Normalization of radial Laguerre-Gauss

[DivGradCurl]

Homework Statement

Normalization of radial Laguerre-Gauss

Normalize \Psi _n (r) = h_n L_n (2\pi r^2) e^{-\pi r^2}

Homework Equations

\int _0 ^{\infty} e^{-x} \, x^k \, L_n ^{(k)} (x) \, L_m ^{(k)} (x) dx = \frac{(n+k)!}{n!} \delta _{m,n}

The Attempt at a Solution

1 = \int _0 ^{\infty} \Psi _m ^{\ast} (r) \Psi _n (r) dr = \int _0 ^{\infty} h_m ^{\ast} L_m (2\pi r^2) e^{-\pi r^2} h_n L_n (2\pi r^2) e^{-\pi r^2} dr

If I let x = 2\pi r^2, then I get dx = (4\pi r) dr. The radial dependence bothers me. I think there's a step I'm missing out.

 

[DivGradCurl]

Just realized dx is not an issue. I don't need to substitute it by a dr, so there's no problem. All I need to do is replace 2\pi r^2 by x. I'm done.