Hermite Polynomials
Hermite Polynomials
In[37]:=
Table[HermiteH[k,x],{k,0,6}]
Out[37]=
2 3 2 4
{1, 2 x, -2 + 4 x , -12 x + 8 x , 12 - 48 x + 16 x ,
3 5 2 4 6
120 x - 160 x + 32 x , -120 + 720 x - 480 x + 64 x }
In[38]:=
Plot[Evaluate[Table[HermiteH[k,x],{k,0,6}]],{x,-6,6}];
Hermite's equation is
y''-2xy'+Ly=0 -Infinity<x<Infinity
If we let y=Exp[-x^2/2]u and substitute we get
u''+[L- (x^2-1) ]u (Check this!)
Here r(x)=1, q(x)=x^2-1, p(x)=1
This is a singular S-L problem with no boundary conditions, but with the
condition that the solution tends to 0 as x->Infinity.
Note that you may also see Hermite's equation as
y''-xy'+Ly=0.
The eigenvalues of Hermite's equation are the nonnegative integers 2n. The eigenfunctions are Exp[-x^2/2] times the Hermite polynomials.
Note that the eigenvalues are real and satisfy L->Infinity.
These polynomials DO solve the differential equation. Check one of them.
In[39]:=
Simplify[D[HermiteH[3,x]Exp[-x^2/2],{x,2}]+(2(3)-(x^2-1))Exp[-x^2/2]HermiteH[3,x]]
Out[39]=
0
Check the first seven:
In[40]:=
Table[D[Exp[-x^2/2]HermiteH[k,x],{x,2}]+(2k-(x^2-1))Exp[-x^2/2]HermiteH[k,x],{k,0,6}]
Out[40]=
2 2 2 3 2
-x /2 x 1 - x -6 x 2 x 2 x (3 - x )
{-E + ----- + ------, ----- + ----- + ------------,
2 2 2 2 2
x /2 x /2 x /2 x /2 x /2
E E E E E
2 2 2 2
8 16 x -2 + 4 x x (-2 + 4 x )
----- - ----- - --------- + -------------- +
2 2 2 2
x /2 x /2 x /2 x /2
E E E E
2 2
(5 - x ) (-2 + 4 x )
--------------------,
2
x /2
E
2 3
48 x 2 x (-12 + 24 x ) -12 x + 8 x
----- - ----------------- - ------------ +
2 2 2
x /2 x /2 x /2
E E E
2 3 2 3
x (-12 x + 8 x ) (7 - x ) (-12 x + 8 x )
----------------- + -----------------------,
2 2
x /2 x /2
E E
2 3 2 4
-96 + 192 x 2 x (-96 x + 64 x ) 12 - 48 x + 16 x
------------ - ------------------- - ------------------ +
2 2 2
x /2 x /2 x /2
E E E
2 2 4 2 2 4
x (12 - 48 x + 16 x ) (9 - x ) (12 - 48 x + 16 x )
----------------------- + -----------------------------,
2 2
x /2 x /2
E E
3 2 4
-960 x + 640 x 2 x (120 - 480 x + 160 x )
--------------- - --------------------------- -
2 2
x /2 x /2
E E
3 5 2 3 5
120 x - 160 x + 32 x x (120 x - 160 x + 32 x )
---------------------- + --------------------------- +
2 2
x /2 x /2
E E
2 3 5
(11 - x ) (120 x - 160 x + 32 x )
----------------------------------,
2
x /2
E
2 4
1440 - 5760 x + 1920 x
------------------------ -
2
x /2
E
3 5
2 x (1440 x - 1920 x + 384 x )
------------------------------- -
2
x /2
E
2 4 6
-120 + 720 x - 480 x + 64 x
------------------------------ +
2
x /2
E
2 2 4 6
x (-120 + 720 x - 480 x + 64 x )
----------------------------------- +
2
x /2
E
2 2 4 6
(13 - x ) (-120 + 720 x - 480 x + 64 x )
------------------------------------------}
2
x /2
E
In[41]:=
Simplify[%]
Out[41]=
{0, 0, 0, 0, 0, 0, 0}
Again, by one of the major Sturm-Liouville theorems, the Hermite polynomials are orthogonal with respect to the weight Exp[-x^2]. This is equivalent to the eigenfunctions Exp[-x^2/2]HermiteH[k,x] being orthogonal with respect to the weight p(x)=1. Check one.
In[42]:=
Integrate[Exp[-x^2]HermiteH[1,x]HermiteH[2,x],{x,-Infinity,Infinity}]
Out[42]=
0
Check many at a time:
In[43]:=
Table[Integrate[Exp[-x^2]HermiteH[k,x]HermiteH[m,x],{x,-Infinity,Infinity}],{k,0,3},{m,0,3}];
MatrixForm[%]
Out[43]=
Sqrt[Pi] 0 0 0 0 2 Sqrt[Pi] 0 0 0 0 8 Sqrt[Pi] 0 0 0 0 48 Sqrt[Pi]
The Hermite polynomials have many interesting properties.
Do you see a pattern? You have to see it now, because on some systems Mathematica won't go any higher!
Do you see a pattern? You have ...
Another property of the Hermite polynomials is the relationship
HermiteH[k,x]=(-1)^n Exp[x^2] D[Exp[-x^2],{x,k}]
Does it seem to work?
In[44]:=
Table[(-1)^k Exp[x^2] D[Exp[-x^2],{x,k}],{k,0,4}]
Simplify[%]
Out[44]=
2 2 2 3
x -2 4 x x 12 x 8 x
{1, 2 x, E (--- + ----), -(E (---- - ----)),
2 2 2 2
x x x x
E E E E
2 2 4
x 12 48 x 16 x
E (--- - ----- + -----)}
2 2 2
x x x
E E E
Out[45]=
2 2 2 4
{1, 2 x, -2 + 4 x , -4 x (3 - 2 x ), 4 (3 - 12 x + 4 x )}
In[46]:=
Table[HermiteH[k,x],{k,0,4}]
Out[46]=
2 3 2 4
{1, 2 x, -2 + 4 x , -12 x + 8 x , 12 - 48 x + 16 x }