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 allBasic Examples (6)
Scope (40)
Numerical Evaluation (5)
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
LaguerreL can be used with Interval and CenteredInterval objects:
Specific Values (5)
Visualization (3)
Plot the LaguerreL polynomial for various orders:
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:
LaguerreL has the mirror property :
LaguerreL threads elementwise over lists:
is an analytic function of
and
:
is not analytic, but it is meromorphic:
is neither non-decreasing nor non-increasing:
Laguerre polynomials are not injective for values other than 1:
LaguerreL is neither non-negative nor non-positive:
has no singularities or discontinuities in
:
TraditionalForm formatting:
Differentiation (3)
Integration (3)
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:
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 :
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:
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