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Algorithm 609 a portable FORTRAN subroutine for the Bickley functions $$Ki_ n(x)$$. (English) Zbl 0545.33001
The author discribes a FORTRAN subroutine for the computation 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. The algorithm consists of two main parts: For 0$$\leq x\leq 2$$ it uses a four-term recurrence relation with starting values for $$n=0,1,2$$ obtained from a power series. For $$2<x<\infty$$ a uniform asymptotic expansion is used for $$n\geq 0$$. The author also discusses the computation of the exponential integral $$E_ n(x)$$, the gamma ratio $$R(x)=\Gamma(x)/\Gamma(x+{1\over2}),$$ and the digamma function $$\psi$$ (x), all used as auxiliary functions. The description of the procedures is fundamentally mathematical and a careful error analysis as well as extensive tests are presented. The author emphazises that the main feature of these subroutines is that they are portable for an accuracy up to 18 digits. He states that, by using the special initialization routines provided, the user has a choice of 15 machine environments representing more than 19 machines and operating systems.
Reviewer: K.S.Kölbig
##### MSC:
 33-04 Software, source code, etc. for problems pertaining to special functions 33B15 Gamma, beta and polygamma functions 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$ 33E99 Other special functions 41A30 Approximation by other special function classes
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