# §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: Annotations for §8.11(i), §8.11 and Ch.8
 8.11.2 $\Gamma\left(a,z\right)=z^{a-1}e^{-z}\left(\sum_{k=0}^{n-1}\frac{u_{k}}{z^{k}}+% R_{n}(a,z)\right),$ $n=1,2,\dots$.

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

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

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 $\gamma\left(a,z\right)=z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{{\left(a% \right)_{k+1}}},$ $a\neq 0,-1,-2,\dots$. ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer A&S Ref: 6.5.33 (corrected) Referenced by: §8.11(ii) Permalink: http://dlmf.nist.gov/8.11.E4 Encodings: TeX, pMML, png See also: Annotations for §8.11(ii), §8.11 and Ch.8

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

Also,

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

## §8.11(iii) Large $a$, Fixed $z/a$

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

 8.11.6 $\gamma\left(a,z\right)\sim-z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{(-a)^{k}b_{k}(% \lambda)}{(z-a)^{2k+1}},$ $0<\lambda<1$, $|\operatorname{ph}a|\leq\frac{\pi}{2}-\delta$. ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function, $\operatorname{ph}$: phase, $x$: real variable, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant, $\lambda$: parameter and $b_{k}(\lambda)$: coefficients Notes: See Paris (2002b) Referenced by: §8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v) Permalink: http://dlmf.nist.gov/8.11.E6 Encodings: TeX, pMML, png Addition (effective with 1.0.14): Originally this equation was stated for real variables $a$ and $x(=\lambda a)$, $0<\lambda<1$. It has been extended to allow for complex variables $a$ and $z(=\lambda a)$, where $0<\lambda<1$ and $|\operatorname{ph}a|\leq\frac{\pi}{2}-\delta$. See also: Annotations for §8.11(iii), §8.11 and Ch.8
 8.11.7 $\Gamma\left(a,z\right)\sim z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{(-a)^{k}b_{k}(% \lambda)}{(z-a)^{2k+1}},$ $\lambda>1$, $|\operatorname{ph}a|\leq\frac{3\pi}{2}-\delta$. ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function, $\operatorname{ph}$: phase, $x$: real variable, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant, $\lambda$: parameter and $b_{k}(\lambda)$: coefficients Referenced by: §8.11(iii), §8.11(iii), §8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v) Permalink: http://dlmf.nist.gov/8.11.E7 Encodings: TeX, pMML, png Addition (effective with 1.0.14): Originally this equation was stated for real variables $a$ and $x(=\lambda a)$, $\lambda>1$. It has been extended to allow for complex variables $a$ and $z(=\lambda a)$, where $\lambda>1$ and $|\operatorname{ph}a|\leq\frac{3\pi}{2}-\delta$. See also: Annotations for §8.11(iii), §8.11 and Ch.8

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: Annotations for §8.11(iii), §8.11 and Ch.8

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 Referenced by: §8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v) Permalink: http://dlmf.nist.gov/8.11.E9 Encodings: TeX, pMML, png See also: Annotations for §8.11(iii), §8.11 and Ch.8

Sharp error bounds and an exponentially-improved extension for (8.11.7) can be found in Nemes (2016). This reference also contains explicit formulas for $b_{k}(\lambda)$ in terms of Stirling numbers and for the case $\lambda>1$ an asymptotic expansion for $b_{k}(\lambda)$ as $k\to\infty$.

The expansion (8.11.7) also applies when $a$ is replaced by $-a$, $\lambda<0$ and $|{\rm ph}\,a|\leq\frac{3\pi}{2}-\omega-\delta$ with $\omega={\rm ph}(-\lambda+\ln\left(-\lambda\right)+\pi i)$, $0<\omega<\pi$. For error bounds and an exponentially-improved extension for this later expansion, see Nemes (2015c).

## §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 $P\left(a+1,x\right)=\tfrac{1}{2}\operatorname{erfc}\left(-y\right)-\frac{1}{3}% \sqrt{\frac{2}{\pi a}}(1+y^{2})e^{-y^{2}}+O\left(a^{-1}\right),$
 8.11.11 $\gamma^{*}\left(1-a,-x\right)=x^{a-1}\left(-\cos\left(\pi a\right)+\frac{\sin% \left(\pi a\right)}{\pi}\left(2\sqrt{\pi}F\left(y\right)+\frac{2}{3}\sqrt{% \frac{2\pi}{a}}\left(1-y^{2}\right)\right)e^{y^{2}}+O\left(a^{-1}\right)\right),$

in both cases uniformly with respect to bounded real values of $y$. For Dawson’s integral $F\left(y\right)$ 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 $\Gamma\left(z,z\right)\sim z^{z-1}e^{-z}\left(\sqrt{\frac{\pi}{2}}z^{\frac{1}{% 2}}-\frac{1}{3}+\frac{\sqrt{2\pi}}{24z^{\frac{1}{2}}}-\frac{4}{135z}+\frac{% \sqrt{2\pi}}{576z^{\frac{3}{2}}}+\frac{8}{2835z^{2}}+\dots\right),$ $|\operatorname{ph}z|\leq 2\pi-\delta$. ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function, $\operatorname{ph}$: phase, $z$: complex variable and $\delta$: small positive constant A&S Ref: 6.5.35 (Modified form) Referenced by: §8.11(v), § ‣ Recent News, Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v) Permalink: http://dlmf.nist.gov/8.11.E12 Encodings: TeX, pMML, png Addition (effective with 1.0.14): The sector of validity for the above equation was increased from $|\operatorname{ph}z|\leq\pi-\delta$ to $|\operatorname{ph}z|\leq 2\pi-\delta$. See also: Annotations for §8.11(v), §8.11 and Ch.8

For sharp error bounds and an exponentially-improved extension, see Nemes (2016). This reference also contains explicit formulas for the coefficients in terms of Stirling numbers.

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: $\mathrm{e}$: base of natural logarithm, $x$: 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: Annotations for §8.11(v), §8.11 and Ch.8

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: $\mathrm{e}$: base of natural logarithm, $!$: 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: Annotations for §8.11(v), §8.11 and Ch.8

then

 8.11.15 $S_{n}(x)=\frac{\gamma\left(n+1,nx\right)}{(nx)^{n}e^{-nx}}.$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function, $x$: real variable, $n$: nonnegative integer and $S_{n}(x)$: function Referenced by: §8.11(v) Permalink: http://dlmf.nist.gov/8.11.E15 Encodings: TeX, pMML, png See also: Annotations for §8.11(v), §8.11 and Ch.8

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: Annotations for §8.11(v), §8.11 and Ch.8

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 $d_{k}(x)=\frac{(-1)^{k}b_{k}(x)}{(1-x)^{2k+1}},$ $k=0,1,2,\dots$, ⓘ Symbols: $x$: real variable, $k$: nonnegative integer, $b_{k}(\lambda)$: coefficients and $d_{k}(x)$: coefficients Permalink: http://dlmf.nist.gov/8.11.E19 Encodings: TeX, pMML, png See also: Annotations for §8.11(v), §8.11 and Ch.8

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).