15 Hypergeometric Function Properties 15.11 Riemann’s Differential Equation 15.13 Zeros

§15.12 Asymptotic Approximations

Permalink:: http://dlmf.nist.gov/15.12

§15.12(i) Large Variable
§15.12(ii) Large
§15.12(iii) Other Large Parameters

§15.12(i) Large Variable

A&S Ref:: 15.7
Keywords:: hypergeometric function
Permalink:: http://dlmf.nist.gov/15.12.i

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

§15.12(ii) Large

Notes:: See Wagner (1988). The region of validity given in Luke (1969a, p. 235) is incorrect.
A&S Ref:: 15.7
Keywords:: hypergeometric function
Referenced by:: §14.15(i), §15.19(iv), §8.18(i)
Permalink:: http://dlmf.nist.gov/15.12.ii

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

(a)

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

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

$\realpart{z}=\tfrac{1}{2}$ and $\left|\mathop{\mathrm{ph}\/}\nolimits c\right|\leq\pi-\delta$ .
(d)

$\realpart{z}>\tfrac{1}{2}$ and $\alpha_{{-}}-\tfrac{1}{2}\pi+\delta\leq\mathop{\mathrm{ph}\/}\nolimits c\leq% \alpha_{{+}}+\tfrac{1}{2}\pi-\delta$ , where

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

Symbols:

$\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : complex variable and $\alpha_{{\pm}}$

Permalink:

http://dlmf.nist.gov/15.12.E1

Encodings:

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

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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : : factorial, : nonnegative integer, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter and : integer
Referenced by:: §15.19(iv)
Permalink:: http://dlmf.nist.gov/15.12.E2
Encodings:: TeX, pMML, png

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

§15.12(iii) Other Large Parameters

Notes:: For (15.12.3) see Luke (1969a, §7.2) and Olver (1997b, p. 162). The sector of validity given in the first reference is incorrect. See the third footnote in the second reference.
Keywords:: hypergeometric function
Referenced by:: §15.19(iv), §2.8(iii), §2.8(iv), §2.8(vi)
Permalink:: http://dlmf.nist.gov/15.12.iii

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

As $\lambda\to\infty$ ,

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\sim$ : asymptotic equality, : nonnegative integer, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter and $q_{s}(z)$ : coefficients
Referenced by:: §15.12(iii), §15.12(iii), §15.19(iv)
Permalink:: http://dlmf.nist.gov/15.12.E3
Encodings:: TeX, pMML, png

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

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

Symbols:: : base of exponential function, : nonnegative integer, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter and $q_{s}(z)$ : coefficients
Permalink:: http://dlmf.nist.gov/15.12.E4
Encodings:: TeX, pMML, png

If $|\mathop{\mathrm{ph}\/}\nolimits\!\left(1-z\right)|<\pi$ , then (15.12.3) applies when $|\mathop{\mathrm{ph}\/}\nolimits\lambda|\leq\tfrac{1}{2}\pi-\delta$ . If $\realpart{z}\leq\tfrac{1}{2}$ , then (15.12.3) applies when $|\mathop{\mathrm{ph}\/}\nolimits\lambda|\leq\pi-\delta$ .

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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/15.12.E5
Encodings:: TeX, pMML, png

where

15.12.6 $\zeta=\mathop{\mathrm{arccosh}\/}\nolimits z.$

Symbols:: $\mathop{\mathrm{arccosh}\/}\nolimits z$ : inverse hyperbolic cosine function, : complex variable and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/15.12.E6
Encodings:: TeX, pMML, png

For $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ 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 $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ , then as $\lambda\to\infty$ with $|\mathop{\mathrm{ph}\/}\nolimits\lambda|\leq\pi-\delta$ ,

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter and $\alpha$
Permalink:: http://dlmf.nist.gov/15.12.E7
Encodings:: TeX, pMML, png

where

15.12.8 $\alpha=\left(-2\mathop{\ln\/}\nolimits\!\left(1-\left(\frac{z-1}{z+1}\right)^{% 2}\right)\right)^{{1/2}},$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : complex variable and $\alpha$
Permalink:: http://dlmf.nist.gov/15.12.E8
Encodings:: TeX, pMML, png

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

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

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, : base of exponential function, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, $\zeta$ : change of variable, $\beta$ and $a_{j}(\zeta)$ : function
Referenced by:: §15.12(iii)
Permalink:: http://dlmf.nist.gov/15.12.E9
Encodings:: TeX, pMML, png

where

15.12.10 $\zeta=\mathop{\mathrm{arccosh}\/}\nolimits\!\left(\tfrac{1}{4}z-1\right),$

Symbols:: $\mathop{\mathrm{arccosh}\/}\nolimits z$ : inverse hyperbolic cosine function, : complex variable and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/15.12.E10
Encodings:: TeX, pMML, png

15.12.11 $\beta=\left(-\frac{3}{2}\zeta+\frac{9}{4}\mathop{\ln\/}\nolimits\!\left(\frac{% 2+e^{\zeta}}{2+e^{{-\zeta}}}\right)\right)^{{1/3}},$

Symbols:: : base of exponential function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\zeta$ : change of variable and $\beta$
Permalink:: http://dlmf.nist.gov/15.12.E11
Encodings:: TeX, pMML, png

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

15.12.12

$a_{0}(\zeta)=\tfrac{1}{2}G_{0}(\beta)+\tfrac{1}{2}G_{0}(-\beta),$

$a_{1}(\zeta)=\left(\tfrac{1}{2}G_{0}(\beta)-\tfrac{1}{2}G_{0}(-\beta)\right)/\beta,$

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

where

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

Symbols:: : base of exponential function, : complex variable, : real or complex parameter, : real or complex parameter, : real or complex parameter, $\zeta$ : change of variable, $\beta$ and $G_{0}(\pm\zeta)$ : function
Permalink:: http://dlmf.nist.gov/15.12.E13
Encodings:: TeX, pMML, png

For $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ 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 $\mathop{F\/}\nolimits\!\left(a+e_{1}\lambda,b+e_{2}\lambda;c+e_{3}\lambda;z\right)$ can be obtained with $e_{j}=\pm 1$ or 0, . For more details see Olde Daalhuis (2010). For other extensions, see Wagner (1986) and Temme (2003).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 15.11 Riemann’s Differential Equation 15.13 Zeros

§15.12 Asymptotic Approximations

Contents

§15.12(i) Large Variable

§15.12(ii) Large

§15.12(iii) Other Large Parameters