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The minimum bandwidth pulse for a given rms value and peak amplitude is found using the Pontryagin Maximum Principle for bounded phase space. The solution is obtained by actually solving the equations of the maximum principle rather than by a verification of a solution arrived at by other means. The resulting optimal pulse shape can be used for comparisons with more easily generated pulses, and may be considered to be the optimal modification of a rectangular pulse from the point of view of minimum bandwidth. Pulses of this sort are useful in pulse communication systems.