8 Incomplete Gamma and Related Functions Incomplete Gamma Functions 8.11 Asymptotic Approximations and Expansions 8.13 Zeros

§8.12 Uniform Asymptotic Expansions for Large Parameter

ⓘ

Keywords:: asymptotic approximations and expansions, exponentially-improved, incomplete gamma functions, uniform for large parameter
Notes:: See Temme (1979b, 1992a, 1996a), Paris (2002b), and Ferreira et al. (2005).
Referenced by:: §8.25(iii)
Permalink:: http://dlmf.nist.gov/8.12
Clarification (effective with 1.1.7):: Just below (8.12.18), $\Re z-a$ was replaced with $\Re\left(z-a\right)$
See also:: Annotations for Ch.8

Define

8.12.1	$\displaystyle\lambda$	$\displaystyle=z/a,$
8.12.1	$\displaystyle\eta$	$\displaystyle=\left(2(\lambda-1-\ln\lambda)\right)^{1/2},$
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function, $z$ : complex variable, $a$ : parameter, $\eta(\lambda)$ and $\lambda$ Permalink: http://dlmf.nist.gov/8.12.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §8.12 and Ch.8

where the branch of the square root is continuous and satisfies $\eta(\lambda)\sim\lambda-1$ as $\lambda\to 1$ . Then

8.12.2	$\displaystyle\tfrac{1}{2}\eta^{2}$	$\displaystyle=\lambda-1-\ln\lambda,$
8.12.2	$\displaystyle\frac{\mathrm{d}\eta}{\mathrm{d}\lambda}$	$\displaystyle=\frac{\lambda-1}{\lambda\eta}.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\ln\NVar{z}$ : principal branch of logarithm function, $\eta(\lambda)$ and $\lambda$ Permalink: http://dlmf.nist.gov/8.12.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §8.12 and Ch.8

Also, denote

8.12.3		$P\left(a,z\right)=\tfrac{1}{2}\operatorname{erfc}\left(-\eta\sqrt{a/2}\right)-% S(a,\eta),$
ⓘ Symbols: $\operatorname{erfc}\NVar{z}$ : complementary error function, $P\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function, $z$ : complex variable, $a$ : parameter, $\eta(\lambda)$ and $S(a,\eta)$ : function Permalink: http://dlmf.nist.gov/8.12.E3 Encodings: TeX, pMML, png See also: Annotations for §8.12 and Ch.8

8.12.4		$Q\left(a,z\right)=\tfrac{1}{2}\operatorname{erfc}\left(\eta\sqrt{a/2}\right)+S% (a,\eta),$
ⓘ Symbols: $\operatorname{erfc}\NVar{z}$ : complementary error function, $Q\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function, $z$ : complex variable, $a$ : parameter, $\eta(\lambda)$ and $S(a,\eta)$ : function Referenced by: §8.12 Permalink: http://dlmf.nist.gov/8.12.E4 Encodings: TeX, pMML, png See also: Annotations for §8.12 and Ch.8

8.12.5

\frac{e^{\pm\pi ia}}{2i\sin\left(\pi a\right)}Q\left(-a,ze^{\pm\pi i}\right)=% \pm\tfrac{1}{2}\operatorname{erfc}\left(\pm i\eta\sqrt{a/2}\right)-iT(a,\eta),

ⓘ

Symbols:

\Gamma\left(\NVar{z}\right)

: gamma function,

\pi

: the ratio of the circumference of a circle to its diameter,

\operatorname{erfc}\NVar{z}

: complementary error function,

\mathrm{e}

: base of natural logarithm,

\mathrm{i}

: imaginary unit,

\Gamma\left(\NVar{a},\NVar{z}\right)

: incomplete gamma function,

Q\left(\NVar{a},\NVar{z}\right)

: normalized incomplete gamma function,

\sin\NVar{z}

: sine function,

z

: complex variable,

a

: parameter,

\eta(\lambda)

and

T(a,\eta)

: function

Referenced by:

Erratum (V1.0.16) for Equation (8.12.5), Erratum (V1.0.16) for Equation (8.12.5)

Permalink:

http://dlmf.nist.gov/8.12.E5

Encodings:

TeX, pMML, png

Change (effective with 1.0.16):

To be consistent with the notation used in (8.12.16), the original equation,

\Gamma\left(a+1\right)\frac{e^{\pm\pi ia}}{2\pi i}\Gamma\left(-a,ze^{\pm\pi i}% \right)=\mp\tfrac{1}{2}\operatorname{erfc}\left(\pm i\eta\sqrt{a/2}\right)+iT(% a,\eta),

was rewritten.

