Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (4.7)

\begin{equation*}
\text{erf}\left(x\right)=\frac{x}{\sqrt{\pi }}{\exp(-{{x}^{2}})}h_{\tfrac{-1}{2}}^{-{{x}^{2}}}.
\end{equation*}Proof:
\begin{align*}
\text{erf}\left( x \right) &=\frac{2}{\sqrt{\pi }}\int_{0}^{x}{\exp \left( -{{t}^{2}} \right)dt} \\
& =\frac{2}{\sqrt{\pi }}\left( \frac{1}{2}t\exp \left( -{{t}^{2}} \right)h_{\tfrac{-1}{2}}^{-{{t}^{2}}}{{\Bigr|}_{t=x}} \right) \\
& =\frac{x}{\sqrt{\pi }}\exp \left( -{{x}^{2}} \right)h_{\tfrac{-1}{2}}^{-{{x}^{2}}}.
\end{align*}
$\square$

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