§8.12 Uniform Asymptotic Expansions for Large Parameter

Notes:: See Temme (1979b, 1992a, 1996a), Paris (2002b), and Ferreira et al. (2005).
Keywords:: asymptotic approximations and expansions, incomplete gamma functions
Referenced by:: §8.25(iii)
Permalink:: http://dlmf.nist.gov/8.12

Define

8.12.1

$\lambda=z/a,$

$\eta=\left(2(\lambda-1-\mathop{\ln\/}\nolimits\lambda)\right)^{{1/2}},$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $z$ : complex variable, $a$ : parameter, $\eta(\lambda)$ and $\lambda$
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where the branch of the square root is continuous and satisfies $\eta(\lambda)\sim\lambda-1$ as $\lambda\to 1$ . Then

8.12.2

$\tfrac{1}{2}\eta^{2}=\lambda-1-\mathop{\ln\/}\nolimits\lambda,$

$\frac{d\eta}{d\lambda}=\frac{\lambda-1}{\lambda\eta}.$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\eta(\lambda)$ and $\lambda$
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Also, denote

8.12.3 $\mathop{P\/}\nolimits\!\left(a,z\right)=\tfrac{1}{2}\mathop{\mathrm{erfc}\/}\nolimits\!\left(-\eta\sqrt{a/2}\right)-S(a,\eta),$

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $\mathop{P\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function, $z$ : complex variable, $a$ : parameter, $\eta(\lambda)$ and $S(a,\eta)$ : function
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8.12.4 $\mathop{Q\/}\nolimits\!\left(a,z\right)=\tfrac{1}{2}\mathop{\mathrm{erfc}\/}\nolimits\!\left(\eta\sqrt{a/2}\right)+S(a,\eta),$

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $\mathop{Q\/}\nolimits\!\left(a,z\right)$ : 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
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8.12.5 $\mathop{\Gamma\/}\nolimits\!\left(a+1\right)\frac{e^{{\pm\pi ia}}}{2\pi i}\mathop{\Gamma\/}\nolimits\!\left(-a,ze^{{\pm\pi i}}\right)=\mp\tfrac{1}{2}\mathop{\mathrm{erfc}\/}\nolimits\!\left(\pm i\eta\sqrt{a/2}\right)+iT(a,\eta),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $e$ : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter, $\eta(\lambda)$ and $T(a,\eta)$ : function
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and

8.12.6 $z^{{-a}}\mathop{\gamma^{{*}}\/}\nolimits\!\left(-a,-z\right)=\mathop{\cos\/}\nolimits\!\left(\pi a\right)-2\mathop{\sin\/}\nolimits\!\left(\pi a\right)\left(\frac{e^{{\frac{1}{2}a\eta^{2}}}}{\sqrt{\pi}}\mathop{F\/}\nolimits\!\left(\eta\sqrt{a/2}\right)+T(a,\eta)\right),$

Symbols:: $\mathop{F\/}\nolimits\!\left(z\right)$ : 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, $z$ : complex variable, $a$ : parameter, $\eta(\lambda)$ and $T(a,\eta)$ : function
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where $\mathop{F\/}\nolimits\!\left(x\right)$ is Dawson’s integral; see §7.2(ii). Then as $a\to\infty$ in the sector $|\mathop{\mathrm{ph}\/}\nolimits a|\leq\pi-\delta(<\pi)$ ,

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:: $e$ : base of exponential function, $\sim$ : asymptotic equality, $a$ : 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
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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:: $e$ : base of exponential function, $\sim$ : asymptotic equality, $a$ : parameter, $k$ : nonnegative integer, $\eta(\lambda)$ , $T(a,\eta)$ : function and $c_{k}(\eta)$ : coefficients
Permalink:: http://dlmf.nist.gov/8.12.E8
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in each case uniformly with respect to $\lambda$ in the sector $|\mathop{\mathrm{ph}\/}\nolimits\lambda|\leq 2\pi-\delta$ ( $<2\pi$ ).

