Lemniscate elliptic functions (English Wikipedia)

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

Last modified: 00:03:50, 13 July 2026 (UTC)

Ref.Un. Ref.Website

Global rank English rank

2723doi.org

2^nd place

1110archive.org

6^th place

106oeis.org

1,188^th place

866^th place

88nist.gov

708^th place

742^nd place

75arxiv.org

49^th place

31^st place

66jstor.org

23^rd place

15^th place

65pacific.edu

low place

44uni-goettingen.de

2,423^rd place

2,569^th place

44web.archive.org

1^st place

33noaa.gov

111^th place

78^th place

31umich.edu

429^th place

307^th place

22researchgate.net

96^th place

89^th place

22unistra.fr

1,779^th place

940^th place

22utpjournals.press

low place

11reed.edu

low place

7,280^th place

11johndcook.com

low place

11osu.edu

2,360^th place

1,459^th place

11hdl.handle.net

66^th place

34^th place

11amherst.edu

low place

6,621^st place

11ias.ac.in

6,516^th place

4,685^th place

11berkeley.edu

621^st place

429^th place

11retronews.fr

2,920^th place

low place

11csiro.au

3,457^th place

2,432^nd place

amherst.edu (Global: low place; English: 6,621^st place)

dacox.people.amherst.edu

Cox (2013) p. 142, Example 7.29(c) Cox, David Archibald (2013). Primes of the Form $x 2 + ny 2$ (Second ed.). Wiley.

archive.org (Global: 6^th place; English: 6^th place)

Fagnano (1718–1723); Euler (1761); Gauss (1917) Fagnano, Giulio Carlo (1718–1723) "Metodo per misurare la lemniscata" [Method for measuring the lemniscate]. Giornale de' letterati d'Italia (in Italian).
"Schediasma primo" [Part 1]. 1718. 29: 258–269.
"Giunte al primo schediasma" [Addendum to part 1]. 1723. 34: 197–207.
"Schediasma secondo" [Part 2]. 1718. 30: 87–111.
Reprinted as Fagnano (1850). "32–34. Metodo per misurare la lemniscata". Opere Matematiche, vol. 2. Allerighi e Segati. pp. 293–313. (Figures) Euler, Leonhard (1761). "Observationes de comparatione arcuum curvarum irrectificibilium" [Observations on the comparison of arcs of irrectifiable curves]. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae (in Latin). 6: 58–84. E 252. (Figures) Gauss, Carl Friedrich (1917). Werke (Band X, Abteilung I) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen.
Gauss (1917) p. 199 used the symbols $sl$ and $cl$ for the lemniscate sine and cosine, respectively, and this notation is most common today: see e.g. Cox (1984) p. 316, Eymard & Lafon (2004) p. 204, and Lemmermeyer (2000) p. 240. Ayoub (1984) uses $sinlem$ and $coslem$ . Whittaker & Watson (1920) use the symbols $sin lemn$ and $cos lemn$ . Some sources use the generic letters $s$ and $c$ . Prasolov & Solovyev (1997) use the letter $φ$ for the lemniscate sine and $φ'$ for its derivative. Gauss, Carl Friedrich (1917). Werke (Band X, Abteilung I) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Cox, David Archibald (1984). "The Arithmetic-Geometric Mean of Gauss". L'Enseignement Mathématique. 30 (2): 275–330. Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. Translated by Wilson, Stephen. American Mathematical Society. ISBN 0-8218-3246-8. Lemmermeyer, Franz (2000). Reciprocity Laws: From Euler to Eisenstein. Springer. ISBN 3-540-66957-4. Ayoub, Raymond (1984). "The Lemniscate and Fagnano's Contributions to Elliptic Integrals". Archive for History of Exact Sciences. 29 (2): 131–149. doi:10.1007/BF00348244. Whittaker, Edmund Taylor; Watson, George Neville (1920) [1st ed. 1902]. "22.8 The lemniscate functions". A Course of Modern Analysis (3rd ed.). Cambridge. pp. 524–528. Prasolov, Viktor; Solovyev, Yuri (1997). "4. Abel's Theorem on Division of Lemniscate". Elliptic functions and elliptic integrals. Translations of Mathematical Monographs. Vol. 170. American Mathematical Society. doi:10.1090/mmono/170. ISBN 978-0-8218-0587-9.
This map was the first ever picture of a Schwarz–Christoffel mapping, in Schwarz (1869) p. 113. Schwarz, Hermann Amandus (1869). "Ueber einige Abbildungsaufgaben" [About some mapping problems]. Crelle's Journal (in German). 70: 105–120. doi:10.1515/crll.1869.70.105.
Euler (1761) §44 p. 79, §47 pp. 80–81 Euler, Leonhard (1761). "Observationes de comparatione arcuum curvarum irrectificibilium" [Observations on the comparison of arcs of irrectifiable curves]. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae (in Latin). 6: 58–84. E 252. (Figures)
Euler (1761) §46 p. 80 Euler, Leonhard (1761). "Observationes de comparatione arcuum curvarum irrectificibilium" [Observations on the comparison of arcs of irrectifiable curves]. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae (in Latin). 6: 58–84. E 252. (Figures)
Euler (1761); Siegel (1969). Prasolov & Solovyev (1997) use the polar-coordinate representation of the Lemniscate to derive differential arc length, but the result is the same. Euler, Leonhard (1761). "Observationes de comparatione arcuum curvarum irrectificibilium" [Observations on the comparison of arcs of irrectifiable curves]. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae (in Latin). 6: 58–84. E 252. (Figures) Siegel, Carl Ludwig (1969). "1. Elliptic Functions". Topics in Complex Function Theory, Vol. I. Wiley-Interscience. pp. 1–89. ISBN 0-471-60844-0. Prasolov, Viktor; Solovyev, Yuri (1997). "4. Abel's Theorem on Division of Lemniscate". Elliptic functions and elliptic integrals. Translations of Mathematical Monographs. Vol. 170. American Mathematical Society. doi:10.1090/mmono/170. ISBN 978-0-8218-0587-9.
Euler (1786); Sridharan (2004); Levien (2008) Euler, Leonhard (1786). "De miris proprietatibus curvae elasticae sub aequatione ${\textstyle y=\int xx\mathop {\mathrm {d} x} {\big /}{\sqrt {1-x^{4}}}}$ contentae" [On the amazing properties of elastic curves contained in equation ${\textstyle y=\int xx\mathop {\mathrm {d} x} {\big /}{\sqrt {1-x^{4}}}}$ ]. Acta Academiae Scientiarum Imperialis Petropolitanae (in Latin). 1782 (2): 34–61. E 605. Sridharan, Ramaiyengar (2004) "Physics to Mathematics: from Lintearia to Lemniscate". Resonance.
"Part I". 9 (4): 21–29. doi:10.1007/BF02834853.
"Part II: Gauss and Landen's Work". 9 (6): 11–20. doi:10.1007/BF02839214. Levien, Raph (2008). The elastica: a mathematical history (PDF) (Technical report). University of California at Berkeley. UCB/EECS-2008-103.
The identity $\operatorname {cl} z={\operatorname {cn} }\left({\sqrt {2}}z;{\tfrac {1}{\sqrt {2}}}\right)$ can be found in Greenhill (1892) p. 33. Greenhill, Alfred George (1892). The Applications of Elliptic Functions. MacMillan.
Peirce (1879). Guyou (1887) and Adams (1925) introduced transverse and oblique aspects of the same projection, respectively. Also see Lee (1976). These authors write their projection formulas in terms of Jacobi elliptic functions, with a square lattice. Peirce, Charles Sanders (1879). "A Quincuncial Projection of the Sphere". American Journal of Mathematics. 2 (4): 394–397. doi:10.2307/2369491. JSTOR 2369491. Guyou, Émile (1887). "Nouveau système de projection de la sphère: Généralisation de la projection de Mercator" [New system of projection of the sphere]. Annales Hydrographiques. Série 2 (in French). 9: 16–35. Adams, Oscar S. (1925). Elliptic Functions Applied to Conformal World Maps (PDF). U.S. Coast and Geodetic Survey. US Government Printing Office. Special Pub. No. 112. Lee, L. P. (1976). Conformal Projections Based on Elliptic Functions. Cartographica Monographs. Vol. 16. Toronto: B. V. Gutsell, York University. ISBN 0-919870-16-3. Supplement No. 1 to The Canadian Cartographer 13.
Adams (1925); Lee (1976). Adams, Oscar S. (1925). Elliptic Functions Applied to Conformal World Maps (PDF). U.S. Coast and Geodetic Survey. US Government Printing Office. Special Pub. No. 112. Lee, L. P. (1976). Conformal Projections Based on Elliptic Functions. Cartographica Monographs. Vol. 16. Toronto: B. V. Gutsell, York University. ISBN 0-919870-16-3. Supplement No. 1 to The Canadian Cartographer 13.

