Laguerre Polynomials
Laguerre Polynomials
In[47]:=
Table[LaguerreL[k,x],{k,0,6}]
Out[47]=
2 2 3
2 - 4 x + x 6 - 18 x + 9 x - x
{1, 1 - x, ------------, --------------------,
2 6
2 3 4
24 - 96 x + 72 x - 16 x + x
------------------------------,
24
2 3 4 5
120 - 600 x + 600 x - 200 x + 25 x - x
------------------------------------------,
120
2 3 4 5 6
720 - 4320 x + 5400 x - 2400 x + 450 x - 36 x + x
------------------------------------------------------}
720
In[48]:=
Plot[Evaluate[Table[LaguerreL[k,x],{k,0,6}]],{x,0,6}];
Laguerre's equation is
xy''+(1-x)y'+Ly=0 0<x<Infinity
If we multiply through by Exp[-x] we get
[ xExp[-x]y' ]' +LExp[-x] y=0 (Check this!)
Here r(x)=xExp[-x], q(x)=0, p(x)=Exp[-x]
This is a singular S-L problem with endpoint conditions y(0+) is bounded and
Exp[-x]y(x)->0 as x->Infinity.
You may also see
(xy')' + [ L+(2-x)/4 ]y=0 0<x<Infinity
with endpoint condition y is bounded as x->Infinity and as x->0.
Here the eigenfunctions are the Laguerre polynomials times Exp[-x/2].
The eigenvalues of Laguerre's equation are the nonnegative integers n. The eigenfunctions are the Laguerre polynomials.
Note that the eigenvalues are real and satisfy L->Infinity.
These polynomials DO solve the differential equation. Check one of them.
In[49]:=
Simplify[x D[LaguerreL[2,x],{x,2}]+(1-x)D[LaguerreL[2,x],x]+2LaguerreL[2,x]]
Out[49]=
0
Check the first seven:
In[50]:=
Table[x D[LaguerreL[2,x],{x,2}]+(1-x)D[LaguerreL[2,x],x]+2LaguerreL[2,x],{k,0,6}]
Out[50]=
2 (1 - x) (-4 + 2 x)
{2 - 3 x + x + ------------------,
2
2 (1 - x) (-4 + 2 x)
2 - 3 x + x + ------------------,
2
2 (1 - x) (-4 + 2 x)
2 - 3 x + x + ------------------,
2
2 (1 - x) (-4 + 2 x)
2 - 3 x + x + ------------------,
2
2 (1 - x) (-4 + 2 x)
2 - 3 x + x + ------------------,
2
2 (1 - x) (-4 + 2 x)
2 - 3 x + x + ------------------,
2
2 (1 - x) (-4 + 2 x)
2 - 3 x + x + ------------------}
2
In[51]:=
Simplify[%]
Out[51]=
{0, 0, 0, 0, 0, 0, 0}
By one of the major Sturm-Liouville theorems, the Laguerre polynomials are orthogonal with respect to the weight Exp[-x]. Check one.
In[52]:=
Integrate[Exp[-x]LaguerreL[1,x]LaguerreL[2,x],{x,0,Infinity}]
Out[52]=
0
Check many at a time:
In[53]:=
Table[Integrate[Exp[-x]LaguerreL[k,x]LaguerreL[m,x],{x,0,Infinity}],{k,0,3},{m,0,3}];
MatrixForm[%]
Out[53]=
1
0
0
0
0
1
0
0
0
0
2 2
(2 - 4 x + x )
Integrate[---------------, {x, 0, Infinity}]
x
E
--------------------------------------------
4
0
0
0
0
2 3 2
(6 - 18 x + 9 x - x )
Integrate[-----------------------, {x, 0, Infinity}]
x
E
----------------------------------------------------
36
The Laguerre polynomials have many interesting properties. Using what is above you can't see one that will prove helpful, so I'll show you....
Integral[Exp[-x](Laguerre[k,x])^2]=(n!)^2
The Laguerre polynomials, like the Legendre and Hermite polynomials, have a relationship involving repeated derivatives.
LaguerreL[k,x]=Exp[x]/n! D[x^n Exp[-x], {x,k}]
Does it seem to work?
In[54]:=
Table[Exp[x]/k! D[x^k Exp[-x], {x,k}],{k,0,4}]
Simplify[%]
Out[54]=
2
x 2 4 x x
E (-- - --- + --)
x x x
x -x x E E E
{1, E (E - --), ------------------,
x 2
E
2 3
x 6 18 x 9 x x
E (-- - ---- + ---- - --)
x x x x
E E E E
--------------------------,
6
2 3 4
x 24 96 x 72 x 16 x x
E (-- - ---- + ----- - ----- + --)
x x x x x
E E E E E
-----------------------------------}
24
Out[55]=
2 2 3
2 - 4 x + x 3 x x
{1, 1 - x, ------------, 1 - 3 x + ---- - --,
2 2 6
3 4
2 2 x x
1 - 4 x + 3 x - ---- + --}
3 24
In[56]:=
Table[LaguerreL[k,x],{k,0,4}]
Out[56]=
2 2 3
2 - 4 x + x 6 - 18 x + 9 x - x
{1, 1 - x, ------------, --------------------,
2 6
2 3 4
24 - 96 x + 72 x - 16 x + x
------------------------------}
24