Ok guys... I was playing around with math, and I've noticed one thing... Considering this integral
$$\int_{-\infty}^{\frac{x^2}{2}} e^{x-\frac{t^2}{2}}dt$$
It looks a lot like the error function. I was trying to manipulate it in a way to make it look like an error function, but I didn't find a way.
Anybody has an idea?
Hint. Using $$\text{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^x e^{-u^2}du$$ you may write $$ \begin{align} \int_{-\infty}^{\frac{x^2}{2}} e^{x-\frac{t^2}{2}}dt&=e^{x}\int_{-\infty}^{\frac{x^2}{2}} e^{-\frac{t^2}{2}}dt\\\\ &=\sqrt{2}\:e^{x}\int_{-\infty}^{\frac{x^2}{2\sqrt{2}}} e^{-u^2}du\\\\ &=\sqrt{2}\:e^{x}\left(\int_{-\infty}^{0} e^{-u^2}du+\int_{0}^{\frac{x^2}{2\sqrt{2}}} e^{-u^2}du\right)\\\\ &=e^x \sqrt{\frac{\pi }{2}} \left(1+\text{erf}\left(\frac{x^2}{2 \sqrt{2}}\right)\right). \end{align} $$