§9.7 Asymptotic Expansions

Keywords:: Airy functions
Referenced by:: §9.11(i), ¶ ‣ §9.12(viii), §9.17(i)
Permalink:: http://dlmf.nist.gov/9.7

§9.7(i) Notation
§9.7(ii) Poincaré-Type Expansions
§9.7(iii) Error Bounds for Real Variables
§9.7(iv) Error Bounds for Complex Variables
§9.7(v) Exponentially-Improved Expansions

§9.7(i) Notation

Notes:: See Olver (1997b, p. 225).
Referenced by:: §10.17(iv), §10.20(i), §10.40(iii), §6.12(i), ¶ ‣ §9.12(viii)
Permalink:: http://dlmf.nist.gov/9.7.i

Here $\delta$ denotes an arbitrary small positive constant and

9.7.1 $\zeta=\tfrac{2}{3}z^{{\ifrac{3}{2}}}.$

Symbols:: $z$ : complex variable and $\zeta(z)$ : change of variable
Permalink:: http://dlmf.nist.gov/9.7.E1
Encodings:: TeX, pMML, png

Also $u_{0}=v_{0}=1$ and for $k=1,2,\ldots,$

9.7.2

$u_{k}=\frac{(2k+1)(2k+3)(2k+5)\cdots(6k-1)}{(216)^{k}(k)!},$

$v_{k}=\frac{6k+1}{1-6k}u_{k}.$

Symbols:: $!$ : $n!$ : factorial, $k$ : nonnegative integer, $u_{s}$ : expansion coefficient and $v_{s}$ : expansion coefficient
Permalink:: http://dlmf.nist.gov/9.7.E2
Encodings:: TeX, TeX, pMML, pMML, png, png

Lastly,

9.7.3 $\chi(n)=\pi^{{\ifrac{1}{2}}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}n+1\right)/\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}n+\tfrac{1}{2}\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function and $\chi(n)$ : function
Permalink:: http://dlmf.nist.gov/9.7.E3
Encodings:: TeX, pMML, png

Numerical values of this function are given in Table 9.7.1 for $n=1(1)20$ to 2D. For large $n$ ,

9.7.4 $\chi(n)\sim(\tfrac{1}{2}\pi n)^{{\ifrac{1}{2}}}.$

Symbols:: $\sim$ : asymptotic equality and $\chi(n)$ : function
Permalink:: http://dlmf.nist.gov/9.7.E4
Encodings:: TeX, pMML, png

Table 9.7.1: $\chi(n)$ .

$n$	$\chi(n)$	$n$	$\chi(n)$	$n$	$\chi(n)$	$n$	$\chi(n)$
1	1.57	6	3.20	11	4.25	16	5.09
2	2.00	7	3.44	12	4.43	17	5.24
3	2.36	8	3.66	13	4.61	18	5.39
4	2.67	9	3.87	14	4.77	19	5.54
5	2.95	10	4.06	15	4.94	20	5.68

Symbols:: $\chi(n)$ : function
Referenced by:: §13.7(ii), §9.7(i)
Permalink:: http://dlmf.nist.gov/9.7.T1

§9.7(ii) Poincaré-Type Expansions

Notes:: See Olver (1997b, pp. 392–393 and 413–414).
Referenced by:: §9.10(ii), §9.11(v), §9.12(iv)
Permalink:: http://dlmf.nist.gov/9.7.ii

As $z\to\infty$ the following asymptotic expansions are valid uniformly in the stated sectors.

9.7.5 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)\sim\frac{e^{{-\zeta}}}{2\sqrt{\pi}z^{{1/4}}}\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{u_{k}}{\zeta^{k}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$ ,

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $e$ : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $u_{s}$ : expansion coefficient
Referenced by:: §9.7(iii), §9.7(iv), §9.7(v)
Permalink:: http://dlmf.nist.gov/9.7.E5
Encodings:: TeX, pMML, png

9.7.6 ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(z\right)\sim-\frac{z^{{1/4}}e^{{-\zeta}}}{2\sqrt{\pi}}\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{v_{k}}{\zeta^{k}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$ ,

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $e$ : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $v_{s}$ : expansion coefficient
Referenced by:: §9.7(iii), §9.7(iv), §9.7(v)
Permalink:: http://dlmf.nist.gov/9.7.E6
Encodings:: TeX, pMML, png

9.7.7 $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)\sim\frac{e^{{\zeta}}}{\sqrt{\pi}z^{{1/4}}}\sum _{{k=0}}^{{\infty}}\frac{u_{k}}{\zeta^{k}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{3}\pi-\delta$ ,

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $e$ : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $u_{s}$ : expansion coefficient
Referenced by:: §9.7(iii), §9.7(iv), §9.7(v)
Permalink:: http://dlmf.nist.gov/9.7.E7
Encodings:: TeX, pMML, png

9.7.8 ${\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(z\right)\sim\frac{z^{{1/4}}e^{{\zeta}}}{\sqrt{\pi}}\sum _{{k=0}}^{{\infty}}\frac{v_{k}}{\zeta^{k}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{3}\pi-\delta$ .

