Expansion of the Green’s function in a cylindrical coordinate system

The expansions of the Green’s function ${e^{ik\left | {r - r’} \right |}}/\left | {r - r’} \right |$ in spherical and spheroidal coordinate systems were studied by some researchers; however, because of its complexity this kind of expansion has not been performed in a cylindrical coordinate system. With the aid of the expressions of the Bessel functions ${J_{ - 1/2}}$ and ${J_{1/2}}$ and of the transformation from these functions to the spherical Bessel functions, the author has analytically derived the expansion of the Green’s function in the cylindrical coordinate system.