# §13.7 Asymptotic Expansions for Large Argument

## §13.7(i) Poincaré-Type Expansions

As $x\to\infty$

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

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

As $z\to\infty$

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

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

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

## §13.7(ii) Error Bounds

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

where

 13.7.5 $\left|\varepsilon_{n}(z)\right|,~{}\beta^{-1}\left|\varepsilon_{n}^{\prime}(z)% \right|\leq 2\alpha C_{n}\left|\frac{{\left(a\right)_{n}}{\left(a-b+1\right)_{% n}}}{n!z^{a+n}}\right|\mathop{\exp\/}\nolimits\!\left(\frac{2\alpha\rho C_{1}}% {|z|}\right),$ Defines: $\varepsilon_{n}(z)$: function (locally) Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\mathop{\exp\/}\nolimits\NVar{z}$: 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)

and with the notation of Figure 13.7.1

 13.7.6 $\displaystyle C_{n}$ $\displaystyle=1$, $\chi(n)$, $\left(\chi(n)+\sigma\nu^{2}n\right)\nu^{n}$, 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, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 13.7(ii)

according as

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

respectively, with

 13.7.8 $\displaystyle\sigma$ $\displaystyle=\left|\ifrac{(b-2a)}{z}\right|$, $\displaystyle\nu$ $\displaystyle=\left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-4\sigma^{2}}\right)^{-% \ifrac{1}{2}}$, $\displaystyle\chi(n)$ $\displaystyle=\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}n+1% \right)/\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}n+\tfrac{1}{2}\right)$. Defines: $\sigma$ (locally), $\nu$ (locally) and $\chi(n)$: function (locally) Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: 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)

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)

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

and

 13.7.11 $R_{n}(a,b,z)=\frac{(-1)^{n}2\pi z^{a-b}}{\mathop{\Gamma\/}\nolimits\!\left(a% \right)\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)}\left(\sum_{s=0}^{m-1}% \frac{{\left(1-a\right)_{s}}{\left(b-a\right)_{s}}}{s!}(-z)^{-s}G_{n+2a-b-s}(z% )+{\left(1-a\right)_{m}}{\left(b-a\right)_{m}}R_{m,n}(a,b,z)\right),$

where $m$ is an arbitrary nonnegative integer, and

 13.7.12 $G_{p}(z)=\frac{e^{z}}{2\pi}\mathop{\Gamma\/}\nolimits\!\left(p\right)\mathop{% \Gamma\/}\nolimits\!\left(1-p,z\right).$ Defines: $G_{p}(z)$: expansion (locally) Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: 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)

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

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