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $e$ : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $v_{s}$ : expansion coefficient
Referenced by:: §9.7(iii)
Permalink:: http://dlmf.nist.gov/9.7.E8
Encodings:: TeX, pMML, png

9.7.9 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(-z\right)\sim\frac{1}{\sqrt{\pi}z^{{1/4}}}\left(\mathop{\cos\/}\nolimits\!\left(\zeta-\tfrac{1}{4}\pi\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{u_{{2k}}}{\zeta^{{2k}}}+\mathop{\sin\/}\nolimits\!\left(\zeta-\tfrac{1}{4}\pi\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{u_{{2k+1}}}{\zeta^{{2k+1}}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{2}{3}\pi-\delta$ ,

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $\mathop{\sin\/}\nolimits z$ : sine function, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $u_{s}$ : expansion coefficient
Referenced by:: §9.7(iii)
Permalink:: http://dlmf.nist.gov/9.7.E9
Encodings:: TeX, pMML, png

9.7.10 ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(-z\right)\sim\frac{z^{{1/4}}}{\sqrt{\pi}}\left(\mathop{\sin\/}\nolimits\!\left(\zeta-\tfrac{1}{4}\pi\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{v_{{2k}}}{\zeta^{{2k}}}-\mathop{\cos\/}\nolimits\!\left(\zeta-\tfrac{1}{4}\pi\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{v_{{2k+1}}}{\zeta^{{2k+1}}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{2}{3}\pi-\delta$ ,

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $\mathop{\sin\/}\nolimits z$ : sine function, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $v_{s}$ : expansion coefficient
Permalink:: http://dlmf.nist.gov/9.7.E10
Encodings:: TeX, pMML, png

9.7.11 $\mathop{\mathrm{Bi}\/}\nolimits\!\left(-z\right)\sim\frac{1}{\sqrt{\pi}z^{{1/4}}}\left(-\mathop{\sin\/}\nolimits\!\left(\zeta-\tfrac{1}{4}\pi\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{u_{{2k}}}{\zeta^{{2k}}}+\mathop{\cos\/}\nolimits\!\left(\zeta-\tfrac{1}{4}\pi\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{u_{{2k+1}}}{\zeta^{{2k+1}}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{2}{3}\pi-\delta$ ,

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $\mathop{\sin\/}\nolimits z$ : sine function, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $u_{s}$ : expansion coefficient
Permalink:: http://dlmf.nist.gov/9.7.E11
Encodings:: TeX, pMML, png

9.7.12 ${\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(-z\right)\sim\frac{z^{{1/4}}}{\sqrt{\pi}}\left(\mathop{\cos\/}\nolimits\!\left(\zeta-\tfrac{1}{4}\pi\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{v_{{2k}}}{\zeta^{{2k}}}+\mathop{\sin\/}\nolimits\!\left(\zeta-\tfrac{1}{4}\pi\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{v_{{2k+1}}}{\zeta^{{2k+1}}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{2}{3}\pi-\delta$ .

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $\mathop{\sin\/}\nolimits z$ : sine function, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $v_{s}$ : expansion coefficient
Referenced by:: §9.7(iii)
Permalink:: http://dlmf.nist.gov/9.7.E12
Encodings:: TeX, pMML, png

9.7.13 $\mathop{\mathrm{Bi}\/}\nolimits\!\left(ze^{{\pm\pi i/3}}\right)\mathrel{\sim}\sqrt{\frac{2}{\pi}}\frac{e^{{\pm\pi i/6}}}{z^{{1/4}}}\*\left(\mathop{\cos\/}\nolimits\!\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2}i\mathop{\ln\/}\nolimits 2\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{u_{{2k}}}{\zeta^{{2k}}}+\mathop{\sin\/}\nolimits\!\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2}i\mathop{\ln\/}\nolimits 2\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{u_{{2k+1}}}{\zeta^{{2k+1}}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{2}{3}\pi-\delta$ ,

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $\mathop{\sin\/}\nolimits z$ : sine function, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $u_{s}$ : expansion coefficient
Permalink:: http://dlmf.nist.gov/9.7.E13
Encodings:: TeX, pMML, png

9.7.14 ${\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(ze^{{\pm\pi i/3}}\right)\mathrel{\sim}\sqrt{\frac{2}{\pi}}e^{{\mp\pi i/6}}z^{{1/4}}\*\left(-\mathop{\sin\/}\nolimits\!\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2}i\mathop{\ln\/}\nolimits 2\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{v_{{2k}}}{\zeta^{{2k}}}+\mathop{\cos\/}\nolimits\!\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2}i\mathop{\ln\/}\nolimits 2\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{v_{{2k+1}}}{\zeta^{{2k+1}}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{2}{3}\pi-\delta$ .

