By Pugh G.R.

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The second is that Spouge gives a simpler yet more accurate version of Stirling’s formula. And finally, Spouge’s approximation and error estimates apply not only to Γ(z + 1), but also to the digamma function Ψ(z + 1) = d/dz [log Γ(z + 1)] and trigamma function Ψ (z). 2) into partial 38 Chapter 2. A Primer on the Gamma Function fractions, 1 z z · · · (z − N + 1) a0 (r) + a1 (r) + · · · + aN (r) + (z) 2 z+1 (z + 1) · · · (z + N) N = b0 (r) + k=1 bk (r) + (z) . 17), the bk (r) obtained from truncating the Lanczos series are the approximate residues of Γ(z+1)(z+a)−(z+1/2) ez+a (2π)−1/2 at z = −k, and the larger N becomes the better the approximation.

The derivation is first retraced more or less along the same lines as Lanczos uses in his paper [14]. This approach uses Fourier series tech40 Chapter 3. The Lanczos Formula niques in a novel albeit complicated way to obtain explicit formulas for the coefficients an (r) as a linear combination of Chebyshev polynomial coefficients. The appearance of Chebyshev coefficients suggests a connection with the orthogonal system of Chebyshev polynomials, and indeed the derivation in the setting of these polynomials provides a slightly cleaner argument.

In practice, unless one is dealing with z values near the origin, one can use much smaller values of N which result in acceptable error bounds. 6 Spouge’s Method To conclude this survey of standard results, special mention is made of the 1994 work of Spouge [27]. 17) which is valid for Re(z + a) > 0. The parameter a is real, N = a − 1, c0 (a) = 1 and ck (a) is the residue of Γ(z + 1)(z + a)−(z+1/2) ez+a (2π)−1/2 at z = −k. Explicitly, for 1 ≤ k ≤ N, this is 1 (−1)k−1 ck (a) = √ (−k + a)k−1/2 e−k+a .