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The residue calculus method of investigation of certain waveguide configurations makes use of the asymptotic properties of the gamma function. Usually the range of the variables concerned is such that this approximation is quite adequate. In a recent investigation of a very narrow waveguide junction peculiar numerical effects were traced to a condition where the variables were much too small to warrant the use of the usual asymptotic formula. A new and very simple modification extends the asymptotic form right down to zero with an error of, at most, only a few percent.