Matrix exponential for a scalar square ($M^2=\kappa I⟹e^M=C(\kappa )I+S(\kappa )M$)

When a square matrix satisfies $M^2=\kappa I$, its powers reduce to $M^{2n}={\kappa }^{n}I$ and $M^{2n+1}={\kappa }^{n}M$. Absolute convergence of the matrix exponential allows even and odd subseries to be separated. Thus $C(\kappa )={\sum }_{n\ge 0}{\kappa }^n/(2n)!$ and $S(\kappa )={\sum }_{n\ge 0}{\kappa }^n/(2n+1)!$. For real $\kappa >0$, these are $\cosh \sqrt{\kappa }$ and $\sinh \sqrt{\kappa }/\sqrt{\kappa }$. For $\kappa <0$, they are $\cos \sqrt{-\kappa }$ and $\sin \sqrt{-\kappa }/\sqrt{-\kappa }$. At $\kappa =0$, $e^M=I+M$, including nonzero nilpotent matrices of square zero. This also explains the limiting sine-over-argument coefficients without dividing by zero.