  We generalize the method of our recent paper on the large-spin expansions of
Gubser-Klebanov-Polyakov (GKP) strings to the large-spin and large-winding
expansions of finite-size giant magnons and finite-size single spikes. By
expressing the energies of long open strings in RxS2 in terms of Lambert's
W-function, we compute the leading, subleading and next-to-subleading series of
classical exponential corrections to the dispersion relations of
Hofman-Maldacena giant magnons and infinite-winding single spikes. We also
compute the corresponding expansions in the doubled regions of giant magnons
and single spikes that are respectively obtained when their angular and linear
velocities become smaller or greater than unity.
