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Periodic magnetic or electrostatic fields are capable of focusing a long electron beam so that the performance normally yielded by a uniform magnetic field is attained. This is the case if a proper choice of the value of the period with respect to the rms value of the field is employed. For high-power traveling-wave tubes, however, the value of the period demanded by either periodic magnetic or electrostatic focusing field is so small that it is almost impossible to realize these fields. By the use of a proper combination of a varying magnetic and electric field, optimum performance is physically attainable for electron beams of high perveance. The general theory of focusing action in such combined or complementary fields is given and is applied to problems which arise in systems involving parallel electron beam flow. It is also shown that in the case of combined fields, the fields which may be employed are much less restricted and critical than is the case when they are used singly.