▁reprodu ced ▁in ▁Mathemat ica ▁with ▁the ▁In verse Series ▁operation . ▁ ▁In verse Series [ Series [ M ▁- ▁Sin [ M ], ▁{ M , ▁ 0 , ▁ 1 0 } ]] ▁ ▁In verse Series [ Series [ M ▁- ▁e ▁Sin [ M ], ▁{ M , ▁ 0 , ▁ 1 0 } ]] ▁ ▁These ▁functions ▁are ▁simple ▁Mac la ur in ▁series . ▁Such ▁Taylor ▁series ▁representations ▁of ▁trans c endent al ▁functions ▁are ▁considered ▁to ▁be ▁definitions ▁of ▁those ▁functions . ▁Therefore , ▁this ▁solution ▁is ▁a ▁formal ▁definition ▁of ▁the ▁inverse ▁Ke pler ▁equation . ▁However , ▁E ▁is ▁not ▁an ▁entire ▁function ▁of ▁M ▁at ▁a ▁given ▁non - zero ▁e . ▁The ▁derivative ▁ ▁goes ▁to ▁zero ▁at ▁an ▁infinite ▁set ▁of ▁complex ▁numbers ▁when ▁e < 1 . ▁There ▁are ▁solutions ▁at ▁ ▁and ▁at ▁those ▁values ▁ ▁( where ▁inverse ▁cos h ▁is ▁taken ▁to ▁be ▁positive ), ▁and ▁d E / d M ▁goes ▁to ▁infinity ▁at ▁these ▁points . ▁This ▁means ▁that ▁the ▁radius ▁of ▁convergence ▁of ▁the ▁Mac la ur in ▁series ▁is ▁ ▁and ▁the ▁series ▁will ▁not ▁conver ge ▁for ▁values ▁of ▁M ▁larger ▁than ▁this . ▁The ▁series ▁can ▁also ▁be ▁used ▁for ▁the ▁hyper bol ic ▁case , ▁in ▁which ▁case ▁the ▁radius ▁of ▁convergence ▁is ▁ ▁The ▁series ▁for ▁when ▁e   =   1 ▁converges ▁when ▁. ▁ ▁While ▁this ▁solution ▁is ▁the ▁simplest ▁in ▁a ▁certain ▁mathematical ▁sense