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Shearing transformation of a linear system at an irregular singular point

Published online by Cambridge University Press:  24 October 2008

Richard C. Gilbert
Affiliation:
California State University, Fullerton

Extract

Consider an nth-order linear ordinary differential equation

Suppose the αj(x) are holomorphic for xS, 0 < x0 ≤ |x| < ∞, where S is an open sector of the complex plane with vertex at the origin and positive central angle not exceeding π. Suppose as x → ∞ in each closed subsector of S. The problem of finding a basis for the solutions of (1) can be reduced by a sequence of transformations (see, for example, Gilbert (1)) to the problem of finding a fundamental matrix for a system of the form

where q is a non-negative integer, A(x) is an m by m matrix which is holomorphic for xS, x0 < ≤ |x| < ∞, and as x → ∞ in each closed subsector of S. If m ≥ 2, A0 is an m by m matrix with elements all zero except for l's on the upper off-diagonal, and the elements of Ar for r ≥ 1 are all zero except possibly in the last row. (There is one such problem (2) for each root, μ of the equation with multiplicity m ≥ 2. Recall that the coefficient αnr, 0 the first term in the asymptotic expansion of the coefficint αnr(x) of equation (1).) We denote the elements of the last row of Ar by 1 ≤ km. The system (2) has an irregular singular point at infinity.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

(1)Gilbert, R. C.Asymptotic formulas for solutions of a singular linear ordinary differential equation. Proc. Roy. Soc. Edinburgh Sect. A 81 (1978), 5770.CrossRefGoogle Scholar
(2)Wasow, W.Asymptotic expansions for ordinary differential equations. Interscience Publishers, New York, 1965.Google Scholar