hermite {EQL} | R Documentation |
Computes the Hermite polynomial H_n(x).
hermite(x, n, prob = TRUE)
x |
a numeric vector or array giving the values at which the Hermite polynomial should be evaluated. |
n |
an integer vector or array giving the degrees of the Hermite
polynomials. If length(x) != 1 , n must be either of the
same length as x or a single value. |
prob |
logical. If TRUE (the default) the probabilistic
version of the Hermite polynomial is evaluated, otherwise the
physicists' Hermite polynomials are used. See the ‘Details’
section below for further information. |
The Hermite polynomials are given by:
and the relationship between the two versions is given by
H_n^Ph(x)=2^(n/2)*H_n^Pr(2^(1/2)*x).
The term ‘probabilistic’ is motivated by the fact that in this case the Hermite polynomial H_n(x) can be as well defined by
H_n(x) = (-1)^n*phi^(n)(x)/phi(x),
where phi(x) denotes the density function of the standard normal distribution and phi^(k)(x) denotes the kth derivative of phi(x) with respect to x.
If the argument n
is a vector it must be of the same
length as the argument x
or the length of the argument x
must be equal to one. The Hermite polynomials are then evaluated
either at x_i with degree n_i or at x with degree
n_i, respectively.
the Hermite polynomial (either the probabilists' or the
physicists' version) evaluated at x
.
Thorn Thaler
Fedoryuk, M.V. (2001). Hermite polynomials. Encyclopaedia of Mathematics, Kluwer Academic Publishers.
2^(3/2)*hermite(sqrt(2)*5, 3) # = 940 hermite(5, 3, FALSE) # = 940 hermite(2:4, 1:3) # H_1(2), H_2(3), H_3(4) hermite(2:4, 2) # H_2(2), H_2(3), H_2(4) hermite(2, 1:3) # H_1(2), H_2(2), H_3(2) ## Not run: hermite(1:3, 1:4) # Error! ## End(Not run)