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Quaternion-valued exponential are derived as follows:

e q = n = 0 1 n ! q n = e Sq cos AVq + UVq sin AVq

where

q = a + bi + cj + dk , Sq = a , Vq = bi + cj + dk , Uq = q Aq , Aq = q = a 2 + b 2 + c 2 + d 2 = q q ¯ , q ¯ = Sq - Vq = a - ( bi + cj + dk )

and

UVq = bi + cj + dk AVq , AVq = b 2 + c 2 + d 2

because

UVq 2 = - 1 , i 2 = j 2 = k 2 = - 1 , i j = - j i = k , j k = - k j = i , k i = - i k = j