Why for cylindrical problems the order of the Bessel function is an integer value?

Question

I read somewhere that for cylindrical problems the order of the Bessel function is an integer value while for spherical problems the order is of half integer value. I know that the Bessel functions are defined for orders of all real values and I just can't understand why for cylindrical problems the order is an integer value.

Answer 1

Because you need to require that the function does not change when the azimuthal angle $\varphi$ increases by $2\pi$.