Hermite Polynomials
Hermite Polynomials
Table[HermiteH[k,x],{k,0,6}]
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.
Simplify[D[HermiteH[3,x]Exp[-x^2/2],{x,2}]+(2(3)-(x^2-1))Exp[-x^2/2]HermiteH[3,x]]Check the first seven:
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}]
Simplify[%]
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.
Integrate[Exp[-x^2]HermiteH[1,x]HermiteH[2,x],{x,-Infinity,Infinity}]Check many at a time:
Table[Integrate[Exp[-x^2]HermiteH[k,x]HermiteH[m,x],{x,-Infinity,Infinity}],{k,0,3},{m,0,3}];
MatrixForm[%]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?
Table[(-1)^k Exp[x^2] D[Exp[-x^2],{x,k}],{k,0,4}]
Simplify[%]
Table[HermiteH[k,x],{k,0,4}]