Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (4.8)

\begin{equation*}
\Phi\left(x\right)=\frac{1}{2}\left[1+\frac{x}{\sqrt{2\pi }}{\exp(\tfrac{-{{x}^{2}}}{2})}h_{\tfrac{-1}{2}}^{\tfrac{-{{x}^{2}}}{2}} \right].
\end{equation*}Proof:
\begin{align*}
\Phi \left( x \right) &=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{\exp \left( \frac{-{{t}^{2}}}{2} \right)}dt \\
& =\frac{1}{2}\left[ 1+\text{erf}\left( \frac{x}{\sqrt{2}} \right) \right] \\
& =\frac{1}{2}\left[ 1+\frac{x}{\sqrt{2\pi }}\exp (\tfrac{-{{x}^{2}}}{2})h_{\tfrac{-1}{2}}^{\tfrac{-{{x}^{2}}}{2}} \right].
\end{align*}
$\square$

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