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Uniform asymptotic expansions for exponential integrals $$E_ n(x)$$ and Bickley functions $$Ki_ n(x)$$. (English) Zbl 0545.33002
The author discusses theoretical and computational aspects of the so- called Bickley functions $Ki_ n(x)=\int^{\infty}_{x}Ki_{n- 1}(t)dt\quad(x\geq 0,n=1,2,...)$ where $$Ki_ 0(x)=K_ 0(x)$$ 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 $$E_ n(x)$$. The author thus investigates both these functions simultaneously. In particular he presents sharp bounds on $$Ki_ n(x)$$ for $$n\geq -1,x\quad x\geq 0,$$ derives new uniform asymptotic expansions for $$E_ n(x)$$ and $$Ki_ n(x)$$ for $$x\geq 0,\quad n\to \infty,$$ and shows how the uniform expansion of $$Ki_ n(x)$$ can be used to start stable recurrence for sequences $$Ki_{n+k-1}(x)$$, $$k=1,...,N$$, $$n\geq - 1$$.
Reviewer: K.S.Kölbig

##### MSC:
 33B15 Gamma, beta and polygamma functions 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 41A30 Approximation by other special function classes 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$ 33E99 Other special functions
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