3,611 research outputs found
Limit curves for zeros of sections of exponential integrals
We are interested in studying the asymptotic behavior of the zeros of partial
sums of power series for a family of entire functions defined by exponential
integrals. The zeros grow on the order of O(n), and after rescaling we
explicitly calculate their limit curve. We find that the rate that the zeros
approach the curve depends on the order of the singularities/zeros of the
integrand in the exponential integrals. As an application of our findings we
derive results concerning the zeros of partial sums of power series for Bessel
functions of the first kind.Comment: 19 pages, 5 figures. arXiv admin note: text overlap with
arXiv:1208.518
Uniform Asymptotic Methods for Integrals
We give an overview of basic methods that can be used for obtaining
asymptotic expansions of integrals: Watson's lemma, Laplace's method, the
saddle point method, and the method of stationary phase. Certain developments
in the field of asymptotic analysis will be compared with De Bruijn's book {\em
Asymptotic Methods in Analysis}. The classical methods can be modified for
obtaining expansions that hold uniformly with respect to additional parameters.
We give an overview of examples in which special functions, such as the
complementary error function, Airy functions, and Bessel functions, are used as
approximations in uniform asymptotic expansions.Comment: 31 pages, 3 figure
Biorthogonal Quantum Systems
Models of PT symmetric quantum mechanics provide examples of biorthogonal
quantum systems. The latter incorporporate all the structure of PT symmetric
models, and allow for generalizations, especially in situations where the PT
construction of the dual space fails. The formalism is illustrated by a few
exact results for models of the form H=(p+\nu)^2+\sum_{k>0}\mu_{k}exp(ikx). In
some non-trivial cases, equivalent hermitian theories are obtained and shown to
be very simple: They are just free (chiral) particles. Field theory extensions
are briefly considered.Comment: 34 pages, 5 eps figures; references added and other changes made to
conform to journal versio
Fredholm Determinants, Differential Equations and Matrix Models
Orthogonal polynomial random matrix models of NxN hermitian matrices lead to
Fredholm determinants of integral operators with kernel of the form (phi(x)
psi(y) - psi(x) phi(y))/x-y. This paper is concerned with the Fredholm
determinants of integral operators having kernel of this form and where the
underlying set is a union of open intervals. The emphasis is on the
determinants thought of as functions of the end-points of these intervals. We
show that these Fredholm determinants with kernels of the general form
described above are expressible in terms of solutions of systems of PDE's as
long as phi and psi satisfy a certain type of differentiation formula. There is
also an exponential variant of this analysis which includes the circular
ensembles of NxN unitary matrices.Comment: 34 pages, LaTeX using RevTeX 3.0 macros; last version changes only
the abstract and decreases length of typeset versio
The reproducing kernel structure arising from a combination of continuous and discrete orthogonal polynomials into Fourier systems
We study mapping properties of operators with kernels defined via a
combination of continuous and discrete orthogonal polynomials, which provide an
abstract formulation of quantum (q-) Fourier type systems. We prove Ismail
conjecture regarding the existence of a reproducing kernel structure behind
these kernels, by establishing a link with Saitoh theory of linear
transformations in Hilbert space. The results are illustrated with Fourier
kernels with ultraspherical weights, their continuous q-extensions and
generalizations. As a byproduct of this approach, a new class of sampling
theorems is obtained, as well as Neumann type expansions in Bessel and q-Bessel
functions.Comment: 16 pages; Title changed, major reformulations. To appear in Constr.
Appro
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