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HermiteH

HermiteH
gives 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^(th) Hermite polynomial:
Compute the 10^(th) Hermite polynomial:
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Evaluate for complex argument and complex orders:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
HermiteH threads element-wise over lists:
Simple cases give exact symbolic results even for symbolic order:
TraditionalForm formatting:
HermiteH can be applied to power series:
HermiteH can deal with real-valued intervals:
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^(th) 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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