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Hh Function

Let

$\displaystyle Z(x)$ $\textstyle \equiv$ $\displaystyle {1\over\sqrt{2\pi}} e^{-x^2/2}$ (1)
$\displaystyle Q(x)$ $\textstyle \equiv$ $\displaystyle {1\over\sqrt{2\pi}} \int_x^\infty e^{-t^2/2}\,dt,$ (2)

where $Z$ and $Q$ are closely related to the Normal Distribution Function, then
$\displaystyle {\rm Hh}_{-n}(x)$ $\textstyle =$ $\displaystyle (-1)^{n-1}\sqrt{2\pi}\,Z^{(n-1)}(x)$ (3)
$\displaystyle {\rm Hh}_n(x)$ $\textstyle =$ $\displaystyle {(-1)^n\over n!} {\rm Hh}_{-1}(x){d^n\over dx^n} \left[{Q(x)\over Z(x)}\right].$ (4)

See also Normal Distribution Function, Tetrachoric Function




© 1996-9 Eric W. Weisstein
1999-05-25