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based on an earlier version of the Wolfram Language.
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# HermiteH

 HermiteHgives the Hermite polynomial .
• Mathematical function, suitable for both symbolic and numerical manipulation.
• Explicit polynomials are given for non-negative integers n.
• The Hermite polynomials satisfy the differential equation .
• They are orthogonal polynomials with weight function in the interval .
• For certain special arguments, HermiteH automatically evaluates to exact values.
• HermiteH can be evaluated to arbitrary numerical precision.
• HermiteH automatically threads over lists.
• HermiteH is an entire function of x with no branch cut discontinuities.
Compute the 10 Hermite polynomial:
Compute the 10 Hermite polynomial:
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 Scope   (6)
Evaluate for complex argument and complex orders:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Simple cases give exact symbolic results even for symbolic order:
HermiteH can be applied to power series:
HermiteH can deal with real-valued intervals:
 Applications   (5)
Solve the Hermite differential equation:
Quantum harmonic oscillator wave functions:
Normalization:
Compute the expectation value of :
Momentum and position wave functions for a harmonic oscillator have the same form:
Solve a recursion relation:
Set up generalized Fourier series based on normalized Hermite functions:
Find series coefficients for :
Compare approximation and exact function:
Gibbs-like phenomenon for approximation of discontinuous function:
Find an integral for symbolic :
Evaluation for non-negative integer values of n requires Limit:
Compare with integration for explicit :
Get the list of coefficients in a Hermite polynomial:
Cancellations in the polynomial form may lead to inaccurate numerical results:
Evaluate the function directly:
Plot the 100 Hermite polynomial:
Machine-precision evaluation of explicit polynomials may be numerically unstable due to cancellations:
Distribution of the zeros of the first 20 Hermite polynomials:
Interpolation between Hermite polynomials:
Comparison of quantum and classical probability distributions for a harmonic oscillator:
Generalized Lissajous figures:
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