The mathematical approach for the calculation of the membrane functions of a coaxial gyrotron cavity with an arbitrary corrugated inner rod is proposed. It is utilized mainly for two aims. First, it is shown that for typical parameters of the coaxial gyrotron cavity with the corrugated inner conductor the shape of corrugations only slightly influences the eigenvalues of competing eigen-modes. However, it can significantly influence the density of ohmic losses in the inner conductor. In particular, it is shown that the density of ohmic losses can be reduced almost twice by the proper choice of the corrugation shape. Second, it is shown that the usual idealizations of the corrugated surface of the inner conductor (the surface with rectangular grooves, having rounded edges, is approximated by a surface with wedged groves that have sharp edges) are correct. The physical interpretation of the obtained results and their practical meaning are discussed.
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