HermiteH

HermiteH[n,x]

gives the Hermite polynomial .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • Explicit polynomials are given for nonnegative 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[n,x] is an entire function of x with no branch cut discontinuities.

Examples

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Basic Examples  (5)

Compute the 10^(th) Hermite polynomial:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope  (35)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number input:

Evaluate efficiently at high precision:

Specific Values  (6)

Values of HermiteH at fixed points:

HermiteH for symbolic n:

Values at zero:

Find the first positive maximum of HermiteH[10,x ]:

Compute the associated HermiteH[7,x] polynomial:

Different HermiteH types give different symbolic forms:

Visualization  (4)

Plot the HermiteH polynomial for various orders:

Plot the real part of :

Plot the imaginary part of :

Plot as real parts of two parameters vary:

Types 2 and 3 of HermiteH function have different branch cut structures:

Function Properties  (7)

HermiteH is defined for all real and complex values:

Approximate function range of HermiteH:

Hermite polynomial of an even order is even:

Hermite polynomial of an odd order is odd:

HermiteH has the mirror property :

HermiteH threads elementwise over lists:

TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z when n=3:

Formula for the ^(th) derivative with respect to z:

Integration  (3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

Series Expansions  (5)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:

Find the series expansion at Infinity:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

Function Identities and Simplifications  (3)

HermiteH may reduce to simpler form:

Exponential generating function of HermiteH:

Recurrence identity:

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  (3)

Get the list of coefficients in a Hermite polynomial:

HermiteH can be represented as a DifferentialRoot:

The exponential generating function for HermiteH:

Possible Issues  (2)

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:

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:

Introduced in 1988
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