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:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around :

Compute the elementwise values of an array:

Or compute the matrix HermiteH function using MatrixFunction :

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:

Plot the HermiteH polynomial for various orders:

Plot the real part of :

Plot the imaginary part of :

Plot the Hermite polynomial as a function of two variables:

HermiteH is defined for all real and complex values:

Approximate function range of TemplateBox[{2, x}, 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:

TemplateBox[{n, x}, HermiteH] is an analytic function of :

TemplateBox[{n, x}, HermiteH] is neither non-decreasing nor non-increasing for :

It is non-decreasing for :

It is non-increasing for :

TemplateBox[{n, x}, HermiteH] is not injective for for :

TemplateBox[{n, x}, HermiteH] is surjective for positive odd values of for :

TemplateBox[{n, x}, HermiteH] is positive for :

It has indefinite sign for :

HermiteH has no singularities or discontinuities:

TemplateBox[{n, x}, HermiteH] is convexfor and :

It is concave for :

TraditionalForm formatting:

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:

Compute the indefinite integral using Integrate :

Verify the anti-derivative:

Definite integral:

More integrals:

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:

HermiteH may reduce to simpler form:

Exponential generating function of HermiteH :

Recurrence identity:

Express HermiteH in terms of LaguerreL :