See also:

Annotations for §8.12 and Ch.8

and

8.12.6		$z^{-a}\gamma^{*}\left(-a,-z\right)=\cos\left(\pi a\right)-2\sin\left(\pi a% \right)\left(\frac{e^{\frac{1}{2}a\eta^{2}}}{\sqrt{\pi}}F\left(\eta\sqrt{a/2}% \right)+T(a,\eta)\right),$
ⓘ Symbols: $F\left(\NVar{z}\right)$ : Dawson’s integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\gamma^{*}\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\sin\NVar{z}$ : sine function, $z$ : complex variable, $a$ : parameter, $\eta(\lambda)$ and $T(a,\eta)$ : function Permalink: http://dlmf.nist.gov/8.12.E6 Encodings: TeX, pMML, png See also: Annotations for §8.12 and Ch.8

where $F\left(x\right)$ is Dawson’s integral; see §7.2(ii). Then as $a\to\infty$ in the sector $|\operatorname{ph}a|\leq\pi-\delta(<\pi)$ ,

8.12.7		$S(a,\eta)\sim\frac{e^{-\frac{1}{2}a\eta^{2}}}{\sqrt{2\pi a}}\sum_{k=0}^{\infty% }c_{k}(\eta)a^{-k},$
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $a$ : parameter, $k$ : nonnegative integer, $\eta(\lambda)$ , $S(a,\eta)$ : function and $c_{k}(\eta)$ : coefficients Referenced by: §8.12 Permalink: http://dlmf.nist.gov/8.12.E7 Encodings: TeX, pMML, png See also: Annotations for §8.12 and Ch.8

8.12.8		$T(a,\eta)\sim\frac{e^{\frac{1}{2}a\eta^{2}}}{\sqrt{2\pi a}}\sum_{k=0}^{\infty}% c_{k}(\eta)(-a)^{-k},$
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $a$ : parameter, $k$ : nonnegative integer, $\eta(\lambda)$ , $T(a,\eta)$ : function and $c_{k}(\eta)$ : coefficients Permalink: http://dlmf.nist.gov/8.12.E8 Encodings: TeX, pMML, png See also: Annotations for §8.12 and Ch.8

in each case uniformly with respect to $\lambda$ in the sector $|\operatorname{ph}\lambda|\leq 2\pi-\delta$ ( $<2\pi$ ).

With $\mu=\lambda-1$ , the coefficients $c_{k}(\eta)$ are given by

8.12.9	$\displaystyle c_{0}(\eta)$	$\displaystyle=\frac{1}{\mu}-\frac{1}{\eta},$
8.12.9	$\displaystyle c_{1}(\eta)$	$\displaystyle=\frac{1}{\eta^{3}}-\frac{1}{\mu^{3}}-\frac{1}{\mu^{2}}-\frac{1}{% 12\mu},$
ⓘ Symbols: $\eta(\lambda)$ , $\mu$ and $c_{k}(\eta)$ : coefficients Referenced by: §8.12 Permalink: http://dlmf.nist.gov/8.12.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §8.12 and Ch.8

8.12.10		$c_{k}(\eta)=\frac{1}{\eta}\frac{\mathrm{d}}{\mathrm{d}\eta}c_{k-1}(\eta)+(-1)^% {k}\frac{g_{k}}{\mu},$
$k=1,2,\dots$ ,
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $k$ : nonnegative integer, $\eta(\lambda)$ , $\mu$ , $c_{k}(\eta)$ : coefficients and $g_{k}$ : coefficients Referenced by: §8.12 Permalink: http://dlmf.nist.gov/8.12.E10 Encodings: TeX, pMML, png See also: Annotations for §8.12 and Ch.8

where $g_{k}$ , $k=0,1,2,\dots$ , are the coefficients that appear in the asymptotic expansion (5.11.3) of $\Gamma\left(z\right)$ . The right-hand sides of equations (8.12.9), (8.12.10) have removable singularities at $\eta=0$ , and the Maclaurin series expansion of $c_{k}(\eta)$ is given by

8.12.11		$c_{k}(\eta)=\sum_{n=0}^{\infty}d_{k,n}\eta^{n},$
$\|\eta\|<2\sqrt{\pi}$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $k$ : nonnegative integer, $n$ : nonnegative integer, $\eta(\lambda)$ , $d(\pm\chi)$ and $c_{k}(\eta)$ : coefficients Permalink: http://dlmf.nist.gov/8.12.E11 Encodings: TeX, pMML, png See also: Annotations for §8.12 and Ch.8

where $d_{0,0}=-\tfrac{1}{3}$ ,

8.12.12		$\displaystyle d_{0,n}$	$\displaystyle=(n+2)\alpha_{n+2},$
	$n\geq 1$ ,
		$\displaystyle d_{k,n}$	$\displaystyle=(-1)^{k}g_{k}d_{0,n}+(n+2)d_{k-1,n+2},$
	$n\geq 0$ , $k\geq 1$ ,
ⓘ Symbols: $k$ : nonnegative integer, $n$ : nonnegative integer, $d(\pm\chi)$ , $g_{k}$ : coefficients and $\alpha_{n+2}$ Permalink: http://dlmf.nist.gov/8.12.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §8.12 and Ch.8

