Lemniscatic constant

The lemniscatic constant is a mathematical constant introduced by Carl Friedrich Gauß in 1798 . It is defined as the value of the elliptic integral

${\displaystyle \varpi =2\int _{0}^{1}{\frac {\mathrm {d} t}{\sqrt {1-t^{4}}}}}$= 2.62205 75542 92119 81046 48395 89891 11941 36827 54951 43162 ... ( continuation A062539 in OEIS )

and occurs when calculating the arc length of the entire lemniscate . Currently (as of March 17, 2020) 600,000,000,000 decimal places of the lemniscate constant are known. They were calculated by Seungmin Kim and Ian Cutress.

designation

Gauss consciously chose the Greek minuscule (spoken: script-pi or varpi) for the lemniscatic constant , an alternative spelling of , in order to draw on the analogy of the circle with its half circumference ${\displaystyle \varpi }$ ${\displaystyle \pi }$

${\displaystyle \pi =2\int _{0}^{1}{\frac {\mathrm {d} t}{\sqrt {1-t^{2}}}}}$

to remember. Ludwig Schlesinger probably first clarified the origin of this designation in Gauss : At first Gauss used the symbol to denote the lemniscate period , and from July 1798 he consistently used this size . ${\displaystyle \Pi }$ ${\displaystyle {\tfrac {\varpi }{2}}}$

In English there is also the (misleading) term pomega for the minuscule . ${\displaystyle \varpi }$

In the English-speaking world,

${\displaystyle G=\varpi /\pi }$= 0.83462 68416 74073 18628 14297 32799 04680 89939 93013 49034 ... (sequence A014549 in OEIS )

referred to as the Gaussian constant .

properties

With the beta function and the gamma function, the following applies ${\displaystyle \mathrm {B} }$ ${\displaystyle \Gamma }$

${\displaystyle \varpi ={\tfrac {1}{2}}\,\mathrm {B} ({\tfrac {1}{4}},{\tfrac {1}{2}})=\Gamma ({\tfrac {1}{4}})^{2}/{\bigl (}2{\sqrt {2\pi }}{\bigr )}.}$

Gauss found the relationship

${\displaystyle \varpi =\pi /M(1,{\sqrt {2}})}$

with the arithmetic-geometric mean and also gave a rapidly converging series ${\displaystyle M}$

${\displaystyle \varpi ={\frac {\pi }{\sqrt {2}}}\sum _{k=0}^{\infty }{\binom {2k}{k}}^{\!2}{\frac {1}{2^{5k}}}}$

with summands of the order of magnitude . The evaluation ${\displaystyle {\frac {1}{k2^{k}}}}$

${\displaystyle \varpi =2\sum _{k=0}^{\infty }{\binom {2k}{k}}{\frac {1}{(4k+1)2^{2k}}}}$

of the elliptic integral gives a similar series, but converges much more slowly because the terms are of the order of magnitude . The series converges in very quickly ${\displaystyle {\frac {1}{k^{3/2}}}}$

${\displaystyle \varpi =\pi {\biggl (}\sum _{k=-\infty }^{\infty }(-1)^{k}\,e^{-\pi k^{2}}{\biggr )}^{2}}$

with summands of the order of magnitude . ${\displaystyle e^{-\pi k^{2}}}$

In 1906 Niels Nielsen established a connection with Euler's constant with the help of the Kummer series of the gamma function : ${\displaystyle \gamma }$

${\displaystyle \log \varpi ={\tfrac {1}{2}}\gamma -{\tfrac {1}{2}}\log 2+\log \pi +{\frac {2}{\pi }}\sum _{k=1}^{\infty }(-1)^{k}{\frac {\log(2k+1)}{2k+1}}\,.}$

In 1937 Theodor Schneider proved the transcendence of . Gregory Chudnovsky showed in 1975 that and therefore also algebraically independent of . ${\displaystyle \varpi }$ ${\displaystyle \Gamma (1/4)}$ ${\displaystyle \varpi }$ ${\displaystyle \pi }$

See also

• Lemniscate sine

literature

• Theodor Schneider : Introduction to the transcendent numbers . Springer-Verlag, Berlin 1957, p. 64.
• Carl Ludwig Siegel : Transcendent Numbers . Bibliographisches Institut, Mannheim 1967, pp. 81–84.
• AI Markuschewitsch : Analytic Functions . Chapter 2 in: AN Kolmogorov , AP Juschkewitsch (Ed.): Mathematics of the 19th Century. Geometry, analytic function theory . Birkhäuser, Basel 1996, ISBN 3-7643-5048-2 , pp. 133-136.
• Jörg Arndt, Christoph Haenel: π. Algorithms, computers, arithmetic . 2nd edition, Springer, 2000, pp. 94–96 (here the Greek minuscule ϖ is typographically correct)
• Steven R. Finch: Gauss' Lemniscate constant , Chapter 6.1 in Mathematical constants , Cambridge University Press, Cambridge 2003, ISBN 0-521-81805-2 , pp. 420-423 (English)
• Hans Wußing , Olaf Neumann: Mathematical diary 1796–1814 by Carl Friedrich Gauß . With a historical introduction by Kurt-R. Beer man . Translated into German by Elisabeth Schuhmann. Reviewed and annotated by Hans Wußing and Olaf Neumann. 5th edition, 2005, entry [91a].

Web links

• Eric W. Weisstein: Lemniscate Constant and Gauss's Constant . In: MathWorld. (English)
• Follow A062540 in OEIS ( continued fraction expansion of π)
• Follow A053002 in OEIS ( continued fraction expansion of π / π)

Individual evidence

1. ↑ Alexander Jih-Hing Yee: Lemniscate Constant. September 8, 2019, accessed on March 17, 2020 . 
2. ↑ Niels Nielsen : Handbook of the theory of the gamma function . Teubner, Leipzig 1906, p. 201 (the correct factor before the sum is 2 / π instead of 2)
3. ^ Theodor Schneider : Arithmetic investigations of elliptical integrals (March 11, 1936), Mathematische Annalen 113, 1937, pp. 1–13
4. ↑ GV Choodnovsky : Algebraic independence of constants connected with the functions of analysis , Notices of the AMS 22, 1975, A-486 (English; preliminary report)
5. ↑ Gregory V. Chudnovsky : Contributions to the theory of transcendental numbers , American Mathematical Society, 1984, ISBN 0-8218-1500-8 , p. 8 (English)