# §15.12 Asymptotic Approximations

## §15.12(i) Large Variable

For the asymptotic behavior of $\mathbf{F}\left(a,b;c;z\right)$ as $z\to\infty$ with $a$, $b$, $c$ fixed, combine (15.2.2) with (15.8.2) or (15.8.8).

## §15.12(ii) Large $c$

Let $\delta$ denote an arbitrary small positive constant. Also let $a,b,z$ be real or complex and fixed, and at least one of the following conditions be satisfied:

• (a)

$a$ and/or $b\in\{0,-1,-2,\dots\}$.

• (b)

$\Re z<\tfrac{1}{2}$ and $|c+n|\geq\delta$ for all $n\in\{0,1,2,\dots\}$.

• (c)

$\Re z=\tfrac{1}{2}$ and $\left|\operatorname{ph}c\right|\leq\pi-\delta$.

• (d)

$\Re z>\tfrac{1}{2}$ and $\alpha_{-}-\tfrac{1}{2}\pi+\delta\leq\operatorname{ph}c\leq\alpha_{+}+\tfrac{1% }{2}\pi-\delta$, where

 15.12.1 $\alpha_{\pm}=\operatorname{arctan}\left(\frac{\operatorname{ph}z-\operatorname% {ph}\left(1-z\right)\mp\pi}{\ln|1-z^{-1}|}\right),$

with $z$ restricted so that $\pm\alpha_{\pm}\in[0,\tfrac{1}{2}\pi)$.

Then for fixed $m\in\{0,1,2,\dots\}$,

 15.12.2 $F\left(a,b;c;z\right)=\sum_{s=0}^{m-1}\frac{{\left(a\right)_{s}}{\left(b\right% )_{s}}}{{\left(c\right)_{s}}s!}z^{s}+O\left(c^{-m}\right),$ $|c|\to\infty$.

Similar results for other sectors are given in Wagner (1988). For the more general case in which $a^{2}=o\left(c\right)$ and $b^{2}=o\left(c\right)$ see Wagner (1990).

For large $b$ and $c$ with $c>b+1$ see López and Pagola (2011).

## §15.12(iii) Other Large Parameters

Again, throughout this subsection $\delta$ denotes an arbitrary small positive constant, and $a,b,c,z$ are real or complex and fixed.

As $\lambda\to\infty$,

 15.12.3 $F\left({a,b\atop c+\lambda};z\right)\sim\frac{\Gamma\left(c+\lambda\right)}{% \Gamma\left(c-b+\lambda\right)}\sum_{s=0}^{\infty}q_{s}(z){\left(b\right)_{s}}% \lambda^{-s-b},$

where $q_{0}(z)=1$ and $q_{s}(z)$, $s=1,2,\dots$, are defined by the generating function

 15.12.4 $\left(\frac{{\mathrm{e}}^{t}-1}{t}\right)^{b-1}{\mathrm{e}}^{t(1-c)}\left(1-z+% z{\mathrm{e}}^{-t}\right)^{-a}=\sum_{s=0}^{\infty}q_{s}(z)t^{s}.$

If $|\operatorname{ph}\left(1-z\right)|<\pi$, then (15.12.3) applies when $|\operatorname{ph}\lambda|\leq\tfrac{1}{2}\pi-\delta$. If $\Re z\leq\tfrac{1}{2}$, then (15.12.3) applies when $|\operatorname{ph}\lambda|\leq\pi-\delta$.

