  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.
