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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.

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

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