I am trying to understand the connection between the 2D Fourier and Hankel transform in a book. In the meantime, I came across the following derivations:
In the polar coordinate, the 2D Fourier transform can be written as $$F(\rho, \psi)=\int_{0}^{\infty}\int_{-\pi}^{\pi}f(r,\theta)e^{-ir\rho\cos(\psi-\theta)}r drd\theta $$
which can be easily shown by the change of variables. However, the question comes up in the following. The book says, if we assume that the function is radially symmetric ($f(r,\theta)=f(r)$), then the Fourier transform can be written as:
$$F(\rho, \psi) = \int_{0}^{\infty} rf(r) dr \int_{-\pi}^{\pi}e^{-ir\rho\cos(\psi-\theta)} d\theta$$
Why can we separate the two integrals even though the second integrand is also a function of $r$?
This question has been answered by @march and @md2perpe.
The notation actually represents a nested integral, which is also described by the following notation: $$F(\rho,\psi)=\int_0^\infty dr \ r f(r) \int_{-\pi}^{\pi} d\theta \ e^{-ir\rho\cos(\psi-\theta)} $$