# An analysis of the Lanczos Gamma approximation by Pugh G.R.

By Pugh G.R.

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1 that N = 4 with n = 3 would also have given the desired accuracy, in which case only n = 3 terms of the series would be needed. The reader has no doubt noticed the ad-hoc manner in which n and N were selected in this example. The question of how best to choose n and N given z and absolute error is not an easy one, and will not be addressed here. 33 Chapter 2. A Primer on the Gamma Function It is a simpler matter, however, to determine the maximum accuracy that can be expected for a given choice of N, both for a fixed z and uniformly, and this is done in the following sections.

From the periodicity of this last function he then argues that the interpolating Fourier cosine series is a better choice in terms of convergence versus the extrapolating Taylor series. 45]. There he derives an expansion for Bessel functions of arbitrary order in terms of Bessel functions of even integer order. Part of that derivation involves a clever method for computing 50 Chapter 3. The Lanczos Formula the coefficients of the expansion which will be described in the sequel. S. Army at the University of Wisconsin in Madison.

It turns out, however, that thanks to a handy identity, the resulting integrals evaluate explicitly to the rational functions Hk (z) which exhibit the poles of Γ(z + 1). 2. 8) to replace 46 Chapter 3. The Lanczos Formula fE,r (θ), we obtain ∞ π/2 Γ(z + 1/2) = Pr (z) 2z cos θ −π/2 k=0 ∞ = Pr (z) ak (r) cos (2kθ) dθ π/2 cos2z θ cos (2kθ) dθ ak (r) −π/2 k=0 √ Γ(z + 1/2) = Pr (z) π Γ(z + 1) ∞ ak (r)Hk (z) . 11) k=0 where again, Re(z) > −1/2, and r ≥ 0. We will see later that convergence of this series can be extended to the left of Re(z) = −1/2.