Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 072, 34 pages      arXiv:1107.3625      http://dx.doi.org/10.3842/SIGMA.2011.072
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

Appell Transformation and Canonical Transforms

Amalia Torre
ENEA UTAPRAD-MAT Laboratorio di Modellistica Matematica, via E. Fermi 45, 00044 Frascati (Rome), Italy

Received January 31, 2011, in final form July 11, 2011; Published online July 19, 2011

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.

Key words: heat equation; paraxial wave equation; Appell transformation.

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