Lemniscate constant (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Lemniscate constant" in English language version.

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Carlson, B. C. (2010), "Elliptic Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.

ams.org

Adlaj, Semjon (2012). "An Eloquent Formula for the Perimeter of an Ellipse" (PDF). American Mathematical Society. p. 1097. One might also observe that the length of the "sine" curve over half a period, that is, the length of the graph of the function ⁠ $\sin(t)$ ⁠ from the point where ⁠ $t=0$ ⁠ to the point where ⁠ $t=\pi$ ⁠ , is ${\sqrt {2}}l(1/{\sqrt {2}})=L+M$ . In this paper $M=1/G=\pi /\varpi$ and $L=\pi /M=G\pi =\varpi$ .

books.google.com

Finch 2003, p. 420. Finch, Steven R. (18 August 2003). Mathematical Constants. Cambridge University Press. pp. 420–422. ISBN 978-0-521-81805-6.
Cremona, J. E. (1997). Algorithms for Modular Elliptic Curves (2nd ed.). Cambridge University Press. ISBN 0521598206. p. 31, formula (2.8.10)

deepdyve.com

In particular, Schneider proved that the beta function $\mathrm {B} (a,b)$ is transcendental for all $a,b\in \mathbb {Q} \setminus \mathbb {Z}$ such that $a+b\notin \mathbb {Z} _{0}^{-}$ . The fact that $\varpi$ is transcendental follows from $\varpi ={\tfrac {1}{2}}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{2}}{\bigr )}$ and similarly for $B$ and $G$ from $\mathrm {B} {\bigl (}{\tfrac {1}{2}},{\tfrac {3}{4}}{\bigr )}.$
Schneider, Theodor (1941). "Zur Theorie der Abelschen Funktionen und Integrale". Journal für die reine und angewandte Mathematik. 183 (19): 110–128. doi:10.1515/crll.1941.183.110. S2CID 118624331.

doi.org

Neither of these proofs was rigorous from the modern point of view. See Cox 1984, p. 281 Cox, David A. (January 1984). "The Arithmetic-Geometric Mean of Gauss" (PDF). L'Enseignement Mathématique. 30 (2): 275–330. doi:10.5169/seals-53831. Retrieved 25 June 2022.
Todd, John (January 1975). "The lemniscate constants". Communications of the ACM. 18 (1): 14–19. doi:10.1145/360569.360580. S2CID 85873.
In particular, Siegel proved that if $\operatorname {G} _{4}(\omega _{1},\omega _{2})$ and $\operatorname {G} _{6}(\omega _{1},\omega _{2})$ with $\operatorname {Im} (\omega _{2}/\omega _{1})>0$ are algebraic, then $\omega _{1}$ or $\omega _{2}$ is transcendental. Here, $\operatorname {G} _{4}$ and $\operatorname {G} _{6}$ are Eisenstein series. The fact that $\varpi$ is transcendental follows from $\operatorname {G} _{4}(\varpi ,\varpi i)=1/15$ and $\operatorname {G} _{6}(\varpi ,\varpi i)=0.$
Apostol, T. M. (1990). Modular Functions and Dirichlet Series in Number Theory (Second ed.). Springer. p. 12. ISBN 0-387-97127-0.
Siegel, C. L. (1932). "Über die Perioden elliptischer Funktionen". Journal für die reine und angewandte Mathematik (in German). 167: 62–69. doi:10.1515/crll.1932.167.62.
In particular, Schneider proved that the beta function $\mathrm {B} (a,b)$ is transcendental for all $a,b\in \mathbb {Q} \setminus \mathbb {Z}$ such that $a+b\notin \mathbb {Z} _{0}^{-}$ . The fact that $\varpi$ is transcendental follows from $\varpi ={\tfrac {1}{2}}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{2}}{\bigr )}$ and similarly for $B$ and $G$ from $\mathrm {B} {\bigl (}{\tfrac {1}{2}},{\tfrac {3}{4}}{\bigr )}.$
Schneider, Theodor (1941). "Zur Theorie der Abelschen Funktionen und Integrale". Journal für die reine und angewandte Mathematik. 183 (19): 110–128. doi:10.1515/crll.1941.183.110. S2CID 118624331.
Cox 1984, p. 277. Cox, David A. (January 1984). "The Arithmetic-Geometric Mean of Gauss" (PDF). L'Enseignement Mathématique. 30 (2): 275–330. doi:10.5169/seals-53831. Retrieved 25 June 2022.
Rubin, Karl (1987). "Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication". Inventiones Mathematicae. 89 (3): 528. Bibcode:1987InMat..89..527R. doi:10.1007/BF01388984.
Hyde, Trevor (2014). "A Wallis product on clovers" (PDF). The American Mathematical Monthly. 121 (3): 237–243. doi:10.4169/amer.math.monthly.121.03.237. S2CID 34819500.
Bottazzini, Umberto; Gray, Jeremy (2013). Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory. Springer. doi:10.1007/978-1-4614-5725-1. ISBN 978-1-4614-5724-4. p. 60
Cox 1984, p. 307, eq. 2.21 for the first equality. The second equality can be proved by using the pentagonal number theorem. Cox, David A. (January 1984). "The Arithmetic-Geometric Mean of Gauss" (PDF). L'Enseignement Mathématique. 30 (2): 275–330. doi:10.5169/seals-53831. Retrieved 25 June 2022.
Blagouchine, Iaroslav V. (2014). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results". The Ramanujan Journal. 35 (1): 21–110. doi:10.1007/s11139-013-9528-5. S2CID 120943474.
Cox 1984, p. 313. Cox, David A. (January 1984). "The Arithmetic-Geometric Mean of Gauss" (PDF). L'Enseignement Mathématique. 30 (2): 275–330. doi:10.5169/seals-53831. Retrieved 25 June 2022.
Cox 1984, p. 312. Cox, David A. (January 1984). "The Arithmetic-Geometric Mean of Gauss" (PDF). L'Enseignement Mathématique. 30 (2): 275–330. doi:10.5169/seals-53831. Retrieved 25 June 2022.

