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We provide an introduction to the Fractional Fourier Transform$\mathcal{F}_{\theta}$ and draw a connection between it and the unit complexnumber $e^{i\theta}$. Motivated by this, we define an entirely new objectassociated with any unit quaternion$e^{i\xi_{1}}\cos\eta+e^{i\xi_{2}}j\sin\eta$, which we call the VersorTransform $\mathcal{V}_{(\xi_{1},\eta,\xi_{2})}$. This transform, which hasboth the Fourier and Laplace Transforms as special cases, encourages analternate view of the relationship between them. We also derive severalidentities for both $\mathcal{F}_{\theta}$ and$\mathcal{V}_{(\xi_{1},\eta,\xi_{2})}$.

Identities of the Fractional Fourier Transform and the Versor Transform