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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 + b⁢i + c⁢j + d⁢k , Sq = a , Vq = b⁢i + c⁢j + d⁢k , Uq = q Aq , Aq = q = a 2 + b 2 + c 2 + d 2 = q q ¯ , q ¯ = Sq - Vq = a - ( b⁢i + c⁢j + d⁢k ) and UVq = b⁢i + c⁢j + d⁢k 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
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 + b⁢i + c⁢j + d⁢k , Sq = a , Vq = b⁢i + c⁢j + d⁢k , Uq = q Aq , Aq = q = a 2 + b 2 + c 2 + d 2 = q q ¯ , q ¯ = Sq - Vq = a - ( b⁢i + c⁢j + d⁢k )
and
UVq = b⁢i + c⁢j + d⁢k 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