arxiv.org (Global: 49^th place; English: 31^st place)

Robinson (2019a) starts from this definition and thence derives other properties of the lemniscate functions. Robinson, Paul L. (2019a). The Lemniscatic Functions (Preprint). arXiv:1902.08614.
Prasolov & Solovyev (1997); Robinson (2019a) Prasolov, Viktor; Solovyev, Yuri (1997). "4. Abel's Theorem on Division of Lemniscate". Elliptic functions and elliptic integrals. Translations of Mathematical Monographs. Vol. 170. American Mathematical Society. doi:10.1090/mmono/170. ISBN 978-0-8218-0587-9. Robinson, Paul L. (2019a). The Lemniscatic Functions (Preprint). arXiv:1902.08614.
Cox & Hyde (2014) Cox, David Archibald; Hyde, Trevor (2014). "The Galois theory of the lemniscate" (PDF). Journal of Number Theory. 135: 43–59. arXiv:1208.2653. doi:10.1016/j.jnt.2013.08.006.
Robinson (2019a) Robinson, Paul L. (2019a). The Lemniscatic Functions (Preprint). arXiv:1902.08614.
Levin (2006); Robinson (2019b) Levin, Aaron (2006). "A Geometric Interpretation of an Infinite Product for the Lemniscate Constant" (PDF). The American Mathematical Monthly. 113 (6): 510–520. doi:10.1080/00029890.2006.11920331. JSTOR 27641976. Robinson, Paul L. (2019b). The Elliptic Functions in a First-Order System (Preprint). arXiv:1903.07147.

berkeley.edu (Global: 621^st place; English: 429^th place)

www2.eecs.berkeley.edu

Euler (1786); Sridharan (2004); Levien (2008) Euler, Leonhard (1786). "De miris proprietatibus curvae elasticae sub aequatione ${\textstyle y=\int xx\mathop {\mathrm {d} x} {\big /}{\sqrt {1-x^{4}}}}$ contentae" [On the amazing properties of elastic curves contained in equation ${\textstyle y=\int xx\mathop {\mathrm {d} x} {\big /}{\sqrt {1-x^{4}}}}$ ]. Acta Academiae Scientiarum Imperialis Petropolitanae (in Latin). 1782 (2): 34–61. E 605. Sridharan, Ramaiyengar (2004) "Physics to Mathematics: from Lintearia to Lemniscate". Resonance.
"Part I". 9 (4): 21–29. doi:10.1007/BF02834853.
"Part II: Gauss and Landen's Work". 9 (6): 11–20. doi:10.1007/BF02839214. Levien, Raph (2008). The elastica: a mathematical history (PDF) (Technical report). University of California at Berkeley. UCB/EECS-2008-103.

csiro.au (Global: 3,457^th place; English: 2,432^nd place)

publications.csiro.au

Rančić, Purser & Mesinger (1996); McGregor (2005). Rančić, Miodrag; Purser, R. James; Mesinger, Fedor (1996). "A global shallow-water model using an expanded spherical cube: Gnomonic versus conformal coordinates". Quarterly Journal of the Royal Meteorological Society. 122 (532): 959–982. doi:10.1002/qj.49712253209. McGregor, John L. (2005). C-CAM: Geometric Aspects and Dynamical Formulation (Technical report). CSIRO Atmospheric Research. 70.

doi.org (Global: 2^nd place; English: 2^nd place)

