Built-in Mathematica Symbol

LaguerreL

LaguerreL[n, x]
gives the Laguerre polynomial

.

LaguerreL[n, a, x]
gives the generalized Laguerre polynomial

.

MORE INFORMATION

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.

Compute the 5

Plot the 10

Laguerre polynomial:

Compute the associated Laguerre polynomial

:

Compute the 5

LaguerreL:

In[1]:=

Out[1]=

Plot the 10

Laguerre polynomial:

In[1]:=

Out[1]=

Compute the associated Laguerre polynomial

:

In[1]:=

Out[1]=

Scope (6)

Evaluate for fractional orders:

Evaluate for complex orders and arguments:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

LaguerreL threads element-wise over lists:

TraditionalForm formatting:

Generalizations & Extensions (1)

LaguerreL can be applied to a power series:

Applications (3)

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:

Properties & Relations (2)

Get the list of coefficients in a Laguerre polynomial:

Use FunctionExpand to expand LaguerreL functions into simpler functions: