This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# LaguerreL

 LaguerreLgives the Laguerre polynomial . LaguerreLgives 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 is an entire function of x with no branch cut discontinuities.
Compute the 5 LaguerreL:
Plot the 10 Laguerre polynomial:
Compute the associated Laguerre polynomial :
Compute the 5 LaguerreL:
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Plot the 10 Laguerre polynomial:
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Compute the associated Laguerre polynomial :
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 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 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 :
Get the list of coefficients in a Laguerre polynomial:
Use FunctionExpand to expand LaguerreL functions into simpler functions:
Cancellations in the polynomial form may lead to inaccurate numerical results:
Evaluate the function directly:
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