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It is known that at $x=1$, the following continued fraction represents $\frac{4}{\pi}$ and can be approximated rapidly using Gauss' Arithmetic Geometric mean. $$C(x) = x + \frac{1^{2}}{2x + \frac{3^{2}}{2x + \frac{5^{2}}{2x + \frac{7^{2}}{2x + \cdots}}}}$$

Are there any other $x$ that $C(x)$ can be approximated through AGM quickly? Is there any connection to elliptic integrals?

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  • $\begingroup$ Mathworld says that $C(1) = 4/\pi$, not $\pi$. $\endgroup$
    – S. Carnahan ♦
    Commented Dec 23, 2012 at 5:30
  • $\begingroup$ I meant represents in a crude way. I can correct it. $\endgroup$
    – Turbo
    Commented Dec 23, 2012 at 8:30
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    $\begingroup$ Okay, sorry to jump on you like that. It is a good question. $\endgroup$
    – S. Carnahan ♦
    Commented Dec 23, 2012 at 15:35
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    $\begingroup$ The AGM doesn't make any use of that continued fraction for $4/\pi$, does it? So there's no reason to think there's any connection between AGM and $C(x)$, right? $\endgroup$
    – Gerry Myerson
    Commented Jun 26, 2013 at 1:58
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    $\begingroup$ $C(4n-3),~n\in\mathbb{N}$ is a rational multiple of $1/\pi$. $\endgroup$
    – Nemo
    Commented Feb 23, 2017 at 18:38

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This is not an answer.

As was noted in the comments, the function $C(x)$ can be expressed in terms of the Gamma function, see Corollary 1 on page 145 in Ramanujan's Notebook II which attributes this result to G. Bauer: Von einem Kettenbruche Euler's und einem Theorem von Wallis, Abh. Bayer. Akad. Wiss. 11(1872), 96 - 116 and to the first Ramanujan letter to Hardy.

$$ C(x) = \frac{\Gamma^2(\frac{x+1}{4})}{\Gamma^2(\frac{x+3}{4})} $$

The AGM function can be expressed in terms of complete elliptic integral of the first kind, as is noted on the Wikipedia page, which is given by Gauss hypergeometric function ${}_2F_1(-\frac{1}{2}, \frac{1}{2}; 1; k^2)$ whereas the function $C(x)$ is given by ${}_2F_1(-\frac{1}{2}, -\frac{1}{2};x,1)$.

There is certainly more to the story, see related question on math.stackexchange and equation (99) at MathWorld.

The explicit relation connecting Gamma values at certain rational points and elliptic integral can be found e.g. in the article of Borwein and Zucker, book Pi and the AGM by Borwein, Borwein and a more recent article by Vidunas.

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  • $\begingroup$ Note that the reference you cite attributes this corollary to Bauer 1872. $\endgroup$
    – Noam D. Elkies
    Commented Feb 27, 2017 at 0:53
  • $\begingroup$ @VitTucek thank you. do you know how fast $(99)$ converges? is it as fast as AGM? $\endgroup$
    – Turbo
    Commented Feb 27, 2017 at 7:03
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    $\begingroup$ @Turbo My understanding is that it also converges quadratically. $\endgroup$
    – Vít Tuček
    Commented Feb 27, 2017 at 10:16

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