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A fresh look at Euler's limit formula for the gamma function

  • G. J. O. Jameson (a1)
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Consider the problem of defining a continuous function f(x) which agrees with factorials at integers. There are many possible ways to do this. In fact, such a function can be constructed by taking any continuous definition of f(x) on [0,1] with f(0) = f(1) = 1 (such as f(x) = 1), and then extending the definition to all x > 1 by the formula f(x + 1) = (x + 1)f(x). This construction was discussed by David Fowler in [1] and [2]. For example, the choice f(x) = ½x(x − 1) + 1 results in a function that is differentiable everywhere, including at integers.

However, this approach had already been overtaken in 1729, when Euler obtained the conclusive solution to the problem by defining what we now call the gamma function. Among all the possible functions that reproduce factorials, this is the ‘right’ one, in the sense that it is the only one satisfying a certain smoothness condition which we will specify below. Admittedly, Euler didn't know this. It is known as the Bohr-Mollerup theorem, and was only proved nearly two centuries later.

First, a remark on notation: the notation Γ (x) for the gamma function, introduced by Legendre, is such that Γ (n) is actually (n − 1)! instead of n!. Though this might seem a little perverse, it does result in some formulae becoming slightly neater. Some writers, including Fowler, write x! for Γ (x + 1), and refer to this as the ‘factorial function’. However, the notation Γ (x) is very firmly entrenched, and I will adhere to it here.

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References
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1. Fowler, David, A simple approach to the factorial function, Math. Gaz. 80 (November 1996) pp. 378381.
2. Fowler, David, A simple approach to the factorial function – the next step, Math. Gaz. 83 (March 1999) pp. 5357.
3. Whittaker, E. T. and Watson, G. N., A course of modem analysis (4th edn.), Cambridge University Press (1965).
4. Copson, E. T., An Introduction to the theory of functions of a complex variable, Oxford University Press (1935).
5. Andrews, George E., Askey, Richard and Roy, Ranjam, Special Functions, Cambridge University Press (1999).
6. Bohr, H. and Mollerup, J., Laerebog i Matematisk Analyse, Gjellerup, Copenhagen (1922).
7. Artin, Emil, Einführung in die Theorie der Gammafunktion, Teubner, Leipzig (1931); English translation: The gamma function, Holt, Rinehart and Winston (1964).
8. Jameson, G. J. O., Inequalities for gamma function ratios, Amer. Math. Monthly 120 (2013) pp. 936940.
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The Mathematical Gazette
  • ISSN: 0025-5572
  • EISSN: 2056-6328
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