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An infinite Legendre integral transform and its inverse

Published online by Cambridge University Press:  24 October 2008

P. C. Clemmow
P. C. Clemmow
Affiliation:
Cavendish LaboratoryCambridge
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Abstract

A consideration of Legendre's equation, of order zero, expressed in terms of the angular variable θ, leads heuristically, but methodically, to an infinite integral representation of the delta function in terms of two independent non-periodic solutions Ev−½(θ), E−v−½(θ). This result yields an integral transform and inverse formula, where in the former the kernel is E−v−½(θ) sin θ and the integration over θ runs from − ∞ to ∞. By rigorous contour integral methods, the inverse is verified when the original function is a rectangular pulse, and the standard series expansion in terms of Legendre polynomials is recovered when the original function is even and of period 2π in θ. Analogies with the theory of Fourier transforms are emphasized.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 3 , July 1961 , pp. 547 - 560
Copyright
Copyright © Cambridge Philosophical Society 1961

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References

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