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On Stirling's Theorem as a Definition of the Gamma Function

Published online by Cambridge University Press:  03 November 2016

F. Egan
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Extract

In the following note, Stirling’s formula for a factorial is first found by a very simple process. The expression so obtained has a meaning for non-integral as well as integral values of the variable. The function it defines is easily seen (§§ 3 ff.) to possess the fundamental properties of the Gamma function, and provides an easy and natural avenue of approach to its study.

Type
Research Article
Information
The Mathematical Gazette , Volume 17 , Issue 223 , May 1933 , pp. 114 - 121
Copyright
Copyright © The Mathematical Association 1933

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References

page no 115 note * In the same way we get, by taking the next term in the logarithmic series,

and if we note that

we find

It is easy to get another term or two of the asymptotic fieries in this way, with the appropriate inequality.

page no 118 note * It follows that the conditiorl is satisfied if

page no 119 note † If we use the lemmas

we get easily enough

page no 119 note ‡

page no 120 note * Cf. the results given by Lindelöf, , Calcul des Rksidus (Paris, 1905), Ch. IV.Google Scholar