8.10 Inequalities8.12 Uniform Asymptotic Expansions for Large Parameter

§8.11 Asymptotic Approximations and Expansions

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§8.11(i) Large z, Fixed a

If a is real and z (=x) is positive, then R_{n}(a,x) is bounded in absolute value by the first neglected term u_{n}/x^{n} and has the same sign provided that n\geq a-1. For bounds on R_{n}(a,z) when a is real and z is complex see Olver (1997b, pp. 109–112). For an exponentially-improved asymptotic expansion (§2.11(iii)) see Olver (1991a).

§8.11(ii) Large a, Fixed z

8.11.4 \mathop{\gamma\/}\nolimits\!\left(a,z\right)=z^{a}e^{{-z}}\sum _{{k=0}}^{\infty}\frac{z^{k}}{\left(a\right)_{{k+1}}}, a\neq 0,-1,-2,\dots.

This expansion is absolutely convergent for all finite z, and it can also be regarded as a generalized asymptotic expansion (§2.1(v)) of \mathop{\gamma\/}\nolimits\!\left(a,z\right) as a\to\infty in |\mathop{\mathrm{ph}\/}\nolimits a|\leq\pi-\delta.

§8.11(iii) Large a, Fixed x/a

The expansion (8.11.7) also applies when a\to-\infty with \lambda<0, and in this case Gautschi (1959a) supplies numerical bounds for the remainders in the truncated expansion (8.11.7). For extensions to complex variables see Temme (1994b, §4), and also Mahler (1930), Tricomi (1950b), and Paris (2002b).

§8.11(iv) Large a, Bounded (x-a)/(2a)^{{\frac{1}{2}}}

in both cases uniformly with respect to bounded real values of y. For Dawson’s integral \mathop{F\/}\nolimits\!\left(y\right) see §7.2(ii). See Tricomi (1950b) for these approximations, together with higher terms and extensions to complex variables. For related expansions involving Hermite polynomials see Pagurova (1965).

§8.11(v) Other Approximations

For the function e_{n}(z) defined by (8.4.11),

8.11.13 \lim _{{n\to\infty}}\frac{e_{n}(nx)}{e^{{nx}}}=\begin{cases}0,&x>1,\\
\tfrac{1}{2},&x=1,\\
1,&0\leq x<1.\end{cases}

With x=1, an asymptotic expansion of e_{n}(nx)/e^{{nx}} follows from (8.11.14) and (8.11.16).

For (8.11.18) and extensions to complex values of x see Buckholtz (1963). For a uniformly valid expansion for n\to\infty and x\in[\delta,1], see Wong (1973b).