The regularized gamma functions are defined by
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
and 
are incomplete gamma functions and 
is a complete gamma function. The function 
is implemented in the Wolfram Language
as GammaRegularized[a,
0, z], and 
is implemented as GammaRegularized[a,
z].
and 
satisfy the identity
 |
(3)
|
For a positive integer 
, the Ramanujan sequence
is given directly by ![theta_n=(n!e^n/n^n)[P(n,n)-1/2]](/images/equations/RegularizedGammaFunction/Inline15.svg)
(Choi 1994).
The derivatives of 
and 
are
 |  |  |
(4)
|
 |  |  |
(5)
|
and the second derivatives are
 |  |  |
(6)
|
 |  |  |
(7)
|
The integrals are
 |  |  |
(8)
|
 |  |  |
(9)
|
