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Γ2 (½)is more than just π

Published online by Cambridge University Press:  23 January 2015

Samuel G. Moreno  and
Esther M. García-Caballero
Samuel G. Moreno
Affiliation:
Departamento de Matemáticas, Universidad de Jaén, 23071 Jaén, Spain
Esther M. García-Caballero
Affiliation:
Departamento de Matemáticas, Universidad de Jaén, 23071 Jaén, Spain
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Extract

For a fixed positive integer m, factorial m is defined by

The problem of finding a formula extending the factorial m! to positive real values of m was posed by D. Bernoulli and C. Goldbach and solved by Euler. In his letter of 13 October 1729 to Goldbach [1], Euler defined a function (which we denote as Γ (x + 1)) by means of

and showed that Γ (m + 1) = m! for positive integers m. After that, Euler found representations for the so-called gamma function (1) in terms of either an infinite product or an improper integral. We refer the reader to the classical (and short) treatise [2] for a brief introduction and main properties of the gamma function.

Type
Articles
Information
The Mathematical Gazette , Volume 97 , Issue 540 , November 2013 , pp. 430 - 434
Copyright
Copyright © The Mathematical Association 2013

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References

1. The Euler Archive, http://www.math.dartmouth.edu/~euler/ Google Scholar
2. Artin, E., The gamma function, Holt, Rinehart and Winston, New York (1964).Google Scholar
3. Melzak, Z. A., Infinite products for πe and π/e , Amer. Math. Monthly 68 (1961) pp. 39–41.Google Scholar
4. Adamchik, V. S., The multiple gamma function and its application to computation of series, Ramanujan J. 9 (2005) pp. 271–288.Google Scholar
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6. van der Pol, B., Note on the gamma function, Canadian J. Math. 6 (1954) pp. 18–22.Google Scholar
7. Nanjundiah, T. S., Van der Pol's expressions for the gamma function, Proc. Amer. Math. Soc. 9 (1958) pp. 305–307.Google Scholar
8. Weisstein, E., Euler-Mascheroni constant. From MathWorld, A Wolfram Web Resource. http://mathworld.wolfram.com/Euler-MascheroniConstant.html Google Scholar