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Differential equations for the cuspoid canonical integrals

Connor, J. N. L. and Curtis, P. R. (1984) Differential equations for the cuspoid canonical integrals. Journal of Mathematical Physics, 25 (10). pp. 2895-2902. ISSN 0022-2488. doi:10.1063/1.526035.

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Differential equations satisfied by the cuspoid canonical integrals I_n(a) are obtained for arbitrary values of n≥2, where n−1 is the codimension of the singularity and a=(ɑ_1,ɑ_2,...,ɑ_(n−1)). A set of linear coupled ordinary differential equations is derived for each step in the sequence I_n(0,0,...,0,0) →I_n(0,0,...,0,ɑ_(n−1)) →I_n(0,0,...,ɑ_(n−2),ɑ_(n−1)) →...→I_n(0,ɑ_2,...,ɑ_(n−2),ɑ_(n−1)) →I_n(ɑ_1,ɑ_2,...,ɑ_n−2,ɑ_(n−1)). The initial conditions for a given step are obtained from the solutions of the previous step. As examples of the formalism, the differential equations for n=2 (fold), n=3 (cusp), n=4 (swallowtail), and n=5 (butterfly) are given explicitly. In addition, iterative and algebraic methods are described for determining the parameters a that are required in the uniform asymptotic cuspoid approximation for oscillating integrals with many coalescing saddle points. The results in this paper unify and generalize previous researches on the properties of the cuspoid canonical integrals and their partial derivatives.

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Additional Information:© 1984 American Institute of Physics. Received 23 November 1983; accepted 2 March 1984. J.N.L.C. thanks Professor R. A. Marcus for his hospitality at the A.A. Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California and NATO for a Senior Scientist award. P.R.C. thanks the States of Jersey Education Committee for a research studentship. The algebraic calculations were carried out on the CDC 7600 computer of the University of Manchester Regional Computer Centre.
Subject Keywords:differential equations, integrals, singularity, scattering theory, atoms, molecules, heavy ions, wave propagation, asymptotic solutions
Issue or Number:10
Classification Code:PACS: 02.30.Hq, 02.30.-f, 11.80.-m
Record Number:CaltechAUTHORS:20120629-145921487
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Official Citation: Differential equations for the cuspoid canonical integrals J. N. L. Connor and P. R. Curtis J. Math. Phys. 25, 2895 (1984);
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:32217
Deposited By: Ruth Sustaita
Deposited On:02 Jul 2012 14:48
Last Modified:09 Nov 2021 21:25

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