Gauss (1917) p. 199 used the symbols $sl$ and $cl$ for the lemniscate sine and cosine, respectively, and this notation is most common today: see e.g. Cox (1984) p. 316, Eymard & Lafon (2004) p. 204, and Lemmermeyer (2000) p. 240. Ayoub (1984) uses $sinlem$ and $coslem$ . Whittaker & Watson (1920) use the symbols $sin lemn$ and $cos lemn$ . Some sources use the generic letters $s$ and $c$ . Prasolov & Solovyev (1997) use the letter $φ$ for the lemniscate sine and $φ'$ for its derivative. Gauss, Carl Friedrich (1917). Werke (Band X, Abteilung I) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Cox, David Archibald (1984). "The Arithmetic-Geometric Mean of Gauss". L'Enseignement Mathématique. 30 (2): 275–330. Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. Translated by Wilson, Stephen. American Mathematical Society. ISBN 0-8218-3246-8. Lemmermeyer, Franz (2000). Reciprocity Laws: From Euler to Eisenstein. Springer. ISBN 3-540-66957-4. Ayoub, Raymond (1984). "The Lemniscate and Fagnano's Contributions to Elliptic Integrals". Archive for History of Exact Sciences. 29 (2): 131–149. doi:10.1007/BF00348244. Whittaker, Edmund Taylor; Watson, George Neville (1920) [1st ed. 1902]. "22.8 The lemniscate functions". A Course of Modern Analysis (3rd ed.). Cambridge. pp. 524–528. Prasolov, Viktor; Solovyev, Yuri (1997). "4. Abel's Theorem on Division of Lemniscate". Elliptic functions and elliptic integrals. Translations of Mathematical Monographs. Vol. 170. American Mathematical Society. doi:10.1090/mmono/170. ISBN 978-0-8218-0587-9.
The circle $x^{2}+y^{2}=x$ is the unit-diameter circle centered at ${\textstyle {\bigl (}{\tfrac {1}{2}},0{\bigr )}}$ with polar equation $r=\cos \theta ,$ the degree-2 clover under the definition from Cox & Shurman (2005). This is not the unit-radius circle $x^{2}+y^{2}=1$ centered at the origin. Notice that the lemniscate ${\bigl (}x^{2}+y^{2}{\bigr )}{}^{2}=x^{2}-y^{2}$ is the degree-4 clover. Cox, David Archibald; Shurman, Jerry (2005). "Geometry and number theory on clovers" (PDF). The American Mathematical Monthly. 112 (8): 682–704. doi:10.1080/00029890.2005.11920241.
This map was the first ever picture of a Schwarz–Christoffel mapping, in Schwarz (1869) p. 113. Schwarz, Hermann Amandus (1869). "Ueber einige Abbildungsaufgaben" [About some mapping problems]. Crelle's Journal (in German). 70: 105–120. doi:10.1515/crll.1869.70.105.
Levin (2006) Levin, Aaron (2006). "A Geometric Interpretation of an Infinite Product for the Lemniscate Constant" (PDF). The American Mathematical Monthly. 113 (6): 510–520. doi:10.1080/00029890.2006.11920331. JSTOR 27641976.
Todd (1975) Todd, John (1975). "The lemniscate constants". Communications of the ACM. 18 (1): 14–19. doi:10.1145/360569.360580.
Prasolov & Solovyev (1997); Robinson (2019a) Prasolov, Viktor; Solovyev, Yuri (1997). "4. Abel's Theorem on Division of Lemniscate". Elliptic functions and elliptic integrals. Translations of Mathematical Monographs. Vol. 170. American Mathematical Society. doi:10.1090/mmono/170. ISBN 978-0-8218-0587-9. Robinson, Paul L. (2019a). The Lemniscatic Functions (Preprint). arXiv:1902.08614.
Cox (2012) Cox, David Archibald (2012). "The Lemniscate". Galois Theory. Wiley. pp. 463–514. doi:10.1002/9781118218457.ch15. ISBN 978-1-118-07205-9.
Lindqvist & Peetre (2001) generalizes the first of these forms. Lindqvist, Peter; Peetre, Jaak (2001). "Two Remarkable Identities, Called Twos, for Inverses to Some Abelian Integrals" (PDF). The American Mathematical Monthly. 108 (5): 403–410. doi:10.1080/00029890.2001.11919766. Archived from the original (PDF) on 2022-05-28.
Ayoub (1984); Prasolov & Solovyev (1997) Ayoub, Raymond (1984). "The Lemniscate and Fagnano's Contributions to Elliptic Integrals". Archive for History of Exact Sciences. 29 (2): 131–149. doi:10.1007/BF00348244. Prasolov, Viktor; Solovyev, Yuri (1997). "4. Abel's Theorem on Division of Lemniscate". Elliptic functions and elliptic integrals. Translations of Mathematical Monographs. Vol. 170. American Mathematical Society. doi:10.1090/mmono/170. ISBN 978-0-8218-0587-9.
Cox & Hyde (2014) Cox, David Archibald; Hyde, Trevor (2014). "The Galois theory of the lemniscate" (PDF). Journal of Number Theory. 135: 43–59. arXiv:1208.2653. doi:10.1016/j.jnt.2013.08.006.
Gómez-Molleda & Lario (2019) Gómez-Molleda, M. A.; Lario, Joan-C. (2019). "Ruler and Compass Constructions of the Equilateral Triangle and Pentagon in the Lemniscate Curve". The Mathematical Intelligencer. 41 (4): 17–21. doi:10.1007/s00283-019-09892-w. hdl:10630/40427.
Rosen (1981) Rosen, Michael (1981). "Abel's Theorem on the Lemniscate". The American Mathematical Monthly. 88 (6): 387–395. doi:10.1080/00029890.1981.11995279. JSTOR 2321821.
Euler (1761); Siegel (1969). Prasolov & Solovyev (1997) use the polar-coordinate representation of the Lemniscate to derive differential arc length, but the result is the same. Euler, Leonhard (1761). "Observationes de comparatione arcuum curvarum irrectificibilium" [Observations on the comparison of arcs of irrectifiable curves]. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae (in Latin). 6: 58–84. E 252. (Figures) Siegel, Carl Ludwig (1969). "1. Elliptic Functions". Topics in Complex Function Theory, Vol. I. Wiley-Interscience. pp. 1–89. ISBN 0-471-60844-0. Prasolov, Viktor; Solovyev, Yuri (1997). "4. Abel's Theorem on Division of Lemniscate". Elliptic functions and elliptic integrals. Translations of Mathematical Monographs. Vol. 170. American Mathematical Society. doi:10.1090/mmono/170. ISBN 978-0-8218-0587-9.
Abel (1827–1828); Rosen (1981); Prasolov & Solovyev (1997) Rosen, Michael (1981). "Abel's Theorem on the Lemniscate". The American Mathematical Monthly. 88 (6): 387–395. doi:10.1080/00029890.1981.11995279. JSTOR 2321821. Prasolov, Viktor; Solovyev, Yuri (1997). "4. Abel's Theorem on Division of Lemniscate". Elliptic functions and elliptic integrals. Translations of Mathematical Monographs. Vol. 170. American Mathematical Society. doi:10.1090/mmono/170. ISBN 978-0-8218-0587-9.
Euler (1786); Sridharan (2004); Levien (2008) Euler, Leonhard (1786). "De miris proprietatibus curvae elasticae sub aequatione ${\textstyle y=\int xx\mathop {\mathrm {d} x} {\big /}{\sqrt {1-x^{4}}}}$ contentae" [On the amazing properties of elastic curves contained in equation ${\textstyle y=\int xx\mathop {\mathrm {d} x} {\big /}{\sqrt {1-x^{4}}}}$ ]. Acta Academiae Scientiarum Imperialis Petropolitanae (in Latin). 1782 (2): 34–61. E 605. Sridharan, Ramaiyengar (2004) "Physics to Mathematics: from Lintearia to Lemniscate". Resonance.
"Part I". 9 (4): 21–29. doi:10.1007/BF02834853.
"Part II: Gauss and Landen's Work". 9 (6): 11–20. doi:10.1007/BF02839214. Levien, Raph (2008). The elastica: a mathematical history (PDF) (Technical report). University of California at Berkeley. UCB/EECS-2008-103.
Bottazzini & Gray (2013) p. 58 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.
Levin (2006); Robinson (2019b) Levin, Aaron (2006). "A Geometric Interpretation of an Infinite Product for the Lemniscate Constant" (PDF). The American Mathematical Monthly. 113 (6): 510–520. doi:10.1080/00029890.2006.11920331. JSTOR 27641976. Robinson, Paul L. (2019b). The Elliptic Functions in a First-Order System (Preprint). arXiv:1903.07147.
Levin (2006) p. 515 Levin, Aaron (2006). "A Geometric Interpretation of an Infinite Product for the Lemniscate Constant" (PDF). The American Mathematical Monthly. 113 (6): 510–520. doi:10.1080/00029890.2006.11920331. JSTOR 27641976.
Cox (2012) p. 508, 509 Cox, David Archibald (2012). "The Lemniscate". Galois Theory. Wiley. pp. 463–514. doi:10.1002/9781118218457.ch15. ISBN 978-1-118-07205-9.
Katz, Nicholas M. (1975). "The congruences of Clausen — von Staudt and Kummer for Bernoulli-Hurwitz numbers". Mathematische Annalen. 216 (1): 1–4. doi:10.1007/BF02547966. See eq. (9)
Ogawa (2005) Ogawa, Takuma (2005). "Similarities between the trigonometric function and the lemniscate function from arithmetic view point". Tsukuba Journal of Mathematics. 29 (1). doi:10.21099/tkbjm/1496164894.
Peirce (1879). Guyou (1887) and Adams (1925) introduced transverse and oblique aspects of the same projection, respectively. Also see Lee (1976). These authors write their projection formulas in terms of Jacobi elliptic functions, with a square lattice. Peirce, Charles Sanders (1879). "A Quincuncial Projection of the Sphere". American Journal of Mathematics. 2 (4): 394–397. doi:10.2307/2369491. JSTOR 2369491. Guyou, Émile (1887). "Nouveau système de projection de la sphère: Généralisation de la projection de Mercator" [New system of projection of the sphere]. Annales Hydrographiques. Série 2 (in French). 9: 16–35. Adams, Oscar S. (1925). Elliptic Functions Applied to Conformal World Maps (PDF). U.S. Coast and Geodetic Survey. US Government Printing Office. Special Pub. No. 112. Lee, L. P. (1976). Conformal Projections Based on Elliptic Functions. Cartographica Monographs. Vol. 16. Toronto: B. V. Gutsell, York University. ISBN 0-919870-16-3. Supplement No. 1 to The Canadian Cartographer 13.
Rančić, Purser & Mesinger (1996); McGregor (2005). Rančić, Miodrag; Purser, R. James; Mesinger, Fedor (1996). "A global shallow-water model using an expanded spherical cube: Gnomonic versus conformal coordinates". Quarterly Journal of the Royal Meteorological Society. 122 (532): 959–982. doi:10.1002/qj.49712253209. McGregor, John L. (2005). C-CAM: Geometric Aspects and Dynamical Formulation (Technical report). CSIRO Atmospheric Research. 70.

