The hypergeometric function $_2F_1(\large \frac{1}{2},\frac{1}{2},1;\frac{1-\frac{u}{\Lambda^2}} {2} \large)$ at large $\mid u\mid$ can be approximated by $$ -\frac{\Lambda}{\pi} \sqrt{\frac{2}{u}} \ln(\frac{u}{\Lambda^2})$$

How to approximate hypergeometric function at large argument ?


  • $\begingroup$ Unless I misunderstand the question, it seems you already gave the answer. Are you looking for a proof then? $\endgroup$ Nov 29, 2013 at 8:51
  • $\begingroup$ Yes, I do not know how to get it. $\endgroup$
    – user9527
    Nov 29, 2013 at 8:56
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    $\begingroup$ The ${}_2F_1$ series has radius of convergence $1$, so to discuss values $z>1$ you must use the analytic continuation of that series. Aren't there some Gaussian relations that write such a ${}_2F_1$ in terms of another ${}_2F_1$ with other arguments, that will help near $z=\infty$? $\endgroup$ Nov 29, 2013 at 14:12
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    $\begingroup$ By the way, Maple says ${}_2F_1(1/2,1/2;1;z) = (2/\pi)K(\sqrt{z})$ in terms of complete elliptic integral $K$. $\endgroup$ Nov 29, 2013 at 14:15
  • $\begingroup$ functions.wolfram.com has formulas for behavior at infinity: functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/… But like the original question, no references or proofs are included. $\endgroup$ Nov 29, 2013 at 14:23

1 Answer 1


According to the wikipedia article, this case (of integer third argument to the hypergeometric function) is discussed in Abramowitz and Stegun.

EDIT While the comment is correct, the answer is in Abramowitz and Stegun. In particular, check out equations 15.4.8 and 15.4.9. See also 15.5.16-17, and see also the transformation formulas in 15.3, which transform the point at infinity to your favorite other singular points.

  • $\begingroup$ It is about $z \sim 0$ in Wiki, and for discussion about $z \sim \infty$, it requires $a-b$ is a integer which is not satisfied here. $\endgroup$
    – user9527
    Nov 30, 2013 at 3:13
  • $\begingroup$ @user24304 See the edit. $\endgroup$
    – Igor Rivin
    Dec 1, 2013 at 3:32
  • $\begingroup$ @IgorRivin I'm following along and not really seeing a log in Abramowitz & Stegun. OTOH, their short section 15.7 "Asymptotic Expansions" does refer to Erdélyi et al., Higher transcendental functions, vol. 1, which says most clearly in its section 2.3.2 "Asymptotic expansions" (pp. 75-76):... $\endgroup$ Dec 1, 2013 at 4:37
  • $\begingroup$ ..."The behavior of the solutions [F(a,b,c,z)] of the hypergeometric equation for large values of $|z|$ can be fully described by means of the formulas for the analytic continuation into the neighborhood of the point at $z = \infty$. Unless $a — b$ is an integer, every solution $u(z)$ can be put into the form $u(z) = \lambda_1z^{-a} +\lambda_2z^{-b} + O(z^{-a-1}) + O(z^{-b-1})$ where $\lambda_1$ and $\lambda_2$ are constants; if $a-b$ is an integer, $z^{-a}$ or $z^{-b}$ has to be multiplied by a factor $\log z$." $\endgroup$ Dec 1, 2013 at 4:42
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    $\begingroup$ For a more modern (and freely available online) resource, see the corresponding chapter in the DLMF, A&S's successor. $\endgroup$ Feb 1, 2015 at 11:32

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