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Extension of Airy's equation Headley, Velmer Bentley

Abstract

We consider the differential equation d²u/dz² - zⁿu = 0 (z, u complex variables; n a positive integer), which is the simplest second order ordinary differential equation with a turning point of order n. The solutions which we study, herein called Aռ functions, are generalizations of Airy functions. Most of their properties are then deduced from those of related Bessel functions of order [formula omitted], but in the discussion of the zeros in section 3, results are deduced directly from the differential equation. It is easy to see that the Aռ functions are special cases of functions studied by Turrittin [9]. The relation of the former to Bessel functions, however,, enables us to use methods not available in [9] to obtain uniform asymptotic representations for large z. We obtain new results on the distribution of the zeros which extend a property [6] of Airy functions, that is, of A₁functions,, to all positive integers n. A similar remark applies to bounds [8] for Airy functions and their reciprocals.

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