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Asymptotics of solutions of a differential equation of second order with two turning points and a complex parameter. II

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Abstract

Asymptotic formulas are constructed and rigorously justified for linearly independent solutions of a second-order differential equation with a coefficient possessing the property of finite smoothness and containing a complex parameter ζ (forTmζ=0 the equation has two real turning points). A perturbation method is applied which consists in extending the coefficient of the equation to the complex Z plane and approximating it in an ε-neighborhood of the real axis of this plane by a quadratic polynomial. It is proved that the leading terms of the constructed formulas expressed in terms of parabolic cylinder functions are uniform with respect to arg ζ and that the error admitted under the approximation indicated above can be estimated by the quantityO(K−1/2, (K→∞ is the second parameter, in addition to S, on which the coefficient of the differential equation depends).

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Literature cited

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Authors
  1. Z. A. Yanson
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 78, pp. 220–245, 1978.

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Yanson, Z.A. Asymptotics of solutions of a differential equation of second order with two turning points and a complex parameter. II. J Math Sci 22, 1150–1170 (1983). https://doi.org/10.1007/BF01305298

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