Does the definition of the upper case incomplete gamma function $\Gamma(a,x)=\int_{x}^{\infty}e^{-t}\ t^{a-1}$ hold for any integer $a$?

Question

The question is about the definition of upper and lower incomplete gamma functions. In [1] we can see :

Lower case: $$ \gamma(a,x)=\int_{0}^{x}e^{-t} \ t^{a-1}\;dt \quad;\quad \text{Re}(a)>0 $$

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

Now,

Question: Does the Upper case relation hold for any integer $a$ ?

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Reference :

[1] Higher Transcendental functions, vol 2 page 133, Erdelyi, A., et al.

Answer 1

Yes the definition of $\Gamma$ holds for all $a$, as the singularity at $t=0$ is avoided.

Also check the comments about the integration path in NIST.

$\gamma$ can be extended by means of $$\gamma(a,z)+\Gamma(a,z)=\Gamma(a).$$