Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (1.1)

$$\begin{equation*} \int{{\exp(-{{u}^{2}})}}du=\frac{1}{2}g\left( \frac{1}{2},-1,u^{2} \right). \end{equation*}$$ Proof:
Let $t={{u}^{2}}$, and hence $u={{t}^{1/2}}$ and $dt=2udu$. Therefore,
$$\begin{align*} \int{\exp \left( -{{u}^{2}} \right)du}&=\frac{1}{2}\int{{{t}^{-1/2}}\exp \left( -t \right)dt} \\ & =\frac{1}{2}g\left( \frac{1}{2},-1,t \right). \end{align*}$$
We can replace $t$ with $u^{2}$ back and derive
$$\begin{equation*} \int{{\exp(-{{u}^{2}})}}du=\frac{1}{2}g\left( \frac{1}{2},-1,u^{2} \right). \end{equation*}$$ $\square$

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