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $\mathop{\sin\/}\nolimits z$ : sine function, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $v_{s}$ : expansion coefficient
Referenced by:: §9.7(iv), §9.7(v)
Permalink:: http://dlmf.nist.gov/9.7.E14
Encodings:: TeX, pMML, png

§9.7(iii) Error Bounds for Real Variables

Notes:: See Olver (1997b, pp. 392–393 and 413–414).
Keywords:: Airy functions
Permalink:: http://dlmf.nist.gov/9.7.iii

In (9.7.5) and (9.7.6) the $n$ th error term, that is, the error on truncating the expansion at $n$ terms, is bounded in magnitude by the first neglected term and has the same sign, provided that the following term is of opposite sign, that is, if $n\geq 0$ for (9.7.5) and $n\geq 1$ for (9.7.6).

In (9.7.7) and (9.7.8) the $n$ th error term is bounded in magnitude by the first neglected term multiplied by $2\chi(n)\mathop{\exp\/}\nolimits\left(\sigma\pi/(72\zeta)\right)$ where $\sigma=5$ for (9.7.7) and $\sigma=7$ for (9.7.8), provided that $n\geq 1$ in both cases.

In (9.7.9)–(9.7.12) the $n$ th error term in each infinite series is bounded in magnitude by the first neglected term and has the same sign, provided that the following term in the series is of opposite sign.

As special cases, when $0<x<\infty$

9.7.15

$\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)\leq\frac{e^{{-\xi}}}{2\sqrt{\pi}x^{{1/4}}}$ ,

$|{\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(x\right)|\leq\frac{x^{{1/4}}e^{{-\xi}}}{2\sqrt{\pi}}\left(1+\frac{7}{72\xi}\right)$ ,

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $e$ : base of exponential function, $x$ : real variable and $xi$ : change of variable
Permalink:: http://dlmf.nist.gov/9.7.E15
Encodings:: TeX, TeX, pMML, pMML, png, png

9.7.16

$\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)\leq\frac{e^{{\xi}}}{\sqrt{\pi}x^{{1/4}}}\left(1+\frac{5\pi}{72\xi}\mathop{\exp\/}\nolimits\left(\frac{5\pi}{72\xi}\right)\right),$

${\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(x\right)\leq\frac{x^{{1/4}}e^{{\xi}}}{\sqrt{\pi}}\left(1+\frac{7\pi}{72\xi}\mathop{\exp\/}\nolimits\left(\frac{7\pi}{72\xi}\right)\right),$

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\exp\/}\nolimits z$ : exponential function, $e$ : base of exponential function, $x$ : real variable and $xi$ : change of variable
Permalink:: http://dlmf.nist.gov/9.7.E16
Encodings:: TeX, TeX, pMML, pMML, png, png

where $\xi=\tfrac{2}{3}x^{{3/2}}$ .

§9.7(iv) Error Bounds for Complex Variables

Notes:: These results can be derived from (9.6.1)–(9.6.5) and Olver (1997b, pp. 266–267).
Keywords:: Airy functions
Referenced by:: §9.7(v)
Permalink:: http://dlmf.nist.gov/9.7.iv

When $n\geq 1$ the $n$ th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by

9.7.17

$2\mathop{\exp\/}\nolimits\!\left(\frac{\sigma}{36|\zeta|}\right)$ ,

$2\chi(n)\mathop{\exp\/}\nolimits\!\left(\frac{\sigma\pi}{72|\zeta|}\right)$ or

$\frac{4\chi(n)}{|\mathop{\cos\/}\nolimits\!\left(\mathop{\mathrm{ph}\/}\nolimits\zeta\right)|^{n}}\mathop{\exp\/}\nolimits\!\left(\frac{\sigma\pi}{36|\Re\zeta|}\right)$ ,

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\exp\/}\nolimits z$ : exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : real part, $\zeta(z)$ : change of variable, $\chi(n)$ : function, $n$ : index and $\sigma$ : index
Permalink:: http://dlmf.nist.gov/9.7.E17
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

according as $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{3}\pi$ , $\tfrac{1}{3}\pi\leq|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{2}{3}\pi$ , or $\tfrac{2}{3}\pi\leq|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi$ . Here $\sigma=5$ for (9.7.5) and $\sigma=7$ for (9.7.6).

