12 Parabolic Cylinder Functions Computation 12.19 Tables 12.21 Software

§12.20 Approximations

Keywords:: parabolic cylinder functions
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Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions $\mathop{U\/}\nolimits\!\left(a,b,x\right)$ and $\mathop{M\/}\nolimits\!\left(a,b,x\right)$ (§13.2(i)) whose regions of validity include intervals with endpoints $x=\infty$ and , respectively. As special cases of these results a Chebyshev-series expansion for $\mathop{U\/}\nolimits\!\left(a,x\right)$ valid when $\lambda\leq x<\infty$ follows from (12.7.14), and Chebyshev-series expansions for $\mathop{U\/}\nolimits\!\left(a,x\right)$ and $\mathop{V\/}\nolimits\!\left(a,x\right)$ valid when $0\leq x\leq\lambda$ follow from (12.4.1), (12.4.2), (12.7.12), and (12.7.13). Here $\lambda$ denotes an arbitrary positive constant.