hdl.handle.net (Global: 66^th place; English: 34^th place)

Gómez-Molleda & Lario (2019) Gómez-Molleda, M. A.; Lario, Joan-C. (2019). "Ruler and Compass Constructions of the Equilateral Triangle and Pentagon in the Lemniscate Curve". The Mathematical Intelligencer. 41 (4): 17–21. doi:10.1007/s00283-019-09892-w. hdl:10630/40427.

ias.ac.in (Global: 6,516^th place; English: 4,685^th place)

Euler (1786); Sridharan (2004); Levien (2008) Euler, Leonhard (1786). "De miris proprietatibus curvae elasticae sub aequatione ${\textstyle y=\int xx\mathop {\mathrm {d} x} {\big /}{\sqrt {1-x^{4}}}}$ contentae" [On the amazing properties of elastic curves contained in equation ${\textstyle y=\int xx\mathop {\mathrm {d} x} {\big /}{\sqrt {1-x^{4}}}}$ ]. Acta Academiae Scientiarum Imperialis Petropolitanae (in Latin). 1782 (2): 34–61. E 605. Sridharan, Ramaiyengar (2004) "Physics to Mathematics: from Lintearia to Lemniscate". Resonance.
"Part I". 9 (4): 21–29. doi:10.1007/BF02834853.
"Part II: Gauss and Landen's Work". 9 (6): 11–20. doi:10.1007/BF02839214. Levien, Raph (2008). The elastica: a mathematical history (PDF) (Technical report). University of California at Berkeley. UCB/EECS-2008-103.

johndcook.com (Global: low place; English: low place)

John (2023-10-04). "Lemniscate functions". www.johndcook.com. Retrieved 2026-06-18.

jstor.org (Global: 23^rd place; English: 15^th place)

