AN ANALYSIS OF THE LANCZOS GAMMA APPROXIMATION

This thesis is an analysis of C. Lanczos’ approximation of the classical gamma function Γ(z+1) as given in his 1964 paper A Precision Approximation of the Gamma Function [14]. The purposes of this study are: (i) to explain the details of Lanczos’ paper, including proofs of all claims made by the author; (ii) to address the question of how best to implement the approximation method in practice; and (iii) to generalize the methods used in the derivation of the approximation. At present there are a number of algorithms for approximating the gamma function. The oldest and most well-known is Stirling’s asymptotic series which is still widely used today. Another more recent method is that of Spouge [27], which is similar in form though different in origin than Lanczos’ formula. All three of these approximation methods take the form Γ(z + 1) = √ 2π(z + w)z+1/2e−z−w [sw,n(z) + ǫw,n(z)] (1) where sw,n(z) denotes a series of n + 1 terms and ǫw,n(z) a relative error to be estimated. The real variable w is a free parameter which can be adjusted to control the accuracy of the approximation. Lanczos’ method stands apart from the other two in that, with w ≥ 0 fixed, as n → ∞ the series sw,n(z) converges while ǫw,n(z) → 0 uniformly on Re(z) > −w. Stirling’s and Spouge’s methods do not share this property. What is new here is a simple empirical method for bounding the relative error |ǫw,n(z)| in the right half plane based on the behaviour of this function as |z| → ∞. This method is used to produce pairs (n, w) which give formulas (1) which, in the case of a uniformly bounded error, are more efficient than Stirling’s and Spouge’s methods at comparable accuracy. In the n = 0 case, a variation of Stirling’s formula is given which has an empirically determined uniform error bound of 0.006 on Re(z) ≥ 0. Another result is a proof of the limit formula Γ(z + 1) = 2 lim r→∞ r [ 1 2 − e−12/r z z + 1 + e−2 2/r z(z − 1) (z + 1)(z + 2) + · · · ] as stated without proof by Lanczos at the end of his paper.