If $|\operatorname{ph}\left(z-1\right)|<\pi$, then as $\lambda\to\infty$ with $|\operatorname{ph}\lambda|\leq\pi-\delta$,

 15.12.5 $\mathbf{F}\left({a+\lambda,b-\lambda\atop c};\tfrac{1}{2}-\tfrac{1}{2}z\right)% =2^{(a+b-1)/2}\frac{(z+1)^{(c-a-b-1)/2}}{(z-1)^{c/2}}\sqrt{\zeta\sinh\zeta}% \left(\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b\right)^{1-c}\left(I_{c-1}\left((% \lambda+\tfrac{1}{2}a-\tfrac{1}{2}b)\zeta\right)(1+O(\lambda^{-2}))+\frac{I_{c% -2}\left((\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b)\zeta\right)}{2\lambda+a-b}\left% (\left(c-\tfrac{1}{2}\right)\left(c-\tfrac{3}{2}\right)\left(\frac{1}{\zeta}-% \coth\zeta\right)+\tfrac{1}{2}(2c-a-b-1)(a+b-1)\tanh\left(\tfrac{1}{2}\zeta% \right)+O(\lambda^{-2})\right)\right),$

where

 15.12.6 $\zeta=\operatorname{arccosh}z.$ ⓘ Symbols: $\operatorname{arccosh}\NVar{z}$: inverse hyperbolic cosine function, $z$: complex variable and $\zeta$: change of variable Permalink: http://dlmf.nist.gov/15.12.E6 Encodings: TeX, pMML, png See also: Annotations for §15.12(iii), §15.12 and Ch.15

For $I_{\nu}\left(z\right)$ see §10.25(ii). For this result and an extension to an asymptotic expansion with error bounds see Jones (2001).

See also Dunster (1999) where the asymptotics of Jacobi polynomials is described; compare (15.9.1).

If $|\operatorname{ph}z|<\pi$, then as $\lambda\to\infty$ with $|\operatorname{ph}\lambda|\leq\pi-\delta$,

 15.12.7 $F\left({a,b-\lambda\atop c+\lambda};-z\right)=2^{b-c+(1/2)}\left(\frac{z+1}{2% \sqrt{z}}\right)^{\lambda}\left(\lambda^{a/2}U\left(a-\tfrac{1}{2},-\alpha% \sqrt{\lambda}\right)\left((1+z)^{c-a-b}z^{1-c}\left(\frac{\alpha}{z-1}\right)% ^{1-a}+O(\lambda^{-1})\right)+\frac{\lambda^{\ifrac{(a-1)}{2}}}{\alpha}U\left(% a-\tfrac{3}{2},-\alpha\sqrt{\lambda}\right)\left((1+z)^{c-a-b}z^{1-c}\left(% \frac{\alpha}{z-1}\right)^{1-a}-2^{c-b-(\ifrac{1}{2})}\left(\frac{\alpha}{z-1}% \right)^{a}+O(\lambda^{-1})\right)\right),$

where

 15.12.8 $\alpha=\left(-2\ln\left(1-\left(\frac{z-1}{z+1}\right)^{2}\right)\right)^{1/2},$ ⓘ Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $z$: complex variable and $\alpha$ Permalink: http://dlmf.nist.gov/15.12.E8 Encodings: TeX, pMML, png See also: Annotations for §15.12(iii), §15.12 and Ch.15

with the branch chosen to be continuous and $\Re\alpha>0$ when $\Re\left(\ifrac{(z-1)}{(z+1)}\right)>0$. For $U\left(a,z\right)$ see §12.2, and for an extension to an asymptotic expansion see Olde Daalhuis (2003a).

If $|\operatorname{ph}z|<\pi$, then as $\lambda\to\infty$ with $|\operatorname{ph}\lambda|\leq\tfrac{1}{2}\pi-\delta$,

