We derive a power series representation of an arbitrary electromagnetic field near some axis through the coaxial field components on the axis. The obtained equations are compared with Fourier–Bessel series approach and verified by several examples. It is shown that for each azimuthal mode we need only two real functions on the axis in order to describe the field in a source free region near to it. The representation of dipole mode in a superconducting radio-frequency gun is analyzed.