# §8.11 Asymptotic Approximations and Expansions

## §8.11(i) Large $z$, Fixed $a$

Define

 8.11.1 $u_{k}=(-1)^{k}\left(1-a\right)_{k}=(a-1)(a-2)\cdots(a-k),$ Defines: $u_{k}$ (locally) Symbols: $\left(\NVar{a}\right)_{\NVar{n}}$: Pochhammer’s symbol (or shifted factorial), $a$: parameter and $k$: nonnegative integer Permalink: http://dlmf.nist.gov/8.11.E1 Encodings: TeX, pMML, png See also: info for 8.11(i)
 8.11.2 $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=z^{a-1}e^{-z}\left(\sum_{k=0}^{n-% 1}\frac{u_{k}}{z^{k}}+R_{n}(a,z)\right),$ $n=1,2,\dots$.

Then as $z\to\infty$ with $a$ and $n$ fixed

 8.11.3 $R_{n}(a,z)=\mathop{O\/}\nolimits\!\left(z^{-n}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{3}{2}\pi-\delta$,

where $\delta$ denotes an arbitrary small positive constant.

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 $|\mathop{\mathrm{ph}\/}\nolimits a|\leq\pi-\delta$.

Also,

 8.11.5 $\mathop{P\/}\nolimits\!\left(a,z\right)\sim\frac{z^{a}e^{-z}}{\mathop{\Gamma\/% }\nolimits\!\left(1+a\right)}\sim(2\pi a)^{-\frac{1}{2}}e^{a-z}(z/a)^{a},$ $a\to\infty$, $|\mathop{\mathrm{ph}\/}\nolimits a|\leq\pi-\delta$.

## §8.11(iii) Large $a$, Fixed $x/a$

If $x=\lambda a$, with $\lambda$ fixed, then as $a\to+\infty$

 8.11.6 $\mathop{\gamma\/}\nolimits\!\left(a,x\right)\sim-x^{a}e^{-x}\sum_{k=0}^{\infty% }\frac{(-a)^{k}b_{k}(\lambda)}{(x-a)^{2k+1}},$ $0<\lambda<1$,
 8.11.7 $\mathop{\Gamma\/}\nolimits\!\left(a,x\right)\sim x^{a}e^{-x}\sum_{k=0}^{\infty% }\frac{(-a)^{k}b_{k}(\lambda)}{(x-a)^{2k+1}},$ $\lambda>1$,

where

 8.11.8 $\displaystyle b_{0}(\lambda)$ $\displaystyle=1,$ $\displaystyle b_{1}(\lambda)$ $\displaystyle=\lambda,$ $\displaystyle b_{2}(\lambda)$ $\displaystyle=\lambda(2\lambda+1),$ Symbols: $\lambda$: parameter and $b_{k}(\lambda)$: coefficients Permalink: http://dlmf.nist.gov/8.11.E8 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: info for 8.11(iii)

and for $k=1,2,\dots$,

 8.11.9 $b_{k}(\lambda)=\lambda(1-\lambda)b_{k-1}^{\prime}(\lambda)+(2k-1)\lambda b_{k-% 1}(\lambda).$ Symbols: $k$: nonnegative integer, $\lambda$: parameter and $b_{k}(\lambda)$: coefficients Permalink: http://dlmf.nist.gov/8.11.E9 Encodings: TeX, pMML, png See also: info for 8.11(iii)

