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Zbl 0545.33002
Amos, D.E.
Uniform asymptotic expansions for exponential integrals $E\sb n(x)$ and Bickley functions $Ki\sb n(x)$. (English)
[J] ACM Trans. Math. Softw. 9, 467-479 (1983). ISSN 0098-3500

The author discusses theoretical and computational aspects of the so- called Bickley functions $$ Ki\sb n(x)=\int\sp{\infty}\sb{x}Ki\sb{n- 1}(t)dt\quad(x\ge 0,n=1,2,...) $$ where $Ki\sb 0(x)=K\sb 0(x)$ is the modified Bessel function of the second kind. These functions arise in heat convection problems, neutron transport calculations, and in other fields. They can be represented in terms of a series of exponential integrals $E\sb n(x)$. The author thus investigates both these functions simultaneously. In particular he presents sharp bounds on $Ki\sb n(x)$ for $n\ge -1,x\quad x\ge 0,$ derives new uniform asymptotic expansions for $E\sb n(x)$ and $Ki\sb n(x)$ for $x\ge 0,\quad n\to \infty,$ and shows how the uniform expansion of $Ki\sb n(x)$ can be used to start stable recurrence for sequences $Ki\sb{n+k-1}(x)$, $k=1,...,N$, $n\ge - 1$.
[K.S.Kölbig]

MSC 2000:: *33B15 Gamma-functions, etc.
41A60 Asymptotic problems in approximation
41A30 Approximation by other special function classes
33C10 Cylinder functions, etc.
33E99 Special functions

Keywords: psi function; gamma function; Bickley functions; modified Bessel function; exponential integrals

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