Lemniscate elliptic functions (English Wikipedia)

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

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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

Fagnano (1718–1723); Euler (1761); Gauss (1917) 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. 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

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

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. Levien, Raph (2008). The elastica: a mathematical history (PDF) (Technical report). University of California at Berkeley. UCB/EECS-2008-103.

csiro.au

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

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.
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.
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.

jstor.org

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

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

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

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

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

scholarlycommons.pacific.edu

Fagnano (1718–1723); Euler (1761); Gauss (1917) 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. Levien, Raph (2008). The elastica: a mathematical history (PDF) (Technical report). University of California at Berkeley. UCB/EECS-2008-103.

reed.edu

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

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

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

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

gdz.sub.uni-goettingen.de

Fagnano (1718–1723); Euler (1761); Gauss (1917) 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

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

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

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.