 15.12.9 $(z+1)^{3\lambda/2}(2\lambda)^{c-1}\mathbf{F}\left({a+\lambda,b+2\lambda\atop c% };-z\right)={\lambda^{-1/3}\left({\mathrm{e}}^{\pi\mathrm{i}(a-c+\lambda+(1/3)% )}\mathrm{Ai}\left({\mathrm{e}}^{-\ifrac{2\pi\mathrm{i}}{3}}\lambda^{\ifrac{2}% {3}}\beta^{2}\right)+{\mathrm{e}}^{\pi\mathrm{i}(c-a-\lambda-(1/3))}\mathrm{Ai% }\left({\mathrm{e}}^{\ifrac{2\pi\mathrm{i}}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}% \right)\right)\left(a_{0}(\zeta)+O(\lambda^{-1})\right)}+\lambda^{-2/3}\left({% \mathrm{e}}^{\pi\mathrm{i}(a-c+\lambda+(2/3))}\mathrm{Ai}'\left({\mathrm{e}}^{% -\ifrac{2\pi\mathrm{i}}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)+{\mathrm{e}}% ^{\pi\mathrm{i}(c-a-\lambda-(2/3))}\mathrm{Ai}'\left({\mathrm{e}}^{\ifrac{2\pi% \mathrm{i}}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)\right)\left(a_{1}(\zeta)% +O(\lambda^{-1})\right),$

where

 15.12.10 $\displaystyle\zeta$ $\displaystyle=\operatorname{arccosh}\left(\tfrac{1}{4}z-1\right),$ ⓘ Symbols: $\operatorname{arccosh}\NVar{z}$: inverse hyperbolic cosine function, $z$: complex variable and $\zeta$: change of variable Permalink: http://dlmf.nist.gov/15.12.E10 Encodings: TeX, pMML, png See also: Annotations for §15.12(iii), §15.12 and Ch.15 15.12.11 $\displaystyle\beta$ $\displaystyle=\left(-\frac{3}{2}\zeta+\frac{9}{4}\ln\left(\frac{2+{\mathrm{e}}% ^{\zeta}}{2+{\mathrm{e}}^{-\zeta}}\right)\right)^{1/3},$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\ln\NVar{z}$: principal branch of logarithm function, $\zeta$: change of variable and $\beta$ Permalink: http://dlmf.nist.gov/15.12.E11 Encodings: TeX, pMML, png See also: Annotations for §15.12(iii), §15.12 and Ch.15

with the branch chosen to be continuous and $\beta>0$ when $\zeta>0$. Also,

 15.12.12 $\displaystyle a_{0}(\zeta)$ $\displaystyle=\tfrac{1}{2}G_{0}(\beta)+\tfrac{1}{2}G_{0}(-\beta),$ $\displaystyle a_{1}(\zeta)$ $\displaystyle=\left(\tfrac{1}{2}G_{0}(\beta)-\tfrac{1}{2}G_{0}(-\beta)\right)/\beta,$ ⓘ Symbols: $\zeta$: change of variable, $\beta$, $a_{j}(\zeta)$: function and $G_{0}(\pm\zeta)$: function Permalink: http://dlmf.nist.gov/15.12.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §15.12(iii), §15.12 and Ch.15

where

 15.12.13 $G_{0}(\pm\beta)=\left(2+{\mathrm{e}}^{\pm\zeta}\right)^{c-b-(\ifrac{1}{2})}% \left(1+{\mathrm{e}}^{\pm\zeta}\right)^{a-c+(\ifrac{1}{2})}\left(z-1-{\mathrm{% e}}^{\pm\zeta}\right)^{-a+(\ifrac{1}{2})}\sqrt{\frac{\beta}{{\mathrm{e}}^{% \zeta}-{\mathrm{e}}^{-\zeta}}}.$

For $\mathrm{Ai}\left(z\right)$ see §9.2, and for further information and an extension to an asymptotic expansion see Olde Daalhuis (2003b). (Two errors in this reference are corrected in (15.12.9).)

By combination of the foregoing results of this subsection with the linear transformations of §15.8(i) and the connection formulas of §15.10(ii), similar asymptotic approximations for $F\left(a+e_{1}\lambda,b+e_{2}\lambda;c+e_{3}\lambda;z\right)$ can be obtained with $e_{j}=\pm 1$ or $0$, $j=1,2,3$. For more details see Farid Khwaja and Olde Daalhuis (2014). For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).