The Fourier transforms for a function of two variables are
$$
\begin{aligned}
& F(u, v)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} \int f(x, y) e^{i(u x+v y)} d x d y, \\
& f(x, y)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} \int F(u, v) e^{-i(u x+v y)} d u d v .
\end{aligned}
$$
Using $f(x, y)=f\left(\left[x^2+y^2\right]^{1 / 2}\right)$, show that the zero-order Hankel transforms
$$
\begin{aligned}
& F(\rho)=\int_0^{\infty} r f(r) J_0(\rho r) d r, \\
& f(r)=\int_0^{\infty} \rho F(\rho) J_0(\rho r) d \rho,
\end{aligned}
$$
are a special case of the Fourier transforms.
This technique may be generalized to derive the Hankel transforms of order $v=$ $0, \frac{1}{2}, 1, \frac{1}{2}, \ldots$ (compare I. N. Sneddon, Fourier Transforms, New York: McGraw-Hill (1951)). A more general approach, valid for $v>-\frac{1}{2}$, is presented in Sneddon's The Use of Integral Transforms (New York: McGraw-Hill (1972)). It might also be noted that the Hankel transforms of nonintegral order $v= \pm \frac{1}{2}$ reduce to Fourier sine and cosine transforms.