I stuck with a rather simple question, but i can't fix it. Hope you can help me.
Hankel transform for a cylindrically symmetric problem is given by the following formula $g(\rho) = 2\pi \int \limits_{0}^{\infty} r f(r) J_l(2 \pi r \rho) dr$
Inverse transform looks the same. It is obvious that in a real system we deal with a bounded region $[0,r_{max}]$. Reading an article concerning numerical implementation of HT I met the change of variables: $x=r/b$ and $y=\rho/\beta$
where $b$ and $\beta$ are maximum values of spatial ($r$) and frequency ($\rho$) domain, respectively.
So, authors obtained the following expression
$g(y) = 2\pi \gamma \frac{b}{\beta} \int \limits_0^1 x f(x) J_l (2 \pi \gamma x y)dx$
where $\gamma = b \beta$
It is not obvious for me how to move from $g(\rho) = g(\beta y)$ to $g(y)$. And the same with $f(bx) \to f(x)$ under the integal.