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Hh Function

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HhFunction

The Hh-function is a function closely related to the normal distribution function. It can be defined using the auxilary functions

Z(x)=1/(sqrt(2pi))e^(-x^2/2)
(1)
Q(x)=1/(sqrt(2pi))int_x^inftye^(-t^2/2)dt
(2)
=1/2erfc(x/(sqrt(2))),
(3)

where erfc is the complementary error function. Then

Hh_(-n)(x)=(-1)^(n-1)sqrt(2pi)Z^((n-1))(x)
(4)
Hh_n(x)=((-1)^n)/(n!)Hh_(-1)(x)(d^n)/(dx^n)[(Q(x))/(Z(x))].
(5)

Values for integer indices from -3 to +3 are given by:

Hh_(-3)(x)=e^(-x^2/2)(x^2-1)
(6)
Hh_(-2)(x)=e^(-x^2/2)x
(7)
Hh_(-1)(x)=e^(-x^2/2)
(8)
Hh_0(x)=sqrt(pi/2)erfc(x/(sqrt(2)))
(9)
Hh_1(x)=e^(-x^2/2)-sqrt(pi/2)xerfc(x/(sqrt(2)))
(10)
Hh_2(x)=1/4[-2xe^(-x^2/2)+sqrt(2pi)(x^2+1)erfc(x/(sqrt(2)))]
(11)
Hh_3(x)=1/(12)[2e^(-x^2/2)(x^2+2)-sqrt(2pi)x(x^2+3)erfc(x/(sqrt(2)))].
(12)

See also

Erfc, Normal Distribution Function, Tetrachoric Function

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 300 and 691, 1972.Jeffreys, H. and Jeffreys, B. S. "The Parabolic Cylinder, Hermite, and Hh Functions" et seq. §23.08-23.09 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 620-627, 1988.

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Hh Function

Cite this as:

Weisstein, Eric W. "Hh Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HhFunction.html

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