# §13.7 Asymptotic Expansions for Large Argument

## §13.7(i) Poincaré-Type Expansions

As $x\to\infty$

 13.7.1 ${\mathbf{M}}\left(a,b,x\right)\sim\frac{e^{x}x^{a-b}}{\Gamma\left(a\right)}% \sum_{s=0}^{\infty}\frac{{\left(1-a\right)_{s}}{\left(b-a\right)_{s}}}{s!}x^{-% s},$

provided that $a\neq 0,-1,\dots$.

As $z\to\infty$

 13.7.2 ${\mathbf{M}}\left(a,b,z\right)\sim\frac{e^{z}z^{a-b}}{\Gamma\left(a\right)}% \sum_{s=0}^{\infty}\frac{{\left(1-a\right)_{s}}{\left(b-a\right)_{s}}}{s!}z^{-% s}+\frac{e^{\pm\pi\mathrm{i}a}z^{-a}}{\Gamma\left(b-a\right)}\sum_{s=0}^{% \infty}\frac{{\left(a\right)_{s}}{\left(a-b+1\right)_{s}}}{s!}(-z)^{-s},$ $-\frac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z\leq\frac{3}{2}\pi-\delta$,

unless $a=0,-1,\dots$ and $b-a=0,-1,\dots$. Here $\delta$ denotes an arbitrary small positive constant. Also,

 13.7.3 $U\left(a,b,z\right)\sim z^{-a}\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}{% \left(a-b+1\right)_{s}}}{s!}(-z)^{-s},$ $|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta$.

## §13.7(ii) Error Bounds

 13.7.4 $U\left(a,b,z\right)=z^{-a}\sum_{s=0}^{n-1}\frac{{\left(a\right)_{s}}{\left(a-b% +1\right)_{s}}}{s!}(-z)^{-s}+\varepsilon_{n}(z),$

where

 13.7.5 $\left|\varepsilon_{n}(z)\right|,~{}\beta^{-1}\left|\varepsilon_{n}^{\prime}(z)% \right|\leq 2\alpha C_{n}\left|\frac{{\left(a\right)_{n}}{\left(a-b+1\right)_{% n}}}{n!z^{a+n}}\right|\exp\left(\frac{2\alpha\rho C_{1}}{|z|}\right),$ ⓘ Defines: $\varepsilon_{n}(z)$: function (locally) Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\exp\NVar{z}$: exponential function, $!$: factorial (as in $n!$), $n$: nonnegative integer, $z$: complex variable, $C_{n}$: coefficient, $\alpha$, $\beta$ and $\rho$ Referenced by: §13.2(i) Permalink: http://dlmf.nist.gov/13.7.E5 Encodings: TeX, pMML, png See also: Annotations for §13.7(ii), §13.7 and Ch.13

and with the notation of Figure 13.7.1

 13.7.6 $C_{n}=1,\quad\chi(n),\quad\left(\chi(n)+\sigma\nu^{2}n\right)\nu^{n},$ ⓘ Defines: $C_{n}$: coefficient (locally) Symbols: $n$: nonnegative integer, $\sigma$, $\nu$ and $\chi(n)$: function Permalink: http://dlmf.nist.gov/13.7.E6 Encodings: TeX, pMML, png See also: Annotations for §13.7(ii), §13.7 and Ch.13

according as

 13.7.7 $z\in\textbf{R}_{1},\quad z\in\textbf{R}_{2}\cup\overline{\textbf{R}}_{2},\quad z% \in\textbf{R}_{3}\cup\overline{\textbf{R}}_{3},$ ⓘ Symbols: $\overline{\NVar{z}}$: complex conjugate, $\in$: element of, $\cup$: union and $z$: complex variable Permalink: http://dlmf.nist.gov/13.7.E7 Encodings: TeX, pMML, png See also: Annotations for §13.7(ii), §13.7 and Ch.13

respectively, with

 13.7.8 $\displaystyle\sigma$ $\displaystyle=\left|\ifrac{(b-2a)}{z}\right|$, $\displaystyle\nu$ $\displaystyle=\left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-4\sigma^{2}}\right)^{-% \ifrac{1}{2}}$, $\displaystyle\chi(n)$ $\displaystyle=\sqrt{\pi}\Gamma\left(\tfrac{1}{2}n+1\right)/\Gamma\left(\tfrac{% 1}{2}n+\tfrac{1}{2}\right)$. ⓘ Defines: $\sigma$ (locally), $\nu$ (locally) and $\chi(n)$: function (locally) Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer and $z$: complex variable Permalink: http://dlmf.nist.gov/13.7.E8 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §13.7(ii), §13.7 and Ch.13

