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