In mathematics, the upper and the lower incomplete gamma functions are respectively as follows:
: \Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t ,\,\! \qquad \gamma(s,x) = \int_0^x t^{s-1}\,e^{-t}\,{\rm d}t .\,\!
Properties
In both cases s is a complex parameter, such that the real part of s is positive.
By integration by parts we find the recurrence relations
:\Gamma(s,x)= (s-1)\Gamma(s-1,x) + x^{s-1} e^{-x}
and conversely
: \gamma(s,x) =(s-1)\gamma(s-1,x) - x^{s-1} e^{-x}
Since the ordinary gamma function is defined as
: \Gamma(s) = \int_0^{\infty} t^{s-1}\,e^{-t}\,{\...