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

8.12.9

$c_{0}(\eta)=\frac{1}{\mu}-\frac{1}{\eta},$

$c_{1}(\eta)=\frac{1}{\eta^{3}}-\frac{1}{\mu^{3}}-\frac{1}{\mu^{2}}-\frac{1}{12\mu},$

Symbols:: $\eta(\lambda)$ , $\mu$ and $c_{k}(\eta)$ : coefficients
Referenced by:: §8.12
Permalink:: http://dlmf.nist.gov/8.12.E9
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8.12.10 $c_{k}(\eta)=\frac{1}{\eta}\frac{d}{d\eta}c_{{k-1}}(\eta)+(-1)^{k}\frac{g_{k}}{\mu},$ $k=1,2,\dots$ ,

Symbols:: $\frac{df}{dx}$ : 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
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where $g_{k}$ , $k=0,1,2,\dots$ , are the coefficients that appear in the asymptotic expansion (5.11.3) of $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ . 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:: $k$ : nonnegative integer, $n$ : nonnegative integer, $\eta(\lambda)$ , $d(\pm\chi)$ and $c_{k}(\eta)$ : coefficients
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where $d_{{0,0}}=-\tfrac{1}{3}$ ,

8.12.12

$d_{{0,n}}=(n+2)\alpha _{{n+2}},$ $n\geq 1$ ,

$d_{{k,n}}=(-1)^{k}g_{k}d_{{0,n}}+(n+2)d_{{k-1,n+2}},$ $n\geq 0$ , $k\geq 1$ ,

Symbols:: $k$ : nonnegative integer, $n$ : nonnegative integer, $d(\pm\chi)$ , $g_{k}$ : coefficients and $\alpha _{{n+2}}$
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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:: $n$ : nonnegative integer, $\eta(\lambda)$ , $\lambda$ and $\alpha _{{n+2}}$
Permalink:: http://dlmf.nist.gov/8.12.E13
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In particular,

8.12.14

$\alpha _{3}=\tfrac{1}{36},$

$\alpha _{4}=-\tfrac{1}{270},$

$\alpha _{5}=\tfrac{1}{4320},$

$\alpha _{6}=\tfrac{1}{17010},$

$\alpha _{7}=-\tfrac{139}{54\; 43200},$

$\alpha _{8}=\tfrac{1}{2\; 0 4120}.$

Symbols:: $\alpha _{{n+2}}$
Permalink:: http://dlmf.nist.gov/8.12.E14
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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 $\mathop{Q\/}\nolimits\!\left(a,a\right)\sim\frac{1}{2}+\frac{1}{\sqrt{2\pi a}}\sum _{{k=0}}^{\infty}c_{k}(0)a^{{-k}},$ $|\mathop{\mathrm{ph}\/}\nolimits a|\leq\pi-\delta$ ,

Symbols:: $\mathop{Q\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $a$ : parameter, $k$ : nonnegative integer, $\delta$ : small positive constant and $c_{k}(\eta)$ : coefficients
Referenced by:: §8.11(v)
Permalink:: http://dlmf.nist.gov/8.12.E15
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8.12.16 $\frac{e^{{\pm\pi ia}}}{2i\mathop{\sin\/}\nolimits\!\left(\pi a\right)}\mathop{Q\/}\nolimits\!\left(-a,ae^{{\pm\pi i}}\right)\sim\pm\frac{1}{2}-\frac{i}{\sqrt{2\pi a}}\sum _{{k=0}}^{\infty}c_{k}(0)(-a)^{{-k}},$ $|\mathop{\mathrm{ph}\/}\nolimits a|\leq\pi-\delta$ ,

Symbols:: $e$ : base of exponential function, $\mathop{Q\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $\mathop{\sin\/}\nolimits z$ : sine function, $a$ : parameter, $k$ : nonnegative integer, $\delta$ : small positive constant and $c_{k}(\eta)$ : coefficients
Permalink:: http://dlmf.nist.gov/8.12.E16
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where

