Lemniscate constant

In mathematics, the lemniscate constant $ϖ$ is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of $π$ for the circle.^[1] Equivalently, the perimeter of the lemniscate $(x^{2}+y^{2})^{2}=x^{2}-y^{2}$ is $2 ϖ$ . The lemniscate constant is closely related to the lemniscate elliptic functions and approximately equal to 2.62205755.^[2] It also appears in evaluation of the gamma and beta function at certain rational values. The symbol $ϖ$ is a cursive variant of $π$ ; see Pi § Variant pi.

Sometimes the quantities $2 ϖ$ or $ϖ/2$ are referred to as the lemniscate constant.^[3]^[4]

As of 2024 over 1.2 trillion digits of this constant have been calculated.^[5]

^
See:
- Gauss, C. F. (1866). Werke (Band III) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. p. 404
- Cox 1984, p. 281
- Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 199
- 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. 57
- Arakawa, Tsuneo; Ibukiyama, Tomoyoshi; Kaneko, Masanobu (2014). Bernoulli Numbers and Zeta Functions. Springer. ISBN 978-4-431-54918-5. p. 203
^
See:
- Finch 2003, p. 420
- Kobayashi, Hiroyuki; Takeuchi, Shingo (2019), "Applications of generalized trigonometric functions with two parameters", Communications on Pure & Applied Analysis, 18 (3): 1509–1521, arXiv:1903.07407, doi:10.3934/cpaa.2019072, S2CID 102487670
- Asai, Tetsuya (2007), Elliptic Gauss Sums and Hecke L-values at s=1, arXiv:0707.3711
- "A062539 - Oeis".
^ "A064853 - Oeis".
^ "Lemniscate Constant".
^ "Records set by y-cruncher". numberworld.org. Retrieved 2024-08-20.

[1] See:
Gauss, C. F. (1866). Werke (Band III) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. p. 404

Cox 1984, p. 281

Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 199

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. 57

Arakawa, Tsuneo; Ibukiyama, Tomoyoshi; Kaneko, Masanobu (2014). Bernoulli Numbers and Zeta Functions. Springer. ISBN 978-4-431-54918-5. p. 203

[2] Gauss, C. F. (1866). Werke (Band III) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. p. 404

[3] Cox 1984, p. 281

[4] Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. American Mathematical Society. ISBN 0-8218-3246-8. p. 199

[5] 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. 57

[6] Arakawa, Tsuneo; Ibukiyama, Tomoyoshi; Kaneko, Masanobu (2014). Bernoulli Numbers and Zeta Functions. Springer. ISBN 978-4-431-54918-5. p. 203

[2] See:
Finch 2003, p. 420

Kobayashi, Hiroyuki; Takeuchi, Shingo (2019), "Applications of generalized trigonometric functions with two parameters", Communications on Pure & Applied Analysis, 18 (3): 1509–1521, arXiv:1903.07407, doi:10.3934/cpaa.2019072, S2CID 102487670

Asai, Tetsuya (2007), Elliptic Gauss Sums and Hecke L-values at s=1, arXiv:0707.3711

"A062539 - Oeis".

[8] Finch 2003, p. 420

[9] Kobayashi, Hiroyuki; Takeuchi, Shingo (2019), "Applications of generalized trigonometric functions with two parameters", Communications on Pure & Applied Analysis, 18 (3): 1509–1521, arXiv:1903.07407, doi:10.3934/cpaa.2019072, S2CID 102487670

[10] Asai, Tetsuya (2007), Elliptic Gauss Sums and Hecke L-values at s=1, arXiv:0707.3711

[11] "A062539 - Oeis".

[3] "A064853 - Oeis".

[4] "Lemniscate Constant".

[5] "Records set by y-cruncher". numberworld.org. Retrieved 2024-08-20.

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