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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> Line 1,323: * {{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 }}

Lemniscate elliptic functions: Difference between revisions