# Incomplete gamma-function

From Encyclopedia of Mathematics

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