Abstract This article is concerned with the Titchmarsh–Weyl m α ( λ) function for the differential equation d 2 y/d x 2+[ λ− q( x)] y=0. The test potential q( x)= x 2, for which the relevant m α ( λ) functions are meromorphic, having simple poles at the points λ=4 k+1 and λ=4 k+3, is studied in detail. We are able to calculate the m α ( λ) function both far from and near to these poles. The calculation is then extended to several other potentials, some of which do not have analytical solutions. Numerical data are given for the Titchmarsh–Weyl m α ( λ) function for these potentials to illustrate the computational effectiveness of the method used.