eudml.org

In particular, Siegel proved that if $\operatorname {G} _{4}(\omega _{1},\omega _{2})$ and $\operatorname {G} _{6}(\omega _{1},\omega _{2})$ with $\operatorname {Im} (\omega _{2}/\omega _{1})>0$ are algebraic, then $\omega _{1}$ or $\omega _{2}$ is transcendental. Here, $\operatorname {G} _{4}$ and $\operatorname {G} _{6}$ are Eisenstein series. The fact that $\varpi$ is transcendental follows from $\operatorname {G} _{4}(\varpi ,\varpi i)=1/15$ and $\operatorname {G} _{6}(\varpi ,\varpi i)=0.$
Apostol, T. M. (1990). Modular Functions and Dirichlet Series in Number Theory (Second ed.). Springer. p. 12. ISBN 0-387-97127-0.
Siegel, C. L. (1932). "Über die Perioden elliptischer Funktionen". Journal für die reine und angewandte Mathematik (in German). 167: 62–69. doi:10.1515/crll.1932.167.62.
Rubin, Karl (1987). "Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication". Inventiones Mathematicae. 89 (3): 528. Bibcode:1987InMat..89..527R. doi:10.1007/BF01388984.

harvard.edu

ui.adsabs.harvard.edu

Rubin, Karl (1987). "Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication". Inventiones Mathematicae. 89 (3): 528. Bibcode:1987InMat..89..527R. doi:10.1007/BF01388984.

lmfdb.org

"Elliptic curve with LMFDB label 32.a3 (Cremona label 32a2)". The L-functions and modular forms database.
"Newform orbit 1.12.a.a". The L-functions and modular forms database.

nist.gov

dlmf.nist.gov

Carlson, B. C. (2010), "Elliptic Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.

numberworld.org

"Lemniscate Constant".
"Records set by y-cruncher". numberworld.org. Retrieved 2024-08-20.