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 $(x-a)/(2a)^{\frac{1}{2}}$

If $x=a+(2a)^{\frac{1}{2}}y$ and $a\to+\infty$, then

 8.11.10 $\mathop{P\/}\nolimits\!\left(a+1,x\right)=\tfrac{1}{2}\mathop{\mathrm{erfc}\/}% \nolimits\!\left(-y\right)-\frac{1}{3}\sqrt{\frac{2}{\pi a}}(1+y^{2})e^{-y^{2}% }+\mathop{O\/}\nolimits\!\left(a^{-1}\right),$
 8.11.11 $\mathop{\gamma^{*}\/}\nolimits\!\left(1-a,-x\right)=x^{a-1}\left(-\mathop{\cos% \/}\nolimits\!\left(\pi a\right)+\frac{\mathop{\sin\/}\nolimits\!\left(\pi a% \right)}{\pi}\left(2\sqrt{\pi}\mathop{F\/}\nolimits\!\left(y\right)+\frac{2}{3% }\sqrt{\frac{2\pi}{a}}\left(1-y^{2}\right)\right)e^{y^{2}}+\mathop{O\/}% \nolimits\!\left(a^{-1}\right)\right),$

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

As $z\to\infty$,

 8.11.12 $\mathop{\Gamma\/}\nolimits\!\left(z,z\right)\sim z^{z-1}e^{-z}\left(\sqrt{% \frac{\pi}{2}}z^{\frac{1}{2}}-\frac{1}{3}+\frac{\sqrt{2\pi}}{24z^{\frac{1}{2}}% }-\frac{4}{135z}+\frac{\sqrt{2\pi}}{576z^{\frac{3}{2}}}+\frac{8}{2835z^{2}}+% \dots\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$.

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

 8.11.13 $\lim_{n\to\infty}\frac{e_{n}(nx)}{e^{nx}}=\begin{cases}0,&x>1,\\ \tfrac{1}{2},&x=1,\\ 1,&0\leq x<1.\end{cases}$ Symbols: $e$: base of exponential function, $x$: real variable, $n$: nonnegative integer and $e_{n}(z)$: functions A&S Ref: 6.5.34 (Slightly modified) Permalink: http://dlmf.nist.gov/8.11.E13 Encodings: TeX, pMML, png See also: info for 8.11(v)

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

If $S_{n}(x)$ is defined by

 8.11.14 $e^{nx}=e_{n}(nx)+\frac{(nx)^{n}}{n!}S_{n}(x),$ Defines: $S_{n}(x)$: function (locally) Symbols: $e$: base of exponential function, $!$: factorial (as in $n!$), $x$: real variable, $n$: nonnegative integer and $e_{n}(z)$: functions Referenced by: §8.11(v) Permalink: http://dlmf.nist.gov/8.11.E14 Encodings: TeX, pMML, png See also: info for 8.11(v)

then

 8.11.15 $S_{n}(x)=\frac{\mathop{\gamma\/}\nolimits\!\left(n+1,nx\right)}{(nx)^{n}e^{-nx% }}.$

As $n\to\infty$

 8.11.16 $S_{n}(1)-\frac{1}{2}\frac{n!e^{n}}{n^{n}}\sim-\tfrac{2}{3}+\tfrac{4}{135}n^{-1% }-\tfrac{8}{2835}n^{-2}-\tfrac{16}{8505}n^{-3}+\dots,$
 8.11.17 $S_{n}(-1)\sim-\tfrac{1}{2}+\tfrac{1}{8}n^{-1}+\tfrac{1}{32}n^{-2}-\tfrac{1}{12% 8}n^{-3}-\tfrac{13}{512}n^{-4}+\dots.$ Symbols: $\sim$: Poincaré asymptotic expansion, $n$: nonnegative integer and $S_{n}(x)$: function Referenced by: §8.11(v) Permalink: http://dlmf.nist.gov/8.11.E17 Encodings: TeX, pMML, png See also: info for 8.11(v)

Also,

 8.11.18 $S_{n}(x)\sim\sum_{k=0}^{\infty}d_{k}(x)n^{-k},$ $n\to\infty$,

uniformly for $x\in(-\infty,1-\delta]$, with

 8.11.19 $d_{k}(x)=\frac{(-1)^{k}b_{k}(x)}{(1-x)^{2k+1}},$ $k=0,1,2,\dots$,

and $b_{k}(x)$ as in §8.11(iii).

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).