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We derive a power series representation of an arbitrary electromagnetic fieldnear some axis through the coaxial field components on the axis. The obtainedequations are compared with Fourier-Bessel series approach and verified byseveral examples. It is shown that for each azimuthal mode we need only tworeal functions on the axis in order to describe the field in a source freeregion near to it. The representation of dipole mode in a superconductingradio-frequency gun is analyzed.

Axial representation of external electromagnetic fields for particle tracking in accelerators