8.12.17

$c_{0}(0)=-\tfrac{1}{3},$

$c_{1}(0)=-\tfrac{1}{540},$

$c_{2}(0)=\tfrac{25}{6048},$

$c_{3}(0)=\tfrac{101}{1\; 55520},$

$c_{4}(0)=-\tfrac{31\; 84811}{36951\; 55200},$

$c_{5}(0)=-\tfrac{27\; 45493}{81517\; 36320}.$

Symbols:: $c_{k}(\eta)$ : coefficients
Permalink:: http://dlmf.nist.gov/8.12.E17
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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 $\mathop{Q\/}\nolimits\!\left(a,z\right)$ .

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

8.12.18 $\rselection{\mathop{Q\/}\nolimits\!\left(a,z\right)\\ \mathop{P\/}\nolimits\!\left(a,z\right)}\sim\frac{z^{{a-\frac{1}{2}}}e^{{-z}}}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}{\left(d(\pm\chi)\sum _{{k=0}}^{\infty}\frac{A_{k}(\chi)}{z^{{k/2}}}\pm\sum _{{k=1}}^{\infty}\frac{B_{k}(\chi)}{z^{{k/2}}}\right)},$

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, $\mathop{Q\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function, $\sim$ : asymptotic equality, $z$ : complex variable, $a$ : parameter, $k$ : nonnegative integer, $\chi$ , $d(\pm\chi)$ , $A_{k}(\chi)$ and $B_{k}(\chi)$
Permalink:: http://dlmf.nist.gov/8.12.E18
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for $z\to\infty$ in $\left|\mathop{\mathrm{ph}\/}\nolimits z\right|<\frac{1}{2}\pi$ , with $\realpart{(z-a)}\leq 0$ for $\mathop{P\/}\nolimits\!\left(a,z\right)$ and $\realpart{(z-a)}\geq 0$ for $\mathop{Q\/}\nolimits\!\left(a,z\right)$ . Here

8.12.19

$\chi=(z-a)/\sqrt{z},$

$d(\pm\chi)=\sqrt{\tfrac{1}{2}\pi}e^{{\chi^{2}/2}}\mathop{\mathrm{erfc}\/}\nolimits\!\left(\pm\chi/\sqrt{2}\right),$

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $e$ : base of exponential function, $z$ : complex variable, $a$ : parameter, $\chi$ and $d(\pm\chi)$
Permalink:: http://dlmf.nist.gov/8.12.E19
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and

8.12.20

$A_{0}(\chi)=1,$

$A_{1}(\chi)=\tfrac{1}{2}\chi+\tfrac{1}{6}\chi^{3},$

$B_{1}(\chi)=\tfrac{1}{3}+\tfrac{1}{6}\chi^{2}.$

Symbols:: $k$ : nonnegative integer, $\chi$ , $A_{k}(\chi)$ and $B_{k}(\chi)$
Permalink:: http://dlmf.nist.gov/8.12.E20
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Higher coefficients $A_{k}(\chi)$ , $B_{k}(\chi)$ , up to $k=8$ , are given in Paris (2002b).

Lastly, a uniform approximation for $\mathop{\Gamma\/}\nolimits\!\left(a,ax\right)$ 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 $\mathop{\mathrm{erfc}\/}\nolimits$ see Paris (2002b) and Dunster (1996a).

¶ Inverse Function

Keywords:: asymptotic approximations and expansions, incomplete gamma functions, inverse incomplete gamma function

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

8.12.21 $\mathop{Q\/}\nolimits\!\left(a,x\right)=q$

Symbols:: $\mathop{Q\/}\nolimits\!\left(a,z\right)$ : normalized incomplete gamma function and $a$ : parameter
Permalink:: http://dlmf.nist.gov/8.12.E21
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see Temme (1992a). These expansions involve the inverse error function $\mathop{\mathrm{inverfc}\/}\nolimits\!\left(x\right)$ (§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:: $\sim$ : asymptotic equality and $a$ : parameter
Permalink:: http://dlmf.nist.gov/8.12.E22
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