Levin (2006) Levin, Aaron (2006). "A Geometric Interpretation of an Infinite Product for the Lemniscate Constant" (PDF). The American Mathematical Monthly. 113 (6): 510–520. doi:10.1080/00029890.2006.11920331. JSTOR 27641976.
Rosen (1981) Rosen, Michael (1981). "Abel's Theorem on the Lemniscate". The American Mathematical Monthly. 88 (6): 387–395. doi:10.1080/00029890.1981.11995279. JSTOR 2321821.
Abel (1827–1828); Rosen (1981); Prasolov & Solovyev (1997) Rosen, Michael (1981). "Abel's Theorem on the Lemniscate". The American Mathematical Monthly. 88 (6): 387–395. doi:10.1080/00029890.1981.11995279. JSTOR 2321821. Prasolov, Viktor; Solovyev, Yuri (1997). "4. Abel's Theorem on Division of Lemniscate". Elliptic functions and elliptic integrals. Translations of Mathematical Monographs. Vol. 170. American Mathematical Society. doi:10.1090/mmono/170. ISBN 978-0-8218-0587-9.
Levin (2006); Robinson (2019b) Levin, Aaron (2006). "A Geometric Interpretation of an Infinite Product for the Lemniscate Constant" (PDF). The American Mathematical Monthly. 113 (6): 510–520. doi:10.1080/00029890.2006.11920331. JSTOR 27641976. Robinson, Paul L. (2019b). The Elliptic Functions in a First-Order System (Preprint). arXiv:1903.07147.
Levin (2006) p. 515 Levin, Aaron (2006). "A Geometric Interpretation of an Infinite Product for the Lemniscate Constant" (PDF). The American Mathematical Monthly. 113 (6): 510–520. doi:10.1080/00029890.2006.11920331. JSTOR 27641976.
Peirce (1879). Guyou (1887) and Adams (1925) introduced transverse and oblique aspects of the same projection, respectively. Also see Lee (1976). These authors write their projection formulas in terms of Jacobi elliptic functions, with a square lattice. Peirce, Charles Sanders (1879). "A Quincuncial Projection of the Sphere". American Journal of Mathematics. 2 (4): 394–397. doi:10.2307/2369491. JSTOR 2369491. Guyou, Émile (1887). "Nouveau système de projection de la sphère: Généralisation de la projection de Mercator" [New system of projection of the sphere]. Annales Hydrographiques. Série 2 (in French). 9: 16–35. Adams, Oscar S. (1925). Elliptic Functions Applied to Conformal World Maps (PDF). U.S. Coast and Geodetic Survey. US Government Printing Office. Special Pub. No. 112. Lee, L. P. (1976). Conformal Projections Based on Elliptic Functions. Cartographica Monographs. Vol. 16. Toronto: B. V. Gutsell, York University. ISBN 0-919870-16-3. Supplement No. 1 to The Canadian Cartographer 13.

nist.gov (Global: 708^th place; English: 742^nd place)

dlmf.nist.gov

Reinhardt & Walker (2010a) §22.12.6, §22.12.12 Reinhardt, William P.; Walker, Peter L. (2010a). "22. Jacobian Elliptic Functions". In Olver, Frank; et al. (eds.). NIST Handbook of Mathematical Functions. Cambridge.
Reinhardt & Walker (2010a) §22.18.E6 Reinhardt, William P.; Walker, Peter L. (2010a). "22. Jacobian Elliptic Functions". In Olver, Frank; et al. (eds.). NIST Handbook of Mathematical Functions. Cambridge.
Reinhardt & Walker (2010a) §22.20(ii) Reinhardt, William P.; Walker, Peter L. (2010a). "22. Jacobian Elliptic Functions". In Olver, Frank; et al. (eds.). NIST Handbook of Mathematical Functions. Cambridge.
Carlson (2010) §19.8 Carlson, Billie C. (2010). "19. Elliptic Integrals". In Olver, Frank; et al. (eds.). NIST Handbook of Mathematical Functions. Cambridge.
Reinhardt & Walker (2010a) §22.12.12 Reinhardt, William P.; Walker, Peter L. (2010a). "22. Jacobian Elliptic Functions". In Olver, Frank; et al. (eds.). NIST Handbook of Mathematical Functions. Cambridge.
Reinhardt & Walker (2010a) §22.11 Reinhardt, William P.; Walker, Peter L. (2010a). "22. Jacobian Elliptic Functions". In Olver, Frank; et al. (eds.). NIST Handbook of Mathematical Functions. Cambridge.
Reinhardt & Walker (2010a) §22.2.E7 Reinhardt, William P.; Walker, Peter L. (2010a). "22. Jacobian Elliptic Functions". In Olver, Frank; et al. (eds.). NIST Handbook of Mathematical Functions. Cambridge.
Reinhardt & Walker (2010a) §22.11.E1 Reinhardt, William P.; Walker, Peter L. (2010a). "22. Jacobian Elliptic Functions". In Olver, Frank; et al. (eds.). NIST Handbook of Mathematical Functions. Cambridge.

noaa.gov (Global: 111^th place; English: 78^th place)

geodesy.noaa.gov

Peirce (1879). Guyou (1887) and Adams (1925) introduced transverse and oblique aspects of the same projection, respectively. Also see Lee (1976). These authors write their projection formulas in terms of Jacobi elliptic functions, with a square lattice. Peirce, Charles Sanders (1879). "A Quincuncial Projection of the Sphere". American Journal of Mathematics. 2 (4): 394–397. doi:10.2307/2369491. JSTOR 2369491. Guyou, Émile (1887). "Nouveau système de projection de la sphère: Généralisation de la projection de Mercator" [New system of projection of the sphere]. Annales Hydrographiques. Série 2 (in French). 9: 16–35. Adams, Oscar S. (1925). Elliptic Functions Applied to Conformal World Maps (PDF). U.S. Coast and Geodetic Survey. US Government Printing Office. Special Pub. No. 112. Lee, L. P. (1976). Conformal Projections Based on Elliptic Functions. Cartographica Monographs. Vol. 16. Toronto: B. V. Gutsell, York University. ISBN 0-919870-16-3. Supplement No. 1 to The Canadian Cartographer 13.
Adams (1925) Adams, Oscar S. (1925). Elliptic Functions Applied to Conformal World Maps (PDF). U.S. Coast and Geodetic Survey. US Government Printing Office. Special Pub. No. 112.
Adams (1925); Lee (1976). Adams, Oscar S. (1925). Elliptic Functions Applied to Conformal World Maps (PDF). U.S. Coast and Geodetic Survey. US Government Printing Office. Special Pub. No. 112. Lee, L. P. (1976). Conformal Projections Based on Elliptic Functions. Cartographica Monographs. Vol. 16. Toronto: B. V. Gutsell, York University. ISBN 0-919870-16-3. Supplement No. 1 to The Canadian Cartographer 13.

oeis.org (Global: 1,188^th place; English: 866^th place)

