hermite_polynomial


hermite_polynomial, an Octave code which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

The physicist's Hermite polynomial H(i,x) can be defined by:

        H(i,x) = (-1)^i exp(x^2/2) * d^i/dx^i ( exp(-x^2/2) )
      

The normalized physicist's Hermite polynomial Hn(i,x) is scaled so that

        Integral ( -oo < x < +oo ) exp ( - x^2 ) * Hn(i,x) Hn(j,x) dx = delta ( i, j )
      

The probabilist's Hermite polynomial He(i,x) is related to H(i,x) by:

        He(i,x) = H(i,x/sqrt(2)) / sqrt ( 2^in )
      

The normalized probabilist's Hermite polynomial Hen(i,x) is scaled so that

        Integral ( -oo < x < +oo ) exp ( - 0.5*x^2 ) * Hen(i,x) Hen(j,x) dx = delta ( i, j )
      

The Hermite function Hf(i,x) is related to H(i,x) by:

        Hf(i,x) = H(i,x) * exp(-x^2/2) / sqrt ( 2^i * i! * sqrt ( pi ) )
      

The Hermite function Hf(i,x) is scaled so that:

        Integral ( -oo < x < +oo ) Hf(i,x) Hf(j,x) dx = delta ( i, j )
      

Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license.

Languages:

hermite_polynomial is available in a C version and a C++ version and a Fortran90 version and a Fortran77 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

hermite_polynomial_test

hermite_rule, an Octave code which computes and prints a Gauss-Hermite quadrature rule.

octave_polynomial, an Octave code which analyzes a variety of polynomial families, returning the polynomial values, coefficients, derivatives, integrals, roots, or other information.

pce_ode_hermite, an Octave code which sets up a simple scalar ODE for exponential decay with an uncertain decay rate, using a polynomial chaos expansion in terms of Hermite polynomials.

polpak, an Octave code which evaluates a variety of mathematical functions.

polynomial_conversion, an Octave code which converts representations of a polynomial between monomial, Bernstein, Chebyshev, Hermite, Lagrange, Laguerre and other forms.

test_values, an Octave code which supplies test values of various mathematical functions.

Reference:

  1. Theodore Chihara,
    An Introduction to Orthogonal Polynomials,
    Gordon and Breach, 1978,
    ISBN: 0677041500,
    LC: QA404.5 C44.
  2. Walter Gautschi,
    Orthogonal Polynomials: Computation and Approximation,
    Oxford, 2004,
    ISBN: 0-19-850672-4,
    LC: QA404.5 G3555.
  3. Frank Olver, Daniel Lozier, Ronald Boisvert, Charles Clark,
    NIST Handbook of Mathematical Functions,
    Cambridge University Press, 2010,
    ISBN: 978-0521192255,
    LC: QA331.N57.
  4. Gabor Szego,
    Orthogonal Polynomials,
    American Mathematical Society, 1992,
    ISBN: 0821810235,
    LC: QA3.A5.v23.

Source Code:


Last modified on 31 March 2024.