§8.11 Asymptotic Approximations and Expansions

Keywords:: asymptotic approximations and expansions, incomplete gamma functions
Permalink:: http://dlmf.nist.gov/8.11

§8.11(i) Large $z$ , Fixed $a$
§8.11(ii) Large $a$ , Fixed $z$
§8.11(iii) Large $a$ , Fixed $x/a$
§8.11(iv) Large $a$ , Bounded $(x-a)/(2a)^{{\frac{1}{2}}}$
§8.11(v) Other Approximations

§8.11(i) Large $z$ , Fixed $a$

Notes:: See Olver (1997b, pp. 66 and 109–112), Temme (1996b, p. 280).
Keywords:: asymptotic approximations and expansions, incomplete gamma functions
Permalink:: http://dlmf.nist.gov/8.11.i

Define

8.11.1 $u_{k}=(-1)^{k}\left(1-a\right)_{{k}}=(a-1)(a-2)\cdots(a-k),$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $a$ : parameter, $k$ : nonnegative integer and $u_{k}$
Permalink:: http://dlmf.nist.gov/8.11.E1
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8.11.2 $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)=z^{{a-1}}e^{{-z}}\left(\sum _{{k=0}}^{{n-1}}\frac{u_{k}}{z^{k}}+R_{n}(a,z)\right),$ $n=1,2,\dots$ .

Symbols:: $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter, $k$ : nonnegative integer, $n$ : nonnegative integer, $u_{k}$ and $R_{n}(a,z)$ : remainder
A&S Ref:: 6.5.32
Permalink:: http://dlmf.nist.gov/8.11.E2
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Then as $z\to\infty$ with $a$ and $n$ fixed

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable, $a$ : parameter, $n$ : nonnegative integer, $\delta$ : small positive constant and $R_{n}(a,z)$ : remainder
Permalink:: http://dlmf.nist.gov/8.11.E3
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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 $z$

Notes:: See (8.5.1) or (8.7.1). (8.11.5) follows from the leading terms of (8.11.4) and (5.11.3).
Permalink:: http://dlmf.nist.gov/8.11.ii

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

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $e$ : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : 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
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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 $\mathop{P\/}\nolimits\!\left(a,z\right)\sim\frac{z^{a}e^{{-z}}}{\mathop{\Gamma\/}\nolimits\!\left(1+a\right)}\sim(2\pi a)^{{-\frac{1}{2}}}e^{{a-z}}(z/a)^{a},$ $a\to\infty$ , $|\mathop{\mathrm{ph}\/}\nolimits a|\leq\pi-\delta$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential function, $\mathop{P\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $z$ : complex variable, $a$ : parameter and $\delta$ : small positive constant
Referenced by:: §8.11(ii)
Permalink:: http://dlmf.nist.gov/8.11.E5
Encodings:: TeX, pMML, png

§8.11(iii) Large $a$ , Fixed $x/a$

Notes:: See Gautschi (1959a) and Temme (1994b).
Referenced by:: §8.11(v)
Permalink:: http://dlmf.nist.gov/8.11.iii

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

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

Symbols:: $e$ : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\sim$ : asymptotic equality, $x$ : real variable, $a$ : parameter, $k$ : nonnegative integer, $\lambda$ : parameter and $b_{k}(\lambda)$ : coefficients
Permalink:: http://dlmf.nist.gov/8.11.E6
Encodings:: TeX, pMML, png

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

Symbols:: $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\sim$ : asymptotic equality, $x$ : real variable, $a$ : parameter, $k$ : nonnegative integer, $\lambda$ : parameter and $b_{k}(\lambda)$ : coefficients
Referenced by:: §8.11(iii)
Permalink:: http://dlmf.nist.gov/8.11.E7
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where

