BUILT-IN MATHEMATICA SYMBOL

HermiteH

gives the Hermite polynomial

.

MORE INFORMATION

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.

EXAMPLES

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Basic Examples (2)

Compute the 10

Hermite polynomial:

Compute the 10

Hermite polynomial:

In[1]:=

Out[1]=

In[1]:=

Out[1]=

Scope (6)

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:

Generalizations & Extensions (2)

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

:

Properties & Relations (1)

Get the list of coefficients in a Hermite polynomial:

Possible Issues (2)

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:

Neat Examples (4)

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: