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*} $\square$

Welcome all math lovers!