Schappacher (1997). OEIS sequence A062539 lists the lemniscate constant's decimal digits. Schappacher, Norbert (1997). "Some milestones of lemniscatomy" (PDF). In Sertöz, S. (ed.). Algebraic Geometry (Proceedings of Bilkent Summer School, August 7–19, 1995, Ankara, Turkey). Marcel Dekker. pp. 257–290.
Such numbers are OEIS sequence A003401.
"A104203". The On-Line Encyclopedia of Integer Sequences.
"A193543 - Oeis".
"A289695 - Oeis".
Sloane, N. J. A. (ed.). "Sequence A175576". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

osu.edu (Global: 2,360^th place; English: 1,459^th place)

people.math.osu.edu

Lindqvist & Peetre (2001) generalizes the first of these forms. Lindqvist, Peter; Peetre, Jaak (2001). "Two Remarkable Identities, Called Twos, for Inverses to Some Abelian Integrals" (PDF). The American Mathematical Monthly. 108 (5): 403–410. doi:10.1080/00029890.2001.11919766. Archived from the original (PDF) on 2022-05-28.

pacific.edu (Global: low place; English: low place)

scholarlycommons.pacific.edu

Fagnano (1718–1723); Euler (1761); Gauss (1917) Fagnano, Giulio Carlo (1718–1723) "Metodo per misurare la lemniscata" [Method for measuring the lemniscate]. Giornale de' letterati d'Italia (in Italian).
"Schediasma primo" [Part 1]. 1718. 29: 258–269.
"Giunte al primo schediasma" [Addendum to part 1]. 1723. 34: 197–207.
"Schediasma secondo" [Part 2]. 1718. 30: 87–111.
Reprinted as Fagnano (1850). "32–34. Metodo per misurare la lemniscata". Opere Matematiche, vol. 2. Allerighi e Segati. pp. 293–313. (Figures) Euler, Leonhard (1761). "Observationes de comparatione arcuum curvarum irrectificibilium" [Observations on the comparison of arcs of irrectifiable curves]. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae (in Latin). 6: 58–84. E 252. (Figures) Gauss, Carl Friedrich (1917). Werke (Band X, Abteilung I) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen.
Euler (1761) §44 p. 79, §47 pp. 80–81 Euler, Leonhard (1761). "Observationes de comparatione arcuum curvarum irrectificibilium" [Observations on the comparison of arcs of irrectifiable curves]. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae (in Latin). 6: 58–84. E 252. (Figures)
Euler (1761) §46 p. 80 Euler, Leonhard (1761). "Observationes de comparatione arcuum curvarum irrectificibilium" [Observations on the comparison of arcs of irrectifiable curves]. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae (in Latin). 6: 58–84. E 252. (Figures)
Euler (1761); Siegel (1969). Prasolov & Solovyev (1997) use the polar-coordinate representation of the Lemniscate to derive differential arc length, but the result is the same. Euler, Leonhard (1761). "Observationes de comparatione arcuum curvarum irrectificibilium" [Observations on the comparison of arcs of irrectifiable curves]. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae (in Latin). 6: 58–84. E 252. (Figures) Siegel, Carl Ludwig (1969). "1. Elliptic Functions". Topics in Complex Function Theory, Vol. I. Wiley-Interscience. pp. 1–89. ISBN 0-471-60844-0. Prasolov, Viktor; Solovyev, Yuri (1997). "4. Abel's Theorem on Division of Lemniscate". Elliptic functions and elliptic integrals. Translations of Mathematical Monographs. Vol. 170. American Mathematical Society. doi:10.1090/mmono/170. ISBN 978-0-8218-0587-9.
Euler (1786); Sridharan (2004); Levien (2008) Euler, Leonhard (1786). "De miris proprietatibus curvae elasticae sub aequatione ${\textstyle y=\int xx\mathop {\mathrm {d} x} {\big /}{\sqrt {1-x^{4}}}}$ contentae" [On the amazing properties of elastic curves contained in equation ${\textstyle y=\int xx\mathop {\mathrm {d} x} {\big /}{\sqrt {1-x^{4}}}}$ ]. Acta Academiae Scientiarum Imperialis Petropolitanae (in Latin). 1782 (2): 34–61. E 605. Sridharan, Ramaiyengar (2004) "Physics to Mathematics: from Lintearia to Lemniscate". Resonance.
"Part I". 9 (4): 21–29. doi:10.1007/BF02834853.
"Part II: Gauss and Landen's Work". 9 (6): 11–20. doi:10.1007/BF02839214. Levien, Raph (2008). The elastica: a mathematical history (PDF) (Technical report). University of California at Berkeley. UCB/EECS-2008-103.

reed.edu (Global: low place; English: 7,280^th place)

people.reed.edu

The circle $x^{2}+y^{2}=x$ is the unit-diameter circle centered at ${\textstyle {\bigl (}{\tfrac {1}{2}},0{\bigr )}}$ with polar equation $r=\cos \theta ,$ the degree-2 clover under the definition from Cox & Shurman (2005). This is not the unit-radius circle $x^{2}+y^{2}=1$ centered at the origin. Notice that the lemniscate ${\bigl (}x^{2}+y^{2}{\bigr )}{}^{2}=x^{2}-y^{2}$ is the degree-4 clover. Cox, David Archibald; Shurman, Jerry (2005). "Geometry and number theory on clovers" (PDF). The American Mathematical Monthly. 112 (8): 682–704. doi:10.1080/00029890.2005.11920241.

researchgate.net (Global: 96^th place; English: 89^th place)

Gauss (1917) p. 199 used the symbols $sl$ and $cl$ for the lemniscate sine and cosine, respectively, and this notation is most common today: see e.g. Cox (1984) p. 316, Eymard & Lafon (2004) p. 204, and Lemmermeyer (2000) p. 240. Ayoub (1984) uses $sinlem$ and $coslem$ . Whittaker & Watson (1920) use the symbols $sin lemn$ and $cos lemn$ . Some sources use the generic letters $s$ and $c$ . Prasolov & Solovyev (1997) use the letter $φ$ for the lemniscate sine and $φ'$ for its derivative. Gauss, Carl Friedrich (1917). Werke (Band X, Abteilung I) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Cox, David Archibald (1984). "The Arithmetic-Geometric Mean of Gauss". L'Enseignement Mathématique. 30 (2): 275–330. Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. Translated by Wilson, Stephen. American Mathematical Society. ISBN 0-8218-3246-8. Lemmermeyer, Franz (2000). Reciprocity Laws: From Euler to Eisenstein. Springer. ISBN 3-540-66957-4. Ayoub, Raymond (1984). "The Lemniscate and Fagnano's Contributions to Elliptic Integrals". Archive for History of Exact Sciences. 29 (2): 131–149. doi:10.1007/BF00348244. Whittaker, Edmund Taylor; Watson, George Neville (1920) [1st ed. 1902]. "22.8 The lemniscate functions". A Course of Modern Analysis (3rd ed.). Cambridge. pp. 524–528. Prasolov, Viktor; Solovyev, Yuri (1997). "4. Abel's Theorem on Division of Lemniscate". Elliptic functions and elliptic integrals. Translations of Mathematical Monographs. Vol. 170. American Mathematical Society. doi:10.1090/mmono/170. ISBN 978-0-8218-0587-9.
Cox (1984) Cox, David Archibald (1984). "The Arithmetic-Geometric Mean of Gauss". L'Enseignement Mathématique. 30 (2): 275–330.

