# §8.12 Uniform Asymptotic Expansions for Large Parameter

Define

 8.12.1 $\displaystyle\lambda$ $\displaystyle=z/a,$ $\displaystyle\eta$ $\displaystyle=\left(2(\lambda-1-\ln\lambda)\right)^{1/2},$ ⓘ 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 $\displaystyle\tfrac{1}{2}\eta^{2}$ $\displaystyle=\lambda-1-\ln\lambda,$ $\displaystyle\frac{\mathrm{d}\eta}{\mathrm{d}\lambda}$ $\displaystyle=\frac{\lambda-1}{\lambda\eta}.$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $\ln\NVar{z}$: 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 $P\left(a,z\right)=\tfrac{1}{2}\operatorname{erfc}\left(-\eta\sqrt{a/2}\right)-% S(a,\eta),$
 8.12.4 $Q\left(a,z\right)=\tfrac{1}{2}\operatorname{erfc}\left(\eta\sqrt{a/2}\right)+S% (a,\eta),$
8.12.5 $\frac{e^{\pm\pi ia}}{2i\sin\left(\pi a\right)}Q\left(-a,ze^{\pm\pi i}\right)=% \pm\tfrac{1}{2}\operatorname{erfc}\left(\pm i\eta\sqrt{a/2}\right)-iT(a,\eta),$

and

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

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 $S(a,\eta)\sim\frac{e^{-\frac{1}{2}a\eta^{2}}}{\sqrt{2\pi a}}\sum_{k=0}^{\infty% }c_{k}(\eta)a^{-k},$
 8.12.8 $T(a,\eta)\sim\frac{e^{\frac{1}{2}a\eta^{2}}}{\sqrt{2\pi a}}\sum_{k=0}^{\infty}% c_{k}(\eta)(-a)^{-k},$

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 $\displaystyle c_{0}(\eta)$ $\displaystyle=\frac{1}{\mu}-\frac{1}{\eta},$ $\displaystyle c_{1}(\eta)$ $\displaystyle=\frac{1}{\eta^{3}}-\frac{1}{\mu^{3}}-\frac{1}{\mu^{2}}-\frac{1}{% 12\mu},$ ⓘ 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 $c_{k}(\eta)=\frac{1}{\eta}\frac{\mathrm{d}}{\mathrm{d}\eta}c_{k-1}(\eta)+(-1)^% {k}\frac{g_{k}}{\mu},$ $k=1,2,\dots$,

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 $c_{k}(\eta)=\sum_{n=0}^{\infty}d_{k,n}\eta^{n},$ $|\eta|<2\sqrt{\pi}$,

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 $\lambda-1=\eta+\tfrac{1}{3}\eta^{2}+\sum_{n=3}^{\infty}\alpha_{n}\eta^{n},$ $|\eta|<2\sqrt{\pi}$.

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 $Q\left(a,a\right)\sim\frac{1}{2}+\frac{1}{\sqrt{2\pi a}}\sum_{k=0}^{\infty}c_{% k}(0)a^{-k},$ $|\operatorname{ph}a|\leq\pi-\delta$,
 8.12.16 $\frac{e^{\pm\pi ia}}{2i\sin\left(\pi a\right)}Q\left(-a,ae^{\pm\pi i}\right)% \sim\pm\frac{1}{2}-\frac{i}{\sqrt{2\pi a}}\sum_{k=0}^{\infty}c_{k}(0)(-a)^{-k},$ $|\operatorname{ph}a|\leq\pi-\delta$,

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: 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 $\Re z-a\leq 0$ for $P\left(a,z\right)$ and $\Re\left(z-a\right)\geq 0$ for $Q\left(a,z\right)$. Here

 8.12.19 $\displaystyle\chi$ $\displaystyle=(z-a)/\sqrt{z},$ $\displaystyle d(\pm\chi)$ $\displaystyle=\sqrt{\tfrac{1}{2}\pi}e^{\chi^{2}/2}\operatorname{erfc}\left(\pm% \chi/\sqrt{2}\right),$ ⓘ 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

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