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THE CONSTRUCTION OF RELATED EQUATIONS FOR THE ASYMPTOTIC THEORY OF LINEAR ORDINARY DIFFERENTIAL EQUATIONS ABOUT A TURNING POINT
WISCONSIN UNIV MADISON MATHEMATICS RESEARCH CENTER
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Fully developed theory is extant, and can justifiably be referred to as classical, for the determination of the asymptotic forms of the solutions of a differential equation over any closed zregion which completely excludes turning-points. This theory applies, of course, irrespective of the region, to all equations with constant coefficients. The state of the theory is very different, namely quite fragmentary, when a turningpoint is lodged within the region. For this reason, and also because modern physical theories require it, the stdy of the solution forms of an equaion in a region about a turning-point is of emminent contemporary interest. The classical algorithms fail irretrievably in such a region, a fact which has been shown to be inevitable by results otherwise obtained, because the forms yielded by those algorithms lak adequacy to reflect the intricate functional metamorphoses which characterize the solutions of the differential equation in a turning-point neighborhood. The origin is in this case a turning point, and bout this point the solutions undergo transitions between oscillatory and exponential function types. Author
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