and $\alpha_{3},\alpha_{4},\dots$ are defined by

8.12.13		$\lambda-1=\eta+\tfrac{1}{3}\eta^{2}+\sum_{n=3}^{\infty}\alpha_{n}\eta^{n},$
$\|\eta\|<2\sqrt{\pi}$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer, $\eta(\lambda)$ , $\lambda$ and $\alpha_{n+2}$ Permalink: http://dlmf.nist.gov/8.12.E13 Encodings: TeX, pMML, png See also: Annotations for §8.12 and Ch.8

In particular,

8.12.14	$\displaystyle\alpha_{3}$	$\displaystyle=\tfrac{1}{36},$
	$\displaystyle\alpha_{4}$	$\displaystyle=-\tfrac{1}{270},$
	$\displaystyle\alpha_{5}$	$\displaystyle=\tfrac{1}{4320},$
	$\displaystyle\alpha_{6}$	$\displaystyle=\tfrac{1}{17010},$
	$\displaystyle\alpha_{7}$	$\displaystyle=-\tfrac{139}{54\;43200},$
	$\displaystyle\alpha_{8}$	$\displaystyle=\tfrac{1}{2\;04120}.$
ⓘ Symbols: $\alpha_{n+2}$ Permalink: http://dlmf.nist.gov/8.12.E14 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png See also: Annotations for §8.12 and Ch.8

For numerical values of $d_{k,n}$ to 30D for $k=0(1)9$ and $n=0(1)N_{k}$ , where $N_{k}=28-4\left\lfloor k/2\right\rfloor$ , see DiDonato and Morris (1986).

Special cases are given by

8.12.15		$Q\left(a,a\right)\sim\frac{1}{2}+\frac{1}{\sqrt{2\pi a}}\sum_{k=0}^{\infty}c_{% k}(0)a^{-k},$
$\|\operatorname{ph}a\|\leq\pi-\delta$ ,
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $Q\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function, $\operatorname{ph}$ : phase, $a$ : parameter, $k$ : nonnegative integer, $\delta$ : small positive constant and $c_{k}(\eta)$ : coefficients Permalink: http://dlmf.nist.gov/8.12.E15 Encodings: TeX, pMML, png See also: Annotations for §8.12 and Ch.8

8.12.16		$\frac{e^{\pm\pi ia}}{2i\sin\left(\pi a\right)}Q\left(-a,ae^{\pm\pi i}\right)% \sim\pm\frac{1}{2}-\frac{i}{\sqrt{2\pi a}}\sum_{k=0}^{\infty}c_{k}(0)(-a)^{-k},$
$\|\operatorname{ph}a\|\leq\pi-\delta$ ,
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $Q\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, $a$ : parameter, $k$ : nonnegative integer, $\delta$ : small positive constant and $c_{k}(\eta)$ : coefficients Referenced by: (8.12.5), Erratum (V1.0.16) for Equation (8.12.5) Permalink: http://dlmf.nist.gov/8.12.E16 Encodings: TeX, pMML, png See also: Annotations for §8.12 and Ch.8

where

8.12.17	$\displaystyle c_{0}(0)$	$\displaystyle=-\tfrac{1}{3},$
	$\displaystyle c_{1}(0)$	$\displaystyle=-\tfrac{1}{540},$
	$\displaystyle c_{2}(0)$	$\displaystyle=\tfrac{25}{6048},$
	$\displaystyle c_{3}(0)$	$\displaystyle=\tfrac{101}{1\;55520},$
	$\displaystyle c_{4}(0)$	$\displaystyle=-\tfrac{31\;84811}{36951\;55200},$
	$\displaystyle c_{5}(0)$	$\displaystyle=-\tfrac{27\;45493}{81517\;36320}.$
ⓘ Symbols: $c_{k}(\eta)$ : coefficients Permalink: http://dlmf.nist.gov/8.12.E17 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png See also: Annotations for §8.12 and Ch.8

For error bounds for (8.12.7) see Paris (2002a). For the asymptotic behavior of $c_{k}(\eta)$ as $k\to\infty$ see Dunster et al. (1998) and Olde Daalhuis (1998c). The last reference also includes an exponentially-improved version (§2.11(iii)) of the expansions (8.12.4) and (8.12.7) for $Q\left(a,z\right)$ .

