9,840 research outputs found
Computing hypergeometric functions rigorously
We present an efficient implementation of hypergeometric functions in
arbitrary-precision interval arithmetic. The functions , ,
and (or the Kummer -function) are supported for
unrestricted complex parameters and argument, and by extension, we cover
exponential and trigonometric integrals, error functions, Fresnel integrals,
incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre
functions, Jacobi polynomials, complete elliptic integrals, and other special
functions. The output can be used directly for interval computations or to
generate provably correct floating-point approximations in any format.
Performance is competitive with earlier arbitrary-precision software, and
sometimes orders of magnitude faster. We also partially cover the generalized
hypergeometric function and computation of high-order parameter
derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case
E-G in section 8.5 (table 6, figure 2); adjusted paper siz
Drip Paintings and Fractal Analysis
It has been claimed [1-6] that fractal analysis can be applied to
unambiguously characterize works of art such as the drip paintings of Jackson
Pollock. This academic issue has become of more general interest following the
recent discovery of a cache of disputed Pollock paintings. We definitively
demonstrate here, by analyzing paintings by Pollock and others, that fractal
criteria provide no information about artistic authenticity. This work has also
led to two new results in fractal analysis of more general scientific
significance. First, the composite of two fractals is not generally scale
invariant and exhibits complex multifractal scaling in the small distance
asymptotic limit. Second the statistics of box-counting and related staircases
provide a new way to characterize geometry and distinguish fractals from
Euclidean objects
When is .999... less than 1?
We examine alternative interpretations of the symbol described as nought,
point, nine recurring. Is "an infinite number of 9s" merely a figure of speech?
How are such alternative interpretations related to infinite cardinalities? How
are they expressed in Lightstone's "semicolon" notation? Is it possible to
choose a canonical alternative interpretation? Should unital evaluation of the
symbol .999 . . . be inculcated in a pre-limit teaching environment? The
problem of the unital evaluation is hereby examined from the pre-R, pre-lim
viewpoint of the student.Comment: 28 page
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