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 usingInterval andCenteredInterval objects:

Or compute average-case statistical intervals usingAround:

Compute the elementwise values of an array:

Or compute the matrixLaguerreL function usingMatrixFunction:

Values ofLaguerreL at fixed points:

Values at zero:

Find the first positive minimum ofLaguerreL[10,x]:

Compute the associatedLaguerreL[7,x] polynomial:

DifferentLaguerreL types give different symbolic forms:

Plot theLaguerreL polynomial for various orders:

Plot the real part of:

Plot the imaginary part of:

Plot as real parts of two parameters vary:

The primary Laguerre function is defined for all real and complex values:

The associated Laguerre function has restrictions on and, but not:

achieves all real and complex values:

So do all associated:

Function range of:

LaguerreL has the mirror property:

LaguerreL threads elementwise over lists:

is an analytic function of and:

is not analytic, but it is meromorphic:

is a decreasing function:

is neither non-decreasing nor non-increasing:

Laguerre polynomials are not injective for values other than 1:

is surjective for odd:

LaguerreL is neither non-negative nor non-positive:

has no singularities or discontinuities in:

is convex:

TraditionalForm formatting:

First derivative with respect tox:

Higher derivatives with respect tox:

Plot the higher derivatives with respect tox whenn=3:

Formula for the derivative with respect tox:

Compute the indefinite integral usingIntegrate:

Definite integral:

More integrals:

Find the Taylor expansion usingSeries:

Plots of the first three approximations around:

General term in the series expansion usingSeriesCoefficient:

Find the series expansion atInfinity:

Find series expansion for an arbitrary symbolic direction:

Taylor expansion at a generic point:

LaguerreL may reduce to simpler form:

Generating function ofLaguerreL:

Recurrence identity: