118 research outputs found
Uniform asymptotic expansions for the zeros of Bessel functions
Reformulated uniform asymptotic expansions are derived for ordinary
differential equations having a large parameter and a simple turning point.
These involve Airy functions, but not their derivatives, unlike traditional
asymptotic expansions. From these, asymptotic expansions are derived for the
zeros of Bessel functions that are valid for large positive values of the
order, uniformly valid for all the zeros. The coefficients in the expansions
are explicitly given elementary functions, and similar expansions are derived
for the zeros of the derivatives of Bessel functions
Conical: an extended module for computing a numerically satisfactory pair of solutions of the differential equation for conical functions
Conical functions appear in a large number of applications in physics and
engineering. In this paper we describe an extension of our module CONICAL for
the computation of conical functions. Specifically, the module includes now a
routine for computing the function , a
real-valued numerically satisfactory companion of the function for . In this way, a natural basis for solving
Dirichlet problems bounded by conical domains is provided.Comment: To appear in Computer Physics Communication
Computation of parabolic cylinder functions having complex argument
Numerical methods for the computation of the parabolic cylinder for
real and complex are presented. The main tools are recent asymptotic
expansions involving exponential and Airy functions, with slowly varying
analytic coefficient functions involving simple coefficients, and stable
integral representations; these two main main methods can be complemented with
Maclaurin series and a Poincar\'e asymptotic expansion. We provide numerical
evidence showing that the combination of these methods is enough for computing
the function with relative accuracy in double precision
floating point arithmetic
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