TL;DR
This paper reviews the optical Appell transformation's interpretation in paraxial optics and explores its connection to the caloric Appell transformation in heat theory, highlighting the role of Fourier, Laplace, and Hankel transforms.
Contribution
It establishes a unified framework linking the optical and caloric Appell transformations through canonical transforms and integral transforms.
Findings
Optical Appell transformation relates to Fourier and cylindrical symmetry.
Caloric Appell transformation involves Laplace and Hankel-type transforms.
Links between optical and heat equation transformations are clarified.
Abstract
The interpretation of the optical Appell transformation, as previously elaborated in relation to the free-space paraxial propagation under both a rectangular and a circular cylindrical symmetry, is reviewed. Then, the caloric Appell transformation, well known in the theory of heat equation, is shown to be amenable for a similar interpretation involving the Laplace transform rather than the Fourier transform, when dealing with the 1D heat equation. Accordingly, when considering the radial heat equation, suitably defined Hankel-type transforms come to be involved in the inherent Appell transformation. The analysis is aimed at outlining the link between the Appell transformation and the canonical transforms.
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