The half power method is a technique commonly used for calculating the system damping using frequency response curves. Past derivations typically assume a small damping ratio but do not keep track of the order of magnitude when simplifying results and focus mainly on displacement frequency response curves. This paper provides two separate and rigorous derivations of the half power bandwidth for displacement and acceleration frequency response functions. The exact expressions are simplified systematically using binomial expansions to include third order effects. The third order and classical approximations are compared with the exact expressions, and the truncation errors are presented for both displacement and acceleration cases. The high order effects are more apparent and the truncation errors are greater for the acceleration case. The classical method is sufficiently accurate for many practical cases where the damping ratio is less than 0.1 but higher order corrections may be used to reduce truncation error for systems where the damping ratio is higher.