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The incomplete gamma functions

Published online by Cambridge University Press:  14 June 2016

G. J. O. Jameson
G. J. O. Jameson*
Affiliation:
Dept. of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF e-mail: g.jameson@lancaster.ac.uk
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Extract

Recall the integral definition of the gamma function: for a > 0. By splitting this integral at a point x ⩾ 0, we obtain the two incomplete gamma functions:

(1)

(2)

Γ(a, x)is sometimes called the complementary incomplete gamma function. These functions were first investigated by Prym in 1877, and Γ(a, x) has also been called Prym's function. Not many books give these functions much space. Massive compilations of results about them can be seen stated without proof in [1, chapter 9] and [2, chapter 8]. Here we offer a small selection of these results, with proofs and some discussion of context. We hope to convince some readers that the functions are interesting enough to merit attention in their own right.

Type
Research Article
Information
The Mathematical Gazette , Volume 100 , Issue 548 , July 2016 , pp. 298 - 306
Copyright
Copyright © Mathematical Association 2016 

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References

1.Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher Transcendental Functions (vol. II), McGraw-Hill (1953)Google Scholar
2.Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press (2010).Google Scholar
3.Jameson, G. J. O., Sine, cosine and exponential integrals, Math. Gaz. 99 (July 2015) pp. 276–289.CrossRefGoogle Scholar