Hh Function
HhFunction
The Hh-function is a function closely related to the normal distribution function. It can be defined using the auxilary functions
Z(x) = [画像:1/(sqrt(2pi))e^(-x^2/2)]
(1)
Q(x) = [画像:1/(sqrt(2pi))int_x^inftye^(-t^2/2)dt]
(2)
= [画像: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) = [画像:((-1)^n)/(n!)Hh_(-1)(x)(d^n)/(dx^n)[(Q(x))/(Z(x))].]
![画像:((-1)^n)/(n!)Hh_(-1)(x)(d^n)/(dx^n)[(Q(x))/(Z(x))].](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/HhFunction/Inline15.svg){kind=link}
(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) = [画像:sqrt(pi/2)erfc(x/(sqrt(2)))]
(9)
Hh_1(x) = [画像:e^(-x^2/2)-sqrt(pi/2)xerfc(x/(sqrt(2)))]
(10)
Hh_2(x) = [画像:1/4[-2xe^(-x^2/2)+sqrt(2pi)(x^2+1)erfc(x/(sqrt(2)))]]
![画像:1/4[-2xe^(-x^2/2)+sqrt(2pi)(x^2+1)erfc(x/(sqrt(2)))]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/HhFunction/Inline35.svg){kind=link}
(11)
Hh_3(x) = [画像:1/(12)[2e^(-x^2/2)(x^2+2)-sqrt(2pi)x(x^2+3)erfc(x/(sqrt(2)))].]
![画像:1/(12)[2e^(-x^2/2)(x^2+2)-sqrt(2pi)x(x^2+3)erfc(x/(sqrt(2)))].](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/HhFunction/Inline38.svg){kind=link}
(12)
See also
Erfc, Normal Distribution Function, Tetrachoric FunctionExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Hh FunctionCite this as:
Weisstein, Eric W. "Hh Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HhFunction.html