Stirling Approximation : A function that approximates factorials.

n! is defined as n * (n-1) * (n-2) * (n-3) …. But have you ever heard of the function called the Stirling Approximation?

Stirling Approximation formula

The formula above is founded by a Scottish Mathematician named James Stirling. Aside from the Stirling Approximation, he is known for his contribution to the theory of Infinite Series and Infinitesimal Calculus. The Stirling Approximation is published in one of his works called: Methodus Differentialis sive Tractatus de Summatione et Interpolatione Serierum Infinitarum (1730; “Differential Method with a Tract on Summation and Interpolation of Infinite Series”), a treatise on infinite series, summation, interpolation, and quadrature. It contains the statement of what is known as Stirling’s formula (Brittanica, 2020).

The approximation is derived using concepts of Gamma function, Laplace Method, Gaussian integral and polar integral.

Figure 1, Factorial function vs Exponential function, from Xkatly.com

Derivation of Stirling’s Approximation using Gamma Function

Figure 2, Local and Global Maxima and Minima, from Quora.com

Stirling’s Approximation accuracy increases as the number of n increases, as shown in figure 4 below.

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Sources:

James Stirling | British mathematician. (2020). From https://www.britannica.com/biography/James-Stirling-British-mathematician

Pérez-Marco, R. (2020). On the definition of Euler Gamma function. arXiv.org. From https://arxiv.org/abs/2001.04445

Weisstein, E.W. Taylor Series (2020). Mathworld.wolfram.com. From https://mathworld.wolfram.com/TaylorSeries.html

Conrad, K. (2020). Kconrad.math.uconn.edu. From https://kconrad.math.uconn.edu/blurbs/analysis/gaussianintegral.pdf