Thesis (doctoral)--Universität Bonn, 1988. / Includes bibliographical references.

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. / Science, Faculty of / Mathematics, Department of / Graduate

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Thesis (Ph. D.)--Ohio State University, 2003. / Title from first page of PDF file. Document formatted into pages; contains x, 227 p.; also includes graphics Includes bibliographical references (p. 224-227). Available online via OhioLINK's ETD Center

This thesis consists of two parts. In the first part, we study a class of special functions Aik (y), k = 2, 4, 6, ··· generalising the classical Airy function Ai(y) to higher orders and in the second part, we apply expressions and properties of Ai4(y) to spectral problem of a specific operator. The first part is however motivated by latter part. We establish regularity properties of Aik (y) and particularly show that Aik (y) is smooth, bounded, and extends to the complex plane as an entire function, and obtain pointwise bounds on Aik (y) for all k. Some analytic properties of Aik (y) are also derived allowing one to express Aik (y) as a finite sum of certain generalised hypergeometric functions. We further obtain full asymptotic expansions of Aik (y) and their first derivative Ai'(y) both for y > 0 and for y < 0. Using these expansions, we derive expressions for the negative real zeroes of Aik (y) and Ai'(y). Using expressions and properties of Ai4(y), we extensively study spectral properties of a non-local operator H whose physical interpretation is the massless relativistic quartic anharmonic oscillator in one dimension. Various spectral results for H are derived including estimates of eigenvalues, spectral gaps and trace formula, and a Weyl-type asymptotic relation. We study asymptotic behaviour, analyticity, and uniform boundedness properties of the eigenfunctions Ψn(x) of H. The Fourier transforms of these eigenfunctions are expressed in two terms, one involving Ai4(y) and another term derived from Ai4(y) denoted by Āi4(y). By investigating the small effect generated by Āi4(y) this work shows that eigenvalues λn of H are exponentially close, with increasing n Ε N, to the negative real zeroes of Ai4(y) and those of its first derivative Ai'4(y) arranged in alternating and increasing order of magnitude. The eigenfunctions Ψ(x) are also shown to be exponentially well-approximated by the inverse Fourier transform of Ai4(|y| - λn) in its normalised form.

M.V. Cerrillo, W.H. Kautz. / "November 15, 1951." / Bibliography: p. 11. / Army Signal Corps Contract DA36-039 SC-64637 Project 102B. Dept. of the Army Projects 3-99-10-022 and DA3-99-10-000.