retronews.fr (Global: 2,920^th place; English: low place)

Peirce (1879). Guyou (1887) and Adams (1925) introduced transverse and oblique aspects of the same projection, respectively. Also see Lee (1976). These authors write their projection formulas in terms of Jacobi elliptic functions, with a square lattice. Peirce, Charles Sanders (1879). "A Quincuncial Projection of the Sphere". American Journal of Mathematics. 2 (4): 394–397. doi:10.2307/2369491. JSTOR 2369491. Guyou, Émile (1887). "Nouveau système de projection de la sphère: Généralisation de la projection de Mercator" [New system of projection of the sphere]. Annales Hydrographiques. Série 2 (in French). 9: 16–35. Adams, Oscar S. (1925). Elliptic Functions Applied to Conformal World Maps (PDF). U.S. Coast and Geodetic Survey. US Government Printing Office. Special Pub. No. 112. Lee, L. P. (1976). Conformal Projections Based on Elliptic Functions. Cartographica Monographs. Vol. 16. Toronto: B. V. Gutsell, York University. ISBN 0-919870-16-3. Supplement No. 1 to The Canadian Cartographer 13.

umich.edu (Global: 429^th place; English: 307^th place)

www-personal.umich.edu

Cox & Hyde (2014) Cox, David Archibald; Hyde, Trevor (2014). "The Galois theory of the lemniscate" (PDF). Journal of Number Theory. 135: 43–59. arXiv:1208.2653. doi:10.1016/j.jnt.2013.08.006.

uni-goettingen.de (Global: 2,423^rd place; English: 2,569^th place)

gdz.sub.uni-goettingen.de

Fagnano (1718–1723); Euler (1761); Gauss (1917) Fagnano, Giulio Carlo (1718–1723) "Metodo per misurare la lemniscata" [Method for measuring the lemniscate]. Giornale de' letterati d'Italia (in Italian).
"Schediasma primo" [Part 1]. 1718. 29: 258–269.
"Giunte al primo schediasma" [Addendum to part 1]. 1723. 34: 197–207.
"Schediasma secondo" [Part 2]. 1718. 30: 87–111.
Reprinted as Fagnano (1850). "32–34. Metodo per misurare la lemniscata". Opere Matematiche, vol. 2. Allerighi e Segati. pp. 293–313. (Figures) Euler, Leonhard (1761). "Observationes de comparatione arcuum curvarum irrectificibilium" [Observations on the comparison of arcs of irrectifiable curves]. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae (in Latin). 6: 58–84. E 252. (Figures) Gauss, Carl Friedrich (1917). Werke (Band X, Abteilung I) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen.
Gauss (1917) p. 199 used the symbols $sl$ and $cl$ for the lemniscate sine and cosine, respectively, and this notation is most common today: see e.g. Cox (1984) p. 316, Eymard & Lafon (2004) p. 204, and Lemmermeyer (2000) p. 240. Ayoub (1984) uses $sinlem$ and $coslem$ . Whittaker & Watson (1920) use the symbols $sin lemn$ and $cos lemn$ . Some sources use the generic letters $s$ and $c$ . Prasolov & Solovyev (1997) use the letter $φ$ for the lemniscate sine and $φ'$ for its derivative. Gauss, Carl Friedrich (1917). Werke (Band X, Abteilung I) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Cox, David Archibald (1984). "The Arithmetic-Geometric Mean of Gauss". L'Enseignement Mathématique. 30 (2): 275–330. Eymard, Pierre; Lafon, Jean-Pierre (2004). The Number Pi. Translated by Wilson, Stephen. American Mathematical Society. ISBN 0-8218-3246-8. Lemmermeyer, Franz (2000). Reciprocity Laws: From Euler to Eisenstein. Springer. ISBN 3-540-66957-4. Ayoub, Raymond (1984). "The Lemniscate and Fagnano's Contributions to Elliptic Integrals". Archive for History of Exact Sciences. 29 (2): 131–149. doi:10.1007/BF00348244. Whittaker, Edmund Taylor; Watson, George Neville (1920) [1st ed. 1902]. "22.8 The lemniscate functions". A Course of Modern Analysis (3rd ed.). Cambridge. pp. 524–528. Prasolov, Viktor; Solovyev, Yuri (1997). "4. Abel's Theorem on Division of Lemniscate". Elliptic functions and elliptic integrals. Translations of Mathematical Monographs. Vol. 170. American Mathematical Society. doi:10.1090/mmono/170. ISBN 978-0-8218-0587-9.
Gauss, C. F. (1866). Werke (Band III) (in Latin and German). Herausgegeben der Königlichen Gesellschaft der Wissenschaften zu Göttingen. p. 405; there's an error on the page: the coefficient of $\varphi ^{17}$ should be ${\tfrac {107}{7\,410\,154\,752\,000}}$ , not ${\tfrac {107}{207\,484\,333\,056\,000}}$ .
Eisenstein, G. (1846). "Beiträge zur Theorie der elliptischen Functionen". Journal für die reine und angewandte Mathematik (in German). 30. Eisenstein uses $\varphi =\operatorname {sl}$ and $\omega =2\varpi$ .

unistra.fr (Global: 1,779^th place; English: 940^th place)

irma.math.unistra.fr

Schappacher (1997). OEIS sequence A062539 lists the lemniscate constant's decimal digits. Schappacher, Norbert (1997). "Some milestones of lemniscatomy" (PDF). In Sertöz, S. (ed.). Algebraic Geometry (Proceedings of Bilkent Summer School, August 7–19, 1995, Ankara, Turkey). Marcel Dekker. pp. 257–290.
Siegel (1969); Schappacher (1997) Siegel, Carl Ludwig (1969). "1. Elliptic Functions". Topics in Complex Function Theory, Vol. I. Wiley-Interscience. pp. 1–89. ISBN 0-471-60844-0. Schappacher, Norbert (1997). "Some milestones of lemniscatomy" (PDF). In Sertöz, S. (ed.). Algebraic Geometry (Proceedings of Bilkent Summer School, August 7–19, 1995, Ankara, Turkey). Marcel Dekker. pp. 257–290.

