# 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 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(42)

### 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(14)

HermiteH is defined for all real and complex values:

Approximate function range of :

Hermite polynomial of an even order is even:

Hermite polynomial of an odd order is odd:

HermiteH has the mirror property :

is an analytic function of :

is neither non-decreasing nor non-increasing for :

It is non-decreasing for :

It is non-increasing for :

is not injective for for :

is surjective for positive odd values of for :

is positive for :

It has indefinite sign for :

HermiteH has no singularities or discontinuities:

is convexfor and :

It is concave for :

### 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 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 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:

Wolfram Research (1988), HermiteH, Wolfram Language function, https://reference.wolfram.com/language/ref/HermiteH.html.

#### Text

Wolfram Research (1988), HermiteH, Wolfram Language function, https://reference.wolfram.com/language/ref/HermiteH.html.

#### BibTeX

@misc{reference.wolfram_2021_hermiteh, author="Wolfram Research", title="{HermiteH}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/HermiteH.html}", note=[Accessed: 19-September-2021 ]}

#### BibLaTeX

@online{reference.wolfram_2021_hermiteh, organization={Wolfram Research}, title={HermiteH}, year={1988}, url={https://reference.wolfram.com/language/ref/HermiteH.html}, note=[Accessed: 19-September-2021 ]}

#### CMS

Wolfram Language. 1988. "HermiteH." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HermiteH.html.

#### APA

Wolfram Language. (1988). HermiteH. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HermiteH.html