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Date: 16-5-2018
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Stirling's approximation gives an approximate value for the factorial function or the gamma function for . The approximation can most simply be derived for an integer by approximating the sum over the terms of the factorial with an integral, so that
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The equation can also be derived using the integral definition of the factorial,
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Note that the derivative of the logarithm of the integrand can be written
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The integrand is sharply peaked with the contribution important only near . Therefore, let where , and write
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Now,
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so
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Taking the exponential of each side then gives
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Plugging into the integral expression for then gives
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Evaluating the integral gives
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(Wells 1986, p. 45). Taking the logarithm of both sides then gives
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This is Stirling's series with only the first term retained and, for large , it reduces to Stirling's approximation
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Taking successive terms of , where is the floor function, gives the sequence 1, 2, 4, 10, 26, 64, 163, 416, 1067, 2755, ... (OEIS A055775).
Stirling's approximation can be extended to the double inequality
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(Robbins 1955, Feller 1968).
Gosper has noted that a better approximation to (i.e., one which approximates the terms in Stirling's series instead of truncating them) is given by
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Considering a real number so that , the equation (27) also gives a much closer approximation to the factorial of 0, , yielding instead of 0 obtained with the conventional Stirling approximation.
REFERENCES:
Feller, W. "Stirling's Formula." §2.9 in An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, pp. 50-53, 1968.
Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 86-88, 2003.
Robbins, H. "A Remark of Stirling's Formula." Amer. Math. Monthly 62, 26-29, 1955.
Sloane, N. J. A. Sequence A055775 in "The On-Line Encyclopedia of Integer Sequences."
Stirling, J. Methodus differentialis, sive tractatus de summation et interpolation serierum infinitarium. London, 1730. English translation by Holliday, J. The Differential Method: A Treatise of the Summation and Interpolation of Infinite Series. 1749.
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 45, 1986.
Whittaker, E. T. and Robinson, G. "Stirling's Approximation to the Factorial." §70 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 138-140, 1967.
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