# LaguerreL LaguerreL[n,x]

gives the Laguerre polynomial .

LaguerreL[n,a,x]

gives the generalized Laguerre polynomial .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• Explicit polynomials are given when possible.
• .
• The Laguerre polynomials are orthogonal with weight function .
• They satisfy the differential equation .
• For certain special arguments, LaguerreL automatically evaluates to exact values.
• LaguerreL can be evaluated to arbitrary numerical precision.
• LaguerreL automatically threads over lists.
• LaguerreL[n,x] is an entire function of x with no branch cut discontinuities.
• LaguerreL can be used with Interval and CenteredInterval objects. »

# Examples

open allclose all

## Basic Examples(6)

Compute the 5 LaguerreL:

Compute the associated Laguerre 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(40)

### Numerical Evaluation(5)

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:

### Specific Values(5)

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:

### Visualization(3)

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:

### Function Properties(13)

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 : 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:

### Differentiation(3)

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:

### Integration(3)

Compute the indefinite integral using Integrate:

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)

LaguerreL may reduce to simpler form:

Generating function of LaguerreL:

Recurrence identity:

## Generalizations & Extensions(1)

LaguerreL can be applied to a power series:

## Applications(4)

Solve the Laguerre differential equation:

Generalized Fourier series for functions defined on :

Radial wave-function of the hydrogen atom:

Compute the energy eigenvalue from the differential equation:

The energy is independent of the orbital quantum number l:

The number of derangement anagrams for a word with character counts :

Count the number of derangements for the word Mathematica:

Direct count:

## Properties & Relations(7)

Get the list of coefficients in a Laguerre polynomial:

Use FunctionExpand to expand LaguerreL functions into simpler functions:

LaguerreL can be represented as a DifferentialRoot:

LaguerreL can be represented in terms of MeijerG:

LaguerreL can be represented as a DifferenceRoot:

General term in the series expansion of LaguerreL:

The generating function for LaguerreL:

## Possible Issues(1)

Cancellations in the polynomial form may lead to inaccurate numerical results:

Evaluate the function directly: