,, ▁other ▁solutions ▁are ▁prefer able ▁for ▁most ▁applications . ▁Alternatively , ▁Ke pler ' s ▁equation ▁can ▁be ▁solved ▁numer ically . ▁ ▁The ▁solution ▁for ▁e ▁ ≠ ▁ 1 ▁was ▁found ▁by ▁Karl ▁St ump ff ▁in ▁ 1 9 6 8 , ▁but ▁its ▁significance ▁wasn ' t ▁recognized . ▁ ▁One ▁can ▁also ▁write ▁a ▁Mac la ur in ▁series ▁in ▁e . ▁This ▁series ▁does ▁not ▁conver ge ▁when ▁e ▁is ▁larger ▁than ▁the ▁Lap lace ▁limit ▁( about ▁ 0 . 6 6 ), ▁regardless ▁of ▁the ▁value ▁of ▁M ▁( unless ▁M ▁is ▁a ▁multiple ▁of ▁), ▁but ▁it ▁converges ▁for ▁all ▁M ▁if ▁e ▁is ▁less ▁than ▁the ▁Lap lace ▁limit . ▁The ▁coefficients ▁in ▁the ▁series , ▁other ▁than ▁the ▁first ▁( which ▁is ▁simply ▁M ), ▁depend ▁on ▁M ▁in ▁a ▁periodic ▁way ▁with ▁period ▁. ▁ ▁In verse ▁radial ▁Ke pler ▁equation ▁The ▁inverse ▁radial ▁Ke pler ▁equation ▁( e ▁= ▁ 1 ) ▁can ▁also ▁be ▁written ▁as : ▁▁▁ ▁E valu ating ▁this ▁yields : ▁ ▁To ▁obtain ▁this ▁result ▁using ▁Mathemat ica : ▁In verse Series [ Series [ Ar c S in [ S q rt [ t ]] ▁- ▁S q rt [( 1 ▁- ▁t ) ▁t ], ▁{ t , ▁ 0 , ▁ 1 5 } ]] ▁ ▁N umer ical ▁approximation ▁of ▁inverse ▁problem ▁For ▁most ▁applications , ▁the ▁inverse ▁problem ▁can ▁be ▁computed ▁numer ically ▁by ▁finding ▁the ▁root ▁of ▁the