Corresponding bounds for the errors in (9.7.7) to (9.7.14) may be obtained by use of these results and those of §9.2(v) and their differentiated forms.

For other error bounds see Boyd (1993).

§9.7(v) Exponentially-Improved Expansions

Notes:: See Olver (1991b, 1993a).
Defines:: $\mathop{G_{{p}}\/}\nolimits\!\left(z\right)$ : product of gamma and incomplete gamma functions
Keywords:: Airy functions
Referenced by:: §10.17(v), §9.17(i)
Permalink:: http://dlmf.nist.gov/9.7.v

In (9.7.5) and (9.7.6) let

9.7.18 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)=\frac{e^{{-\zeta}}}{2\sqrt{\pi}z^{{1/4}}}\left(\sum _{{k=0}}^{{n-1}}(-1)^{k}\frac{u_{k}}{\zeta^{k}}+R_{n}(z)\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $e$ : base of exponential function, $k$ : nonnegative integer, $z$ : complex variable, $\zeta(z)$ : change of variable, $n$ : index, $R_{n}$ : remainder function and $u_{s}$ : expansion coefficient
Permalink:: http://dlmf.nist.gov/9.7.E18
Encodings:: TeX, pMML, png

9.7.19 ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(z\right)=-\frac{z^{{1/4}}e^{{-\zeta}}}{2\sqrt{\pi}}\left(\sum _{{k=0}}^{{n-1}}(-1)^{k}\frac{v_{k}}{\zeta^{k}}+S_{n}(z)\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $e$ : base of exponential function, $k$ : nonnegative integer, $z$ : complex variable, $\zeta(z)$ : change of variable, $n$ : index, $S_{n}$ : remainder function and $v_{s}$ : expansion coefficient
Permalink:: http://dlmf.nist.gov/9.7.E19
Encodings:: TeX, pMML, png

with $n=\left\lfloor 2|\zeta|\right\rfloor$ . Then

9.7.20 $R_{n}(z)=(-1)^{n}\sum _{{k=0}}^{{m-1}}(-1)^{k}u_{k}\frac{G_{{n-k}}(2\zeta)}{\zeta^{k}}+R_{{m,n}}(z),$

Symbols:: $k$ : nonnegative integer, $z$ : complex variable, $\zeta(z)$ : change of variable, $m$ : index, $n$ : index, $R_{n}$ : remainder function, $G_{p}$ : function and $u_{s}$ : expansion coefficient
Permalink:: http://dlmf.nist.gov/9.7.E20
Encodings:: TeX, pMML, png

9.7.21 $S_{n}(z)=(-1)^{{n-1}}\sum _{{k=0}}^{{m-1}}(-1)^{k}v_{k}\frac{G_{{n-k}}(2\zeta)}{\zeta^{k}}+S_{{m,n}}(z),$

Symbols:: $k$ : nonnegative integer, $z$ : complex variable, $\zeta(z)$ : change of variable, $m$ : index, $n$ : index, $S_{n}$ : remainder function, $G_{p}$ : function and $v_{s}$ : expansion coefficient
Permalink:: http://dlmf.nist.gov/9.7.E21
Encodings:: TeX, pMML, png

where

9.7.22 $G_{p}(z)=\frac{e^{z}}{2\pi}\mathop{\Gamma\/}\nolimits\!\left(p\right)\mathop{\Gamma\/}\nolimits\!\left(1-p,z\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $z$ : complex variable and $G_{p}$ : function
Permalink:: http://dlmf.nist.gov/9.7.E22
Encodings:: TeX, pMML, png

(For the notation see §8.2(i).) And as $z\rightarrow\infty$ with $m$ fixed

9.7.23 $R_{{m,n}}(z),S_{{m,n}}(z)=\mathop{O\/}\nolimits\!\left(e^{{-2|\zeta|}}\zeta^{{-m}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{2}{3}\pi$ .

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $e$ : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable, $\zeta(z)$ : change of variable, $m$ : index, $n$ : index, $R_{n}$ : remainder function and $S_{n}$ : remainder function
Permalink:: http://dlmf.nist.gov/9.7.E23
Encodings:: TeX, pMML, png

For re-expansions of the remainder terms in (9.7.7)–(9.7.14) combine the results of this section with those of §9.2(v) and their differentiated forms, as in §9.7(iv).

For higher re-expansions of the remainder terms see Olde Daalhuis (1995, 1996), and Olde Daalhuis and Olver (1995a).