gives the Laguerre polynomial .
gives the generalized Laguerre polynomial .
- 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.
Examplesopen 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:
Numerical Evaluation (4)
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Complex number input:
Evaluate efficiently at high precision:
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:
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:
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:
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:
Generalizations & Extensions (1)
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:
Properties & Relations (7)
Possible Issues (1)
Cancellations in the polynomial form may lead to inaccurate numerical results:
Evaluate the function directly: