# Zero-order Bessel function as eigenfunction of invariant linear system with circularly symmetric response

This is a problem in Goldman's 'Introduction to Fourier Optics', Chapter 2.

Please show that zero-order Bessel function $$J_0(2\pi\rho_0r$$) is eigenfunction of an invariant linear system with circularly symmetric response.

I have tried two methods. One is introduced on linear time-invariant system by Wikipedia. It proves that complex exponential function is eigenfunction of any invariant linear system. The methods depends on the communitive property of convolution. It works in a pretty clear way in $$\exp(ikx)$$ but not in $$J_0(kx)$$.

Another one is to transform the convolution in spatial domain into multiplication in frequency domain. The transformation of $$J_0$$ is shown here: Fourier transform of Bessel functions. Not very simple though. The fourier transformation of circularly symmetric impulse response involves $$J_0$$ in an integral again. I cannot show that the latter is scalar.

Already tried for one week on this problem. Any feedback is strongly appreciated!