= Generating-function proof of Hermite orthogonality
{title2=$\int_{\mathbb R}e^{-t^2}H_m(t)H_n(t)dt=\sqrt\pi\,2^nn!\,\delta_{mn}$}
Multiply two copies of the <generating function of the physicists' Hermite polynomials> and integrate against $e^{-t^2}$. Completing the square gives
$$
\int_{\mathbb R}e^{-t^2}e^{2ty-y^2}e^{2tz-z^2}dt=\sqrt\pi e^{2yz}.
$$
Comparing coefficients gives the displayed <orthogonality> and normalization. Equivalently differentiation in $y,z$ at zero justifies coefficient extraction by the Gaussian integrable bound on compact parameter ranges. Hence the unnormalized <Hermite functions> $e^{-t^2/2}H_n(t)$ have <norm> $\pi^{1/4}\sqrt{2^nn!}$. This proves their mutual <orthogonality>; <completeness> of that family is an additional assertion.
Back to article page
