The short-scan case in fan-beam computed tomography requires the introduction of a weighting function to handle redundant data. Parker introduced such a weighting function for a scan over pi plus the opening angle of the fan. In this article we derive a general class of weighting functions for arbitrary scan angles between pi plus fan angle and 2pi (over-scan). These weighting functions lead to mathematically exact reconstructions in the continuous case. Parker weights are a special case of a weighting function that belongs to this class. It will be shown that Parker weights are not generally the best choice in terms of noise reduction, especially when there is considerable over-scan. We derive a new weighting function that has a value of 0.5 for most of the redundant data and is smooth at the boundaries.