oeis.org

"A064853 - Oeis".
"A014549 - Oeis".
"A085565 - Oeis". and "A076390 - Oeis".
"A113847 - Oeis".
"A062540 - OEIS". oeis.org. Retrieved 2022-09-14.
"A053002 - OEIS". oeis.org.
"A068467 - Oeis".

researchgate.net

Blagouchine, Iaroslav V. (2014). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results". The Ramanujan Journal. 35 (1): 21–110. doi:10.1007/s11139-013-9528-5. S2CID 120943474.

semanticscholar.org

api.semanticscholar.org

Todd, John (January 1975). "The lemniscate constants". Communications of the ACM. 18 (1): 14–19. doi:10.1145/360569.360580. S2CID 85873.
In particular, Schneider proved that the beta function $\mathrm {B} (a,b)$ is transcendental for all $a,b\in \mathbb {Q} \setminus \mathbb {Z}$ such that $a+b\notin \mathbb {Z} _{0}^{-}$ . The fact that $\varpi$ is transcendental follows from $\varpi ={\tfrac {1}{2}}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{2}}{\bigr )}$ and similarly for $B$ and $G$ from $\mathrm {B} {\bigl (}{\tfrac {1}{2}},{\tfrac {3}{4}}{\bigr )}.$
Schneider, Theodor (1941). "Zur Theorie der Abelschen Funktionen und Integrale". Journal für die reine und angewandte Mathematik. 183 (19): 110–128. doi:10.1515/crll.1941.183.110. S2CID 118624331.
Hyde, Trevor (2014). "A Wallis product on clovers" (PDF). The American Mathematical Monthly. 121 (3): 237–243. doi:10.4169/amer.math.monthly.121.03.237. S2CID 34819500.
Blagouchine, Iaroslav V. (2014). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results". The Ramanujan Journal. 35 (1): 21–110. doi:10.1007/s11139-013-9528-5. S2CID 120943474.

uchicago.edu

math.uchicago.edu

Hyde, Trevor (2014). "A Wallis product on clovers" (PDF). The American Mathematical Monthly. 121 (3): 237–243. doi:10.4169/amer.math.monthly.121.03.237. S2CID 34819500.

umn.edu

www-users.math.umn.edu

Garrett, Paul. "Level-one elliptic modular forms" (PDF). University of Minnesota. p. 11—13

uni-goettingen.de

gdz.sub.uni-goettingen.de

See:
- Gauss, C. F. (1866). Werke (Band III) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen.
- See:
  - Gauss, C. F. (1866). Werke (Band III) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen.

uu.nl

webspace.science.uu.nl

Neither of these proofs was rigorous from the modern point of view. See Cox 1984, p. 281 Cox, David A. (January 1984). "The Arithmetic-Geometric Mean of Gauss" (PDF). L'Enseignement Mathématique. 30 (2): 275–330. doi:10.5169/seals-53831. Retrieved 25 June 2022.
Cox 1984, p. 277. Cox, David A. (January 1984). "The Arithmetic-Geometric Mean of Gauss" (PDF). L'Enseignement Mathématique. 30 (2): 275–330. doi:10.5169/seals-53831. Retrieved 25 June 2022.
Cox 1984, p. 307, eq. 2.21 for the first equality. The second equality can be proved by using the pentagonal number theorem. Cox, David A. (January 1984). "The Arithmetic-Geometric Mean of Gauss" (PDF). L'Enseignement Mathématique. 30 (2): 275–330. doi:10.5169/seals-53831. Retrieved 25 June 2022.
Cox 1984, p. 313. Cox, David A. (January 1984). "The Arithmetic-Geometric Mean of Gauss" (PDF). L'Enseignement Mathématique. 30 (2): 275–330. doi:10.5169/seals-53831. Retrieved 25 June 2022.
Cox 1984, p. 312. Cox, David A. (January 1984). "The Arithmetic-Geometric Mean of Gauss" (PDF). L'Enseignement Mathématique. 30 (2): 275–330. doi:10.5169/seals-53831. Retrieved 25 June 2022.