Stirling’s Original Asymptotic Series from a Formula Like One of Binet’s and its Evaluation by Sequence Acceleration

Abstract

We give an apparently new proof of Stirling's original asymptotic formula for the behavior of ln z! for large z. Stirling's original formula is not the formula widely known as "Stirling's formula", which was actually due to De Moivre. We also show by experiment that this old formula is quite effective for numerical evaluation of ln z! over C, when coupled with the sequence acceleration method known as Levin's u-transform. As an homage to Stirling, who apparently used inverse symbolic computation to identify the constant term in his formula, we do the same in our proof.

A Curious Identity Arising From Stirling's Formula and Saddle-Point Method on Two Different Contours

“…Apart from computational convenience, the numerical advantages of the expansion (9) over (7) are visible because they have the same sequence of coefficients and (n + 1) −m is always smaller than n −m ; see also [5] for Stirling's original expansion for log n! in decreasing powers of n + 1 2 .…”

“…On the other hand, a more effective means of computing c 2m is to make first the change of variables e u − 1 − u = 1 2 v 2 in the rightmost integral in (5), where u = u(v) is positive when v is, and is analytic in |v| 󰃑 ε; see [27, § 3.6.3]. Then…”

Devilish Tricks for Sequence Acceleration

“…No. No floating-point system that we know of preserves this property 6 . For IEEE double precision floats, which are represented in a 64 bit word, not only are there gaps between consecutive floating-point numbers, there is also a minimum positive real number (about 10 −309 ) and a maximum positive real number (about 10 308 ).…”

“…(32) 15 I find it deeply amusing that Stirling invented equation (33) which is now beginning to be called DeMoivre's formula, and DeMoivre invented equation (32), which is called Stirling's formula. See [3,6].…”

“…There are many other examples of the success of evalf/Sum in [5], as well. See [6] for its use for Stirling's original series; it also works well for the formula popularly known as Stirling's.…”