Timeline for Dalzel's integral for $\pi$ and the lemniscate constant
Current License: CC BY-SA 3.0
7 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Aug 27, 2013 at 6:13 | comment | added | Zurab Silagadze | For example, 21/8 approximates $\varpi_4$ with relative accuracy $1.1\cdot 10^{-3}$, and 97/37 -- with relative accuracy $1.7\cdot 10^{-4}$. 22/7 approximates $\pi$ with relative accuracy $4.0\cdot 10^{-4}$. Beukers' article is indeed interesting and informative. Interestingly, it seems 17/7 approximates $\varpi_6\approx 2.42865$ with relative accuracy $3.3\cdot 10^{-5}$! | |
| Aug 26, 2013 at 16:38 | comment | added | Suvrit | You might be interested in the following paper nieuwarchief.nl/serie5/pdf/naw5-2000-01-4-372.pdf which describes (see page 7), more general Dalzel like formulae for $\pi$ (which essentially include the Dalzel formula as the "simplest" case) | |
| Aug 26, 2013 at 14:34 | comment | added | Igor Rivin | The backslashes are linebreaks, sorry. In any case, it seems that to get an unusually good approximation, you have to go to the 168 term. | |
| Aug 26, 2013 at 14:33 | comment | added | Igor Rivin | Here is the first hundred terms of the continued fraction for $\omega_2/2$ (so just the integral, without the factor of two): {1, 3, 4, 1, 1, 1, 5, 2, 1, 4, 1, 6, 1, 1, 4, 4, 3, 4, 4, 1, 1, 1, 1, \ 21, 1, 9, 1, 3, 1, 2, 11, 1, 1, 1, 5, 6, 8, 1, 2, 1, 168, 1, 2, 1, 1, \ 3, 1, 2, 1, 1, 1, 2, 1, 3, 6, 2, 1, 1, 19, 3, 1, 43, 5, 2, 1, 1, 1, \ 3, 1, 1, 3, 1, 4, 1, 4, 1, 19, 1, 5, 3, 1, 3, 1, 4, 1, 3, 2, 1, 40, \ 2, 1, 5, 9, 4, 6, 2, 1, 3, 1, 1} | |
| Aug 26, 2013 at 12:59 | comment | added | Gerry Myerson | One thing that makes the Dalzel formula nice is that $22/7$ is such a good approximation to $\pi$. Do you know a good rational approximation to the lemniscate constant? | |
| Aug 26, 2013 at 7:12 | history | asked | Zurab Silagadze | CC BY-SA 3.0 |