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:

LaguerreL can be used with Interval and CenteredInterval objects:

Values of LaguerreL at fixed points:

Values at zero:

Find the first positive minimum of LaguerreL[10,x ]:

Compute the associated LaguerreL[7,x] polynomial:

Different LaguerreL types give different symbolic forms:

Plot the LaguerreL 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 to x:

Higher derivatives with respect to x:

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

Formula for the derivative with respect to x:

Compute the indefinite integral using Integrate:

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:

LaguerreL may reduce to simpler form:

Generating function of LaguerreL:

Recurrence identity: