Uniform asymptotic expansions for exponential integrals \(E_ n(x)\) and Bickley functions \(Ki_ n(x)\). (English) Zbl 0545.33002

Amos, D. E.

ACM Trans. Math. Softw. 9, 467-479 (1983).

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

Keywords:

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