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Note on the generalized Mehler transform

Published online by Cambridge University Press:  24 October 2008

J. S. Lowndes
J. S. Lowndes
Affiliation:
Royal College of Science and Technology, Glasgow
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Extract

1. The integral transform. One result of recent studies of boundary-value problems of the wave and diffusion equations involving wedge- or conically-shaped boundaries has been the interest shown in integrals in which the variable of integration appears as the order of Bessel or Legendre functions. An integral of this type occurs as the inversion formula for the generalized Mehler transform which is defined by

where ψ(μ, k) = Γ(½ − k + iμ)Γ(½ − k − iμ) and is the associated Legendre function of the first kind. Oberhettinger and Higgins (4) have given a table of transform pairs corresponding to the above transform.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 60 , Issue 1 , January 1964 , pp. 57 - 59
Copyright
Copyright © Cambridge Philosophical Society 1964

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References

REFERENCES

(1)Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.Higher transcendental functions, Vol. 2 (McGraw-Hill, 1953).Google Scholar
(2)Lebedev, N. N.Dokl. Akad. Nauk, SSSR, 68 (1949), 653–656Google Scholar
(3)Lowndes, J. S.SIAM Rev. 3 (1961), 162–164CrossRefGoogle Scholar
(4)Oberhettinger, F. and Higgins, T. P.Tables of Lebedev, Mehler and generalized Mehler Transforms (Boeing Scientific Research Laboratories Mathematical Note 246; Washington, 1961).CrossRefGoogle Scholar