A different type of uniform expansion with coefficients that do not possess a removable singularity at $z=a$ is given by

8.12.18		$\rselection{Q\left(a,z\right)\\ P\left(a,z\right)}\sim\frac{z^{a-\frac{1}{2}}e^{-z}}{\Gamma\left(a\right)}{% \left(d(\pm\chi)\sum_{k=0}^{\infty}\frac{A_{k}(\chi)}{z^{k/2}}\mp\sum_{k=1}^{% \infty}\frac{B_{k}(\chi)}{z^{k/2}}\right)},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : Poincaré asymptotic expansion, $\mathrm{e}$ : base of natural logarithm, $P\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function, $Q\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function, $z$ : complex variable, $a$ : parameter, $k$ : nonnegative integer, $\chi$ , $d(\pm\chi)$ , $A_{k}(\chi)$ and $B_{k}(\chi)$ Referenced by: §8.12, Erratum (V1.0.16) for Equation (8.12.18) Permalink: http://dlmf.nist.gov/8.12.E18 Encodings: TeX, pMML, png Errata (effective with 1.0.16): The original $\pm$ in front of the second summation was replaced by $\mp$ to correct an error in Paris (2002b); for details see https://arxiv.org/abs/1611.00548. Reported 2017-01-28 by Richard Paris See also: Annotations for §8.12 and Ch.8

for $z\to\infty$ in $\left|\operatorname{ph}z\right|<\frac{1}{2}\pi$ , with $\Re\left(z-a\right)\leq 0$ for $P\left(a,z\right)$ and $\Re\left(z-a\right)\geq 0$ for $Q\left(a,z\right)$ . Here

8.12.19	$\displaystyle\chi$	$\displaystyle=(z-a)/\sqrt{z},$
8.12.19	$\displaystyle d(\pm\chi)$	$\displaystyle=\sqrt{\tfrac{1}{2}\pi}e^{\chi^{2}/2}\operatorname{erfc}\left(\pm% \chi/\sqrt{2}\right),$
ⓘ Defines: $\chi$ (locally) and $d(\pm\chi)$ (locally) Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\mathrm{e}$ : base of natural logarithm, $z$ : complex variable and $a$ : parameter Permalink: http://dlmf.nist.gov/8.12.E19 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §8.12 and Ch.8

and

8.12.20	$\displaystyle A_{0}(\chi)$	$\displaystyle=1,$
	$\displaystyle A_{1}(\chi)$	$\displaystyle=\tfrac{1}{2}\chi+\tfrac{1}{6}\chi^{3},$
	$\displaystyle B_{1}(\chi)$	$\displaystyle=\tfrac{1}{3}+\tfrac{1}{6}\chi^{2}.$
ⓘ Defines: $A_{k}(\chi)$ (locally) and $B_{k}(\chi)$ (locally) Symbols: $k$ : nonnegative integer and $\chi$ Permalink: http://dlmf.nist.gov/8.12.E20 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §8.12 and Ch.8

Higher coefficients $A_{k}(\chi)$ , $B_{k}(\chi)$ , up to $k=8$ , are given in Paris (2002b).

Lastly, a uniform approximation for $\Gamma\left(a,ax\right)$ for large $a$ , with error bounds, can be found in Dunster (1996a).

For other uniform asymptotic approximations of the incomplete gamma functions in terms of the function $\operatorname{erfc}$ see Paris (2002b) and Dunster (1996a).

Inverse Function

ⓘ

Keywords:: asymptotic approximations and expansions, for inverse function, incomplete gamma functions, inverse incomplete gamma function, large variable and/or large parameter, uniform for large parameter
See also:: Annotations for §8.12 and Ch.8

For asymptotic expansions, as $a\to\infty$ , of the inverse function $x=x(a,q)$ that satisfies the equation

8.12.21		$Q\left(a,x\right)=q$
ⓘ Symbols: $Q\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function and $a$ : parameter Permalink: http://dlmf.nist.gov/8.12.E21 Encodings: TeX, pMML, png See also: Annotations for §8.12, §8.12 and Ch.8

see Temme (1992a). These expansions involve the inverse error function $\operatorname{inverfc}\left(x\right)$ (§7.17), and are uniform with respect to $q\in[0,1]$ . As a special case,

8.12.22		$x(a,\tfrac{1}{2})\sim a-\tfrac{1}{3}+\tfrac{8}{405}a^{-1}+\tfrac{184}{25515}a^% {-2}+\tfrac{2248}{34\;44525}a^{-3}+\cdots,$
$a\to\infty$ .
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion and $a$ : parameter Permalink: http://dlmf.nist.gov/8.12.E22 Encodings: TeX, pMML, png See also: Annotations for §8.12, §8.12 and Ch.8

8.11 Asymptotic Approximations and Expansions 8.13 Zeros

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§8.12 Uniform Asymptotic Expansions for Large Parameter

Inverse Function

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