors < 0.10% for cases of practical interest.