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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