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The Synchrotron Integrals with Coupling

The synchrotron integrals used to compute emittances, the energy spread, etc., have been analyzed assuming no coupling between the horizontal and vertical planes[1, 2]. With Mobius, these assumptions are not valid and so this paper presents the appropriate generalizations. To simplify matters it will be still be assumed that the bends are in the horizonal plane. In this case I1, I2, I3, and I5 are unchanged (but are given below for completeness). Without proof, the generalized synchrotron integrals are: