LaguerreL

LaguerreL[n,x]

gives the Laguerre polynomial TemplateBox[{n, x}, LaguerreL].

LaguerreL[n,a,x]

gives the generalized Laguerre polynomial TemplateBox[{n, a, x}, LaguerreL3].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • Explicit polynomials are given when possible.
  • TemplateBox[{n, x}, LaguerreL]=TemplateBox[{n, 0, x}, LaguerreL3].
  • 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^(th) 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  (41)

Numerical Evaluation  (6)

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 LaguerreL function using MatrixFunction:

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 TemplateBox[{10}, LucasL](z):

Plot the imaginary part of TemplateBox[{10}, LucasL](z):

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 TemplateBox[{n, a, z}, LaguerreL3] has restrictions on and , but not :

TemplateBox[{1, x}, LaguerreL] achieves all real and complex values:

So do all associated TemplateBox[{1, n, x}, LaguerreL3]:

Function range of TemplateBox[{2, x}, LaguerreL]:

LaguerreL has the mirror property :

LaguerreL threads elementwise over lists:

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

TemplateBox[{n, a, x}, LaguerreL3] is not analytic, but it is meromorphic:

TemplateBox[{1, a, x}, LaguerreL3] is a decreasing function:

TemplateBox[{2, a, x}, LaguerreL3] is neither non-decreasing nor non-increasing:

Laguerre polynomials are not injective for values other than 1:

TemplateBox[{n, a, x}, LaguerreL3] is surjective for odd :

LaguerreL is neither non-negative nor non-positive:

TemplateBox[{n, a, x}, LaguerreL3] has no singularities or discontinuities in :

TemplateBox[{2, a, x}, LaguerreL3] is convex:

TraditionalForm formatting:

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 ^(th) 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 Gauss–Laguerre quadrature rule is based on the roots of the n^(th)-order Laguerre polynomial. Compute the nodes and weights of an n-point Gauss–Laguerre quadrature rule for a given value of :

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

Compare the result of the Gauss–Laguerre 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: 22-December-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: 22-December-2024 ]}