Values of Incomplete gamma function

Question

Look this claim :

Does $\Gamma(0.5,-x^2)= i\alpha$, for $x$ large real number?

i=unity imaginary part

$\alpha$ is real number

I would like someone to prove me this if it's a true claim

Answer 1

According to Maple, $$ \Gamma(1/2,-10000)=\sqrt {\pi\ }{\rm erfc} \left( 100\,i \right) = 1.772453851-{ 8.807258633\times 10^{4340}}\,i $$

So, in fact it is not purely imaginary. But the imaginary part is so much larger than the real part, one can see why you might think it is.

added

Think of it this way: $$ \Gamma(1/2,-x^2) = \int_{-x^2}^\infty \frac{e^{-t}}{\sqrt{t}}\,dt = \int_{-x^2}^0 \frac{e^{-t}}{\sqrt{t}}\,dt+\int_{0}^\infty \frac{e^{-t}}{\sqrt{t}}\,dt $$ The first integral is purely imaginary, and the second integral is $\Gamma(1/2) = \sqrt{\pi}$, which is nonzero.

Answer 2

we assume that :$\displaystyle\int e^{x^2}dx=\frac{(-1)^{^{-\tfrac12}}}2\cdot\Gamma\bigg(\frac12,-x^2\bigg)$, if we accepted :

$f(x)$=$\frac{x }{2x²-1}e^{x²}$ as a good asymptotic approximation to the antiderivative of

${e^{x²}}$ for large $x$ , it's seems that the values of $f(x)$ are real for large $x $,

implies that $\Gamma(0.5,-x²) $ =$i\alpha$ for large $x $ values

,$\alpha $ is real number and $i$=unit imaginary part