Evaluating The Upper Incomplete Gamma Function With A Negative Integration Bound

Question

I need to evaluate:

$$ \Gamma(a,-x, \infty) = \int_{-x}^{\infty} t^{a-1} e^{-t} dt$$

When I plug this into Wolfram, for instance $Gamma[\frac{1}{2}, -4]$ I get a complex number. How is this calculated?

Answer 1

The upper incomplete gamma function is defined as such: $$\Gamma(s,x) = \int_x^\infty t^{s-1}e^{-t}dt$$
In your case you are asking about $$\Gamma(1/2,-4) = \int_{-4}^\infty t^{-1/2}e^{-t}dt$$ Note that this is only real for $t>0$... for $t<0$ we have that $t^{1/2}$ is an imaginary number, therefore making the integral complex over this region. This should generalize to an arbitrary negative lower bound of integration.

Answer 2

Coincidentally there is a paper by A. Gil, D. Ruiz-Antolín, J. Segura, N. M. Temme from Aug. 14, which describes the computation of Tricomi's incomplete gamma function, this function $\gamma^{*}(a, z)$ is real for real positive and negative values of $a$ and $z$ and can be defined as $$\gamma^{*}(a,z) = \frac{z^{-a}}{\Gamma(a)}\gamma(a,z)$$

If you have access to this function you can compute $$\Gamma(a,z)=\Gamma(a)-\gamma(a,z) =\Gamma(a)\Big(1-z^a\gamma^{*}(a,z)\Big)$$

Here is the example with $a=1/2, z=-4$ from my own Pascal library: $\gamma^{*}(1/2,-4) = 9.2824012072877763$ and therefore $$\Gamma(1/2,-4) = \sqrt{\pi}(1-\sqrt{-4}\times 9.2824012072877763) \\= 1.7724538509055160-32.905255531014461 i$$

If you have access to Kummer's confluent hypergeometric $M$ function (sometimes called $_1 F_1$) you can express the entire Tricomi function for all $a,z \in \mathbb{C}$ as (see DLMF) $$ \gamma^{*}(a,z) = e^{-z} \; \frac{M(1,a+1,z)}{\Gamma(a+1)} = e^{-z} \sum\limits_{n=0}^{\infty} \frac{z^n}{\Gamma(a+n+1)}. $$ This confluent hypergeometric form is used by my routine for real $z<0,\,$ i.e. $$\gamma^{*}(1/2,-4)= \frac{e^4 M(1,3/2,-4)}{\Gamma(3/2)}= \\ 54.5981500331442391 \times 0.150670194461895983 / 0.886226925452758014 = 9.2824012072877763$$