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Incomplete gamma-function

From Encyclopedia of Mathematics
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The function defined by the formula

where is the gamma-function. If is an integer, then

Series representation:

Continued fraction representation:

Asymptotic representation for large :

Asymptotic representation for large :

where

Connection with the confluent hypergeometric function:

Connection with the Laguerre polynomials :

Recurrence relation:

References

[1] M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1973)
[2] V.I. Pagurova, "Tables of the incomplete gamma-function" , Moscow (1963) (In Russian)


Comments

The following notations are also used:

with , . The -function is related to the confluent hypergeometric function:

New asymptotic expansions for both and are given in [a1].

References

[a1] N.M. Temme, "The asymptotic expansion of the incomplete gamma functions" SIAM J. Math. Anal. , 10 (1979) pp. 757–766
How to Cite This Entry:
Incomplete gamma-function. V.I. Pagurova (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Incomplete_gamma-function&oldid=11834
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098