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We derive an analytic series solution of the elliptic equations providing the4-tachyon off-shell amplitude in cubic string field theory (CSFT). From such asolution we compute the exact coefficient of the quartic effective actionrelevant for time dependent solutions and we derive the exact coefficient ofthe quartic tachyon coupling. The rolling tachyon solution expressed as aseries of exponentials $e^t$ is studied both using level-truncationcomputations and the exact 4-tachyon amplitude. The results for the leveltruncated coefficients are shown to converge to those derived using the exactstring amplitude. The agreement with previous work on the subject, both on thequartic tachyon coupling and on the CSFT rolling tachyon, is an excellent testfor the accuracy of our off-shell solution.Comment: 26 pages, 5 figure

A solution to the 4-tachyon off-shell amplitude in cubic string field theory