Incomplete Gamma function of negative arguments Playing with Mathematica, I found that $$\Re\left[\Gamma\left(-\frac{n}{2},x\right)\right]=\Gamma\left(-\frac{n}{2}\right)$$ for $x<0$ and $n$ any positive integer. 
Would it be possible to prove that?
On Wikipedia, I found a formula for the incomplete Gamma function when the first argument is a negative integer: $$\Gamma(-n,z)=\frac{1}{n!}\left(\frac{e^{-z}}{z^n}\sum_{k=0}^{n-1}(-1)^k(n-k-1)!z^k+(-1)^k+(-1)^n\Gamma(0,z)\right)$$ 
Maybe there is a similar formula when the first argument is a negative half-integer!
 A: By dlmf 8.4.6 
$$
\Gamma\left(\tfrac{1}{2},z\right)=2\int\limits_{\sqrt{z}}^{\infty}e^{-t^{2}}dt.
$$
For negative arguments we have
$$
\Gamma\left(\tfrac{1}{2},-|z|\right)=2\int\limits_{i\sqrt{|z|}}^{\infty}e^{-t^{2}}dt=2\int\limits_{i\sqrt{|z|}}^{i\sqrt{|z|}\,+\infty}e^{-t^{2}}dt=2\int_{0}^{+\infty}e^{-\left(t+i\sqrt{|z|}\right)^{2}}dt.
$$
$$
\text{Re}\left\{\Gamma\left(\tfrac{1}{2},-|z|\right)\right\}=2\int_{0}^{+\infty}e^{-\left(t^2-|z|\right)}\cos\left(2t\sqrt{|z|}\right)dt=2e^{|z|}\cdot\frac{\sqrt{\pi}}{2}e^{-|z|}=\Gamma\left(\frac12\right).
$$
By dlmf 8.8.2
$$
\Gamma\left(a+1,z\right)=a\Gamma\left(a,z\right)+z^{a}e^{-z}.
$$
From this one can easily deduce by induction that the claim is true for all $n\in\mathbb{Z}$:
$$\text{Re}\left\{\Gamma\left(n+\tfrac{1}{2},-|z|\right)\right\}=\Gamma\left(n+\tfrac{1}{2}\right).$$
Numerical checked confirmed this formula for non-negative $n$ too.
A: This is a partial answer I hope will provide a useful starting point. I confirmed the asker's numerical observations with the web version of Mathematica function evalution and the Keisan high-precision calculator. I found a recent paper which gives information in its introduction that also confirms the asker's conjecture:
Ian Thompson, "Algorithm 926: Incomplete Gamma Functions with Negative Arguments", ACM Transactions on Mathematical Software, Volume 39, Issue 2, February 2013, Article No. 14

This article is concerned with the computation of $\gamma (n+1/2,−x)$,
  where $n$ is an integer and $x>0$. It turns out that these are purely
  imaginary quantities, and so, once the lower function is computed,
  $\Gamma(n+1/2,−x)$ can be obtained from (3) without the loss of any
  significant digits.

The referenced equation (3) is the well-known identity $\gamma(a,t)=\Gamma(a)−\Gamma(a,t)$. The paper doesn't go into details as to why $\gamma(n+1/2,-x)$ is always a purely imaginary quantity (at least not in any form that I, as a non-mathematician, recognize as such). It provides the following expansion which may be useful:
$$
\begin{align}
S_{n}(x) = e^{-x} \sum_{j=0}^{\infty} \frac{x^{j}}{j!(n+\frac{1}{2}+j)}, \enspace \enspace n \in{}\mathbb{Z}, \enspace x \ge 0, \\
\gamma(n+\frac{1}{2},-x) = i(-1)^{n}e^{x}x^{n+1/2}S_{n}(x).
\end{align}
$$
It also notes the recurrence relation
$$(n+\frac{1}{2})S_{n}(x) + xS_{n+1}(x)=1.$$