8.11.8

$b_{0}(\lambda)=1,$

$b_{1}(\lambda)=\lambda,$

$b_{2}(\lambda)=\lambda(2\lambda+1),$

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

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

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

Permalink:: http://dlmf.nist.gov/8.11.iv

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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $e$ : base of exponential function, $\mathop{P\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function, $x$ : real variable, $a$ : parameter and $y$ : change of variable
Permalink:: http://dlmf.nist.gov/8.11.E10
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8.11.11 $\mathop{\gamma^{{*}}\/}\nolimits\!\left(1-a,-x\right)=x^{{a-1}}\left(-\mathop{\cos\/}\nolimits\!\left(\pi a\right)+\frac{\mathop{\sin\/}\nolimits\!\left(\pi a\right)}{\pi}\left(2\sqrt{\pi}\mathop{F\/}\nolimits\!\left(y\right)+\frac{2}{3}\sqrt{\frac{2\pi}{a}}\left(1-y^{2}\right)\right)e^{{y^{2}}}+\mathop{O\/}\nolimits\!\left(a^{{-1}}\right)\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{F\/}\nolimits\!\left(z\right)$ : Dawson’s integral, $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $\mathop{\gamma^{{*}}\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable, $a$ : parameter and $y$ : change of variable
Permalink:: http://dlmf.nist.gov/8.11.E11
Encodings:: TeX, pMML, png

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

Notes:: (8.11.12) can be obtained from (8.12.15), (8.2.4), and (5.11.3). (8.11.15) follows from (8.4.7) with $z=nx$ . For (8.11.16) see Ramanujan (1962, pp. 323–324). For (8.11.17) see Copson (1933).
Keywords:: asymptotic approximations and expansions, incomplete gamma functions, truncated exponential series
Permalink:: http://dlmf.nist.gov/8.11.v

As $z\to\infty$ ,

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

Symbols:: $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $z$ : complex variable and $\delta$ : small positive constant
A&S Ref:: 6.5.35 (Modified form)
Referenced by:: §8.11(v), § ‣ § Recent News
Permalink:: http://dlmf.nist.gov/8.11.E12
Encodings:: TeX, pMML, png

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
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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 $e^{{nx}}=e_{n}(nx)+\frac{(nx)^{n}}{n!}S_{n}(x),$

Symbols:: $e$ : base of exponential function, $!$ : $n!$ : factorial, $x$ : real variable, $n$ : nonnegative integer, $S_{n}(x)$ : function and $e_{n}(z)$ : functions
Referenced by:: §8.11(v)
Permalink:: http://dlmf.nist.gov/8.11.E14
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then

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

Symbols:: $e$ : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : 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
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As $n\to\infty$

8.11.16 $S_{n}(1)-\frac{1}{2}\frac{n!e^{n}}{n^{n}}\sim-\tfrac{2}{3}+\tfrac{4}{135}n^{{-1}}-\tfrac{8}{2835}n^{{-2}}-\tfrac{16}{8505}n^{{-3}}+\dots,$

Symbols:: $e$ : base of exponential function, $!$ : $n!$ : factorial, $\sim$ : asymptotic equality, $n$ : nonnegative integer and $S_{n}(x)$ : function
Referenced by:: §8.11(v), §8.11(v)
Permalink:: http://dlmf.nist.gov/8.11.E16
Encodings:: TeX, pMML, png

8.11.17 $S_{n}(-1)\sim-\tfrac{1}{2}+\tfrac{1}{8}n^{{-1}}+\tfrac{1}{32}n^{{-2}}-\tfrac{1}{128}n^{{-3}}-\tfrac{13}{512}n^{{-4}}+\dots.$

Symbols:: $\sim$ : asymptotic equality, $n$ : nonnegative integer and $S_{n}(x)$ : function
Referenced by:: §8.11(v)
Permalink:: http://dlmf.nist.gov/8.11.E17
Encodings:: TeX, pMML, png

Also,

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

Symbols:: $\sim$ : asymptotic equality, $x$ : real variable, $k$ : nonnegative integer, $n$ : nonnegative integer, $S_{n}(x)$ : function and $d_{k}(x)$ : coefficients
Referenced by:: §8.11(v)
Permalink:: http://dlmf.nist.gov/8.11.E18
Encodings:: TeX, pMML, png

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

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).

§8.11 Asymptotic Approximations and Expansions

Contents

§8.11(i) Large , Fixed

§8.11(ii) Large , Fixed

§8.11(iii) Large , Fixed

§8.11(iv) Large , Bounded

§8.11(v) Other Approximations

§8.11(i) Large $z$ , Fixed $a$

§8.11(ii) Large $a$ , Fixed $z$

§8.11(iii) Large $a$ , Fixed $x/a$

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