Orthogonal Polynomials

1. Nov 13, 2009

In my third year math class we were asked a question to prove that Ho(X) and H1(x) are orthogonal to H2(x), with respect to the weight function e^(-x^2) over the interval negative to positive infinity

where Ho(x) = 1
H1(x) = 2x
H2(x) = (4x^2) - 2

i know that i have to multiply Ho(x) by H2(x) and divide by the weight function and integrate..but i get lost when it comes to integrating by parts with e^(x^2)...

2. Nov 13, 2009

in this question Hn(x) is the herite polynomial...where n = 0, 1, 2 ,3 etc

3. Nov 13, 2009

LCKurtz

Since you have symmetry in x you can do this integral:

$$2\int_0^\infty (4x^2-2)e^{-x^2}\,dx$$

Try breaking it into two parts and on the first part use integration by parts with:

$$u = 2x\ dv = 2xe^{-x^2}dx$$

and if I'm not mistaken, nice things will happen.

4. Nov 13, 2009

well that looks right..but it should be e^(x^2)...

5. Nov 13, 2009

LCKurtz

Why do you say that? The integral won't even converge with a positive exponential in there.

6. Nov 13, 2009