# 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

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## Basic Examples(6)

Compute the 5 Laguerre polynomial:

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 :

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:

### 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(6)

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:

Compare the value of the MarcumQ function for large arguments to its asymptotic formula:

Construct an approximation using the central limit theorem:

Evaluate numerically:

An n-point GaussLaguerre quadrature rule is based on the roots of the n-order Laguerre polynomial. Compute the nodes and weights of an n-point GaussLaguerre quadrature rule for a given value of :

Use the n-point Gaussian quadrature rule to numerically evaluate an integral:

Compare the result of the GaussLaguerre quadrature with the result from NIntegrate:

## 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:

Wolfram Research (1988), LaguerreL, Wolfram Language function, https://reference.wolfram.com/language/ref/LaguerreL.html (updated 2022).

#### Text

Wolfram Research (1988), LaguerreL, Wolfram Language function, https://reference.wolfram.com/language/ref/LaguerreL.html (updated 2022).

#### CMS

Wolfram Language. 1988. "LaguerreL." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/LaguerreL.html.

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_laguerrel, author="Wolfram Research", title="{LaguerreL}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/LaguerreL.html}", note=[Accessed: 29-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_laguerrel, organization={Wolfram Research}, title={LaguerreL}, year={2022}, url={https://reference.wolfram.com/language/ref/LaguerreL.html}, note=[Accessed: 29-May-2024 ]}