Constructive Approximation

, Volume 10, Issue 1, pp 15–30 | Cite as

On the Darling-Mandelbrot probability density and the zeros of some incomplete gamma functions

  • John S. Lew
Article

Abstract

Recently, Mandelbrot has encountered and numerically investigated a probability densitypd(t) on the nonnegative reals, where, 0<D<1. This density has Fourier transform 1/fd(-is), wherefd(z)=−Dzdγ(−D, z) and γ(·.·) is an incomplete gamma function. Previously, Darling had met this density, but had not studied its form. We expressfd(z) as a confluent hypergeometric function, then locate and approximate its zeros, thereby improving some results of Buchholz. Via properties of Laplace transforms, we approximatepd(t) asymptotically ast→0+ and +∞, then note some implications asD→0+ and 1−.

AMS classification

33E70 41A60 60E99 

Key words and phrases

Incomplete gamma function Confluent hypergeometric function Darling-Mandelbrot probability density Asymptotic zeros Zeros 

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Copyright information

© Springer-Verlag New York Inc. 1994

Authors and Affiliations

  • John S. Lew
    • 1
  1. 1.IBM Research Division Mathematical Sciences DepartmentT. J. Watson Research CenterYorktown HeightsUSA

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