The density-functional free energy can be written in a form that is stationary with respect to variations in the occupation function. For this reason it is useful to look for approximate occupation functions that are sufficiently close to the Fermi function that accuracy is not compromised and yet have advantages for computation. From a computational point of view it is useful to reduce the number of poles of the occupation function in the upper half of the complex energy plane and to locate the poles as far from the real axis as possible. A family of approximate occupation functions that economize computation is introduced. Their properties are discussed and illustrated for a model system.