Also, when $z\in\textbf{R}_{1}\cup\textbf{R}_{2}\cup\overline{\textbf{R}}_{2}$

 13.7.9 $\displaystyle\alpha$ $\displaystyle=\frac{1}{1-\sigma}$, $\displaystyle\beta$ $\displaystyle=\frac{1-\sigma^{2}+\sigma|z|^{-1}}{2(1-\sigma)}$, $\displaystyle\rho$ $\displaystyle=\tfrac{1}{2}\left|2a^{2}-2ab+b\right|+\frac{\sigma(1+\frac{1}{4}% \sigma)}{(1-\sigma)^{2}}$, ⓘ Defines: $\alpha$ (locally), $\beta$ (locally) and $\rho$ (locally) Symbols: $z$: complex variable and $\sigma$ Referenced by: §13.7(ii), §13.7(ii) Permalink: http://dlmf.nist.gov/13.7.E9 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §13.7(ii), §13.7 and Ch.13

and when $z\in\textbf{R}_{3}\cup\overline{\textbf{R}}_{3}$ $\sigma$ is replaced by $\nu\sigma$ and $|z|^{-1}$ is replaced by $\nu|z|^{-1}$ everywhere in (13.7.9).

For numerical values of $\chi(n)$ see Table 9.7.1.

Corresponding error bounds for (13.7.2) can be constructed by combining (13.2.41) with (13.7.4)–(13.7.9).

## §13.7(iii) Exponentially-Improved Expansion

Let

 13.7.10 $U\left(a,b,z\right)=z^{-a}\sum_{s=0}^{n-1}\frac{{\left(a\right)_{s}}{\left(a-b% +1\right)_{s}}}{s!}(-z)^{-s}+R_{n}(a,b,z),$

and

 13.7.11 $R_{n}(a,b,z)=\frac{(-1)^{n}2\pi z^{a-b}}{\Gamma\left(a\right)\Gamma\left(a-b+1% \right)}\left(\sum_{s=0}^{m-1}\frac{{\left(1-a\right)_{s}}{\left(b-a\right)_{s% }}}{s!}(-z)^{-s}G_{n+2a-b-s}(z)+{\left(1-a\right)_{m}}{\left(b-a\right)_{m}}R_% {m,n}(a,b,z)\right),$ ⓘ Defines: $R_{n}(a,b,z)$: remainder (locally) Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: integer, $n$: nonnegative integer, $s$: nonnegative integer, $z$: complex variable and $G_{p}(z)$: expansion Referenced by: §13.29(i) Permalink: http://dlmf.nist.gov/13.7.E11 Encodings: TeX, pMML, png See also: Annotations for §13.7(iii), §13.7 and Ch.13

where $m$ is an arbitrary nonnegative integer, and

 13.7.12 $G_{p}(z)=\frac{e^{z}}{2\pi}\Gamma\left(p\right)\Gamma\left(1-p,z\right).$ ⓘ Defines: $G_{p}(z)$: expansion (locally) Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\pi$: 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 and $z$: complex variable Permalink: http://dlmf.nist.gov/13.7.E12 Encodings: TeX, pMML, png See also: Annotations for §13.7(iii), §13.7 and Ch.13

(For the notation see §8.2(i).) Then as $z\to\infty$ with $\left|\left|z\right|-n\right|$ bounded and $a,b,m$ fixed

 13.7.13 $R_{m,n}(a,b,z)=\begin{cases}O\left(e^{-|z|}z^{-m}\right),&|\operatorname{ph}z|% \leq\pi,\\ O\left(e^{z}z^{-m}\right),&\pi\leq|\operatorname{ph}z|\leq\tfrac{5}{2}\pi-% \delta.\\ \end{cases}$

For proofs see Olver (1991b, 1993a). For the special case $\operatorname{ph}z=\pm\pi$ see Paris (2013). For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver (1995a).