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A Very Short Proof of Stirling's Formula
... and as in [6] is easily seen to converge to some finite constant C := lim x→∞ f (x). It remains to determine this constant. ...
... It remains to determine this constant. One can proceed as in [6] and use the delicate Lebesgue dominated convergence theorem to pass under the integral sign. But we can avoid that in the following way C = lim x→∞ ...
We present novel elementary proofs of Stirling's approximation formula and Wallis' product formula, both based on Gautschi's inequality for the Gamma function.
... for x > 0. In [5] a short and direct proof was provided for the assertion that lim x→+∞ r (x) = 0. ...
... (Note that our proof does not rely on the fact that r (x) is strictly decreasing on (0, ∞), a property not provided by [5].) To prove the second inequality in (15), define ...
... Over the years, many proofs of equation 1 have been published, and the literature is replete with ones of many different kinds. A sample of these can be found in [2,4,5,6,7,8,9,10,11,12,13]. With one exception-and the present offering is no different in this regard-these articles contain ad-hoc proofs of equation 1. Reference [6] is the exception: it is Walter Hayman's most cited article, and his proof illustrates a powerful method for asymptotically estimating the coefficients of a class of power series of analytic functions, termed admissible by him. ...
... There are some proofs which only require elementary methods such as that of Mermin (1984), Namias (1986), Diaconis and Freedman (1986), Patin (1989) and Marsaglia and Marsaglia (1990). ...
We examined the properties of the coefficients of the \cite{lanczos1964} approximation of the $\Gamma$-function with complex values of the free parameter together with the convergence properties of the approximation when using these coefficients. We report that for fixed real parts of the free parameter that using complex coefficients both increases the computational cost of the Lanczos approximation while drecreasing the accuracy. We conclude that in practical applications of numerical evaluation of the $\Gamma$-function only coefficients generated with real values of the free parameter should be used.
... In [77] we find a "very short proof" that eschews the Central Limit Theorem as claimed to be used in [12] because that "cannot reasonably be considered elementary." Instead, the author (Patin) uses the Lebesgue dominated convergence theorem! ...
Since its inception in 1894, the Monthly has printed 50 articles on the Γ function or Stirling's asymptotic formula, including the magisterial 1959 paper by Phillip J. Davis, which won the 1963 Chauvenet prize, and the eye-opening 2000 paper by the Fields medalist Manjul Bhargava. In this article, we look back and comment on what has been said, and why, and try to guess what will be said about the Γ function in future Monthly issues.¹1 It is a safe bet that there will be more proofs of Stirling's formula, for instance.View all notes We also identify some gaps, which surprised us: phase plots, Riemann surfaces, and the functional inverse of Γ make their first appearance in the Monthly here. We also give a new elementary treatment of the asymptotics of n! and the first few terms of a new asymptotic formula for invΓ.
... The literature concerning Stirling's formula is very large, counting hundreds of items on JSTOR. See for example [2], [3], [7], [8], [9] and [11]. ...
We present a new short proof of Stirling’s formula for the gamma function. Our approach is based on the Gauss product formula and on a remark concerning the existence of horizontal asymptotes.
A simple proof of Stirling's formula for the gamma function - Volume 99 Issue 544 - G. J. O. Jameson
In this paper, we are concerned with the monotonicity and the estimates of σx and λ x defined by the famous Stirling's formula: (1 + λ x )(x > 0), and improve some old results.