utpjournals.press (Global: low place; English: low place)

Peirce (1879). Guyou (1887) and Adams (1925) introduced transverse and oblique aspects of the same projection, respectively. Also see Lee (1976). These authors write their projection formulas in terms of Jacobi elliptic functions, with a square lattice. Peirce, Charles Sanders (1879). "A Quincuncial Projection of the Sphere". American Journal of Mathematics. 2 (4): 394–397. doi:10.2307/2369491. JSTOR 2369491. Guyou, Émile (1887). "Nouveau système de projection de la sphère: Généralisation de la projection de Mercator" [New system of projection of the sphere]. Annales Hydrographiques. Série 2 (in French). 9: 16–35. Adams, Oscar S. (1925). Elliptic Functions Applied to Conformal World Maps (PDF). U.S. Coast and Geodetic Survey. US Government Printing Office. Special Pub. No. 112. Lee, L. P. (1976). Conformal Projections Based on Elliptic Functions. Cartographica Monographs. Vol. 16. Toronto: B. V. Gutsell, York University. ISBN 0-919870-16-3. Supplement No. 1 to The Canadian Cartographer 13.
Adams (1925); Lee (1976). Adams, Oscar S. (1925). Elliptic Functions Applied to Conformal World Maps (PDF). U.S. Coast and Geodetic Survey. US Government Printing Office. Special Pub. No. 112. Lee, L. P. (1976). Conformal Projections Based on Elliptic Functions. Cartographica Monographs. Vol. 16. Toronto: B. V. Gutsell, York University. ISBN 0-919870-16-3. Supplement No. 1 to The Canadian Cartographer 13.

web.archive.org (Global: 1^st place; English: 1^st place)

Levin (2006) Levin, Aaron (2006). "A Geometric Interpretation of an Infinite Product for the Lemniscate Constant" (PDF). The American Mathematical Monthly. 113 (6): 510–520. doi:10.1080/00029890.2006.11920331. JSTOR 27641976.
Lindqvist & Peetre (2001) generalizes the first of these forms. Lindqvist, Peter; Peetre, Jaak (2001). "Two Remarkable Identities, Called Twos, for Inverses to Some Abelian Integrals" (PDF). The American Mathematical Monthly. 108 (5): 403–410. doi:10.1080/00029890.2001.11919766. Archived from the original (PDF) on 2022-05-28.
Levin (2006); Robinson (2019b) Levin, Aaron (2006). "A Geometric Interpretation of an Infinite Product for the Lemniscate Constant" (PDF). The American Mathematical Monthly. 113 (6): 510–520. doi:10.1080/00029890.2006.11920331. JSTOR 27641976. Robinson, Paul L. (2019b). The Elliptic Functions in a First-Order System (Preprint). arXiv:1903.07147.
Levin (2006) p. 515 Levin, Aaron (2006). "A Geometric Interpretation of an Infinite Product for the Lemniscate Constant" (PDF). The American Mathematical Monthly. 113 (6): 510–520. doi:10.1080/00029890.2006.11920331. JSTOR 27641976.

Lemniscate elliptic functions (English Wikipedia)

amherst.edu (Global: low place; English: 6,621st place)

dacox.people.amherst.edu

archive.org (Global: 6th place; English: 6th place)

arxiv.org (Global: 49th place; English: 31st place)

berkeley.edu (Global: 621st place; English: 429th place)

www2.eecs.berkeley.edu

csiro.au (Global: 3,457th place; English: 2,432nd place)

publications.csiro.au

doi.org (Global: 2nd place; English: 2nd place)

hdl.handle.net (Global: 66th place; English: 34th place)

ias.ac.in (Global: 6,516th place; English: 4,685th place)

johndcook.com (Global: low place; English: low place)

jstor.org (Global: 23rd place; English: 15th place)

nist.gov (Global: 708th place; English: 742nd place)

dlmf.nist.gov

noaa.gov (Global: 111th place; English: 78th place)

geodesy.noaa.gov

oeis.org (Global: 1,188th place; English: 866th place)

osu.edu (Global: 2,360th place; English: 1,459th place)

people.math.osu.edu

pacific.edu (Global: low place; English: low place)

scholarlycommons.pacific.edu

reed.edu (Global: low place; English: 7,280th place)

people.reed.edu

researchgate.net (Global: 96th place; English: 89th place)

retronews.fr (Global: 2,920th place; English: low place)

umich.edu (Global: 429th place; English: 307th place)

www-personal.umich.edu

uni-goettingen.de (Global: 2,423rd place; English: 2,569th place)

gdz.sub.uni-goettingen.de

unistra.fr (Global: 1,779th place; English: 940th place)

irma.math.unistra.fr

utpjournals.press (Global: low place; English: low place)

web.archive.org (Global: 1st place; English: 1st place)

amherst.edu (Global: low place; English: 6,621^st place)

archive.org (Global: 6^th place; English: 6^th place)

arxiv.org (Global: 49^th place; English: 31^st place)

berkeley.edu (Global: 621^st place; English: 429^th place)

csiro.au (Global: 3,457^th place; English: 2,432^nd place)

doi.org (Global: 2^nd place; English: 2^nd place)

hdl.handle.net (Global: 66^th place; English: 34^th place)

ias.ac.in (Global: 6,516^th place; English: 4,685^th place)

jstor.org (Global: 23^rd place; English: 15^th place)

nist.gov (Global: 708^th place; English: 742^nd place)

noaa.gov (Global: 111^th place; English: 78^th place)

oeis.org (Global: 1,188^th place; English: 866^th place)

osu.edu (Global: 2,360^th place; English: 1,459^th place)

reed.edu (Global: low place; English: 7,280^th place)

researchgate.net (Global: 96^th place; English: 89^th place)

retronews.fr (Global: 2,920^th place; English: low place)

umich.edu (Global: 429^th place; English: 307^th place)

uni-goettingen.de (Global: 2,423^rd place; English: 2,569^th place)

unistra.fr (Global: 1,779^th place; English: 940^th place)

web.archive.org (Global: 1^st place; English: 1^st place)