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Quaternion-valued exponential are derived as follows:

${\displaystyle e^q=\sum _{n=0}^{\infty }\frac{1}{n!}{q}^n{=e}^{\mathrm{Sq}}\left(\cos \left(\mathrm{AVq}\right)+\mathrm{UVq}\sin \left(\mathrm{AVq}\right)\right)}$

where

${\displaystyle q=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k},\mathrm{Sq}=a,\mathrm{Vq}=b\mathbf{i}+c\mathbf{j}+d\mathbf{k},\mathrm{Uq}=\frac{q}{\mathrm{Aq}},\mathrm{Aq}=\left|q\right|=\sqrt{a^2+b^2+c^2+d^2}=\sqrt{q\overline{q}},\overline{q}=\mathrm{Sq}-\mathrm{Vq}=a-\left(b\mathbf{i}+c\mathbf{j}+d\mathbf{k}\right)}$

and

${\displaystyle \mathrm{UVq}=\frac{b\mathbf{i}+c\mathbf{j}+d\mathbf{k}}{\mathrm{AVq}},\mathrm{AVq}=\sqrt{b^2+c^2+d^2}}$

because

${\displaystyle {\mathrm{UVq}}^2=-1,{\mathbf{i}}^2={\mathbf{j}}^2={\mathbf{k}}^2=-1,\mathbf{i}\mathbf{j}=-\mathbf{j}\mathbf{i}=\mathbf{k},\mathbf{j}\mathbf{k}=-\mathbf{k}\mathbf{j}=\mathbf{i},\mathbf{k}\mathbf{i}=-\mathbf{i}\mathbf{k}=\mathbf{j}}$