Showing Hermite polynomials are orthogonal

Question

I need to show that two Hermite polynomials are orthogonal, but I'm a little confused.

I have: $H_2(x) = 4x^2-2$ and $H_3(x) = 8x^3-12x$

I know I need to integrate $$\int_{-L}^L (4x^2-2)(8x^3-12x) dx=0,$$ because it says I need to show it's orthogonal on $[-L,L]$, where $L$ is a constant.

Do I just pick random values for $L$, or is there some sort of procedure?

Answer 1

Note that the integrand is an even function--$4x^2-2$--times an odd function--$8x^3-12x$--so is an odd function. This function is a polynomial, and so is continuous. Given any $L>0$, we have that the integral over $[-L,L]$ of an odd function (that's continuous there) is $0$.

Answer 2

Think of $L$ as an arbitrary, but fixed interval length. These functions are orthogonal regardless of the actual value of $L$, as the straightforward integration demonstrates:

\begin{align} &\int_{-L}^L (4x^2-2)(8x^3-12x) dx\\ &=\left[8 \left(\frac{2 x^6}{3}-2 x^4+\frac{3 x^2}{2}\right)\right]_{-L}^{L}\\ &=\left(8 \left(\frac{2 L^6}{3}-2 L^4+\frac{3 L^2}{2}\right)\right)-\left(8 \left(\frac{2 (-L)^6}{3}-2 (-L)^4+\frac{3 (-L)^2}{2}\right)\right)\\ &=0. \end{align}