Lemniscate elliptic functions: Difference between revisions

Revision as of 21:50, 19 June 2025

Revision as of 21:51, 19 June 2025

Line 1,165:

:<math>\mathrm{H}_8=\frac{3}{10},\,\mathrm{H}_{12}=\frac{567}{130},\,\mathrm{H}_{16}=\frac{43\,659}{170},\,\ldots</math>

Also<ref>{{cite journal |last1=Katz |first1=Nicholas M. |date=1975 |title=The congruences of Clausen — von Staudt and Kummer for Bernoulli-Hurwitz numbers |url=https://link.springer.com/article/10.1007/BF02547966 |journal=Mathematische Annalen |volume=216 |issue=1 |pages=1–4|doi=10.1007/BF02547966 |url-access=subscription }} See eq. (9)</ref>

:<math>\operatorname{denom}\mathrm{H}_{4n}=\prod_{(p-1)|4n}p</math>

where <math>p\in\mathbb{P}</math> such that <math>p\not\equiv 3\,(\text{mod}\,4),</math>

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* {{cite journal |last1=Nishimura |first1=Ryo |title=New properties of the lemniscate function and its transformation |journal=Journal of Mathematical Analysis and Applications |date=2015 |volume=427 |issue=1 |pages=460–468 |doi=10.1016/j.jmaa.2015.02.066 |doi-access=free }}

* {{cite journal |last1=Ogawa |first1=Takuma |title=Similarities between the trigonometric function and the lemniscate function from arithmetic view point |journal=Tsukuba Journal of Mathematics |date=2005 |volume=29 |issue=1 |doi=10.21099/tkbjm/1496164894 |url=https://projecteuclid.org/journals/tsukuba-journal-of-mathematics/volume-29/issue-1/Similarities-between-the-trigonometric-function-and-the-lemniscate-function-from/10.21099/tkbjm/1496164894.full }}

* {{cite journal |last1=Peirce |first1=Charles Sanders |authorlink1=Charles Sanders Peirce |date=1879 |title=A Quincuncial Projection of the Sphere |journal=American Journal of Mathematics |volume=2 |issue=4 |pages=394–397 |doi=10.2307/2369491 |doi-access=free |jstor=2369491 |jstor-access=free |url=https://archive.org/details/sim_american-journal-of-mathematics_1879_2/page/n403/mode/2up}}

* {{cite book |last=Popescu-Pampu |first=Patrick |date=2016 |title=What is the Genus? |series=Lecture Notes in Mathematics |volume=2162 |publisher=Springer |doi=10.1007/978-3-319-42312-8 |isbn=978-3-319-42311-1 }}

* {{cite book |last1=Prasolov |first1=Viktor |last2=Solovyev |first2=Yuri |date=1997 |chapter=4. Abel's Theorem on Division of Lemniscate |title=Elliptic functions and elliptic integrals |series=Translations of Mathematical Monographs |volume=170 |publisher=American Mathematical Society. |doi=10.1090/mmono/170 |isbn=978-0-8218-0587-9 }}