Editing Lemniscate of Bernoulli (section)

==Arc length and elliptic functions==
{{main|Lemniscate elliptic functions}}
[[File:The lemniscate sine and cosine related to the arclength of the lemniscate of Bernoulli.png|thumb|upright=1.8|The [[lemniscate elliptic functions|lemniscate sine and cosine]] relate the arc length of an arc of the lemniscate to the distance of one endpoint from the origin.]]
The determination of the [[arc length]] of arcs of the lemniscate leads to [[elliptic integral]]s, as was discovered in the eighteenth century. Around 1800, the [[elliptic function]]s inverting those integrals were studied by [[C. F. Gauss]] (largely unpublished at the time, but allusions in the notes to his ''[[Disquisitiones Arithmeticae]]''). The [[period lattice]]s are of a very special form, being proportional to the [[Gaussian integer]]s. For this reason the case of elliptic functions with [[complex multiplication]] by [[square root of minus one|{{sqrt|−1}}]] is called the ''[[lemniscatic case]]'' in some sources.

Using the elliptic integral
:<math>\operatorname{arcsl}x \stackrel{\text{def}}{{}={}} \int_0^x\frac{dt}{\sqrt{1-t^4}}</math>
the formula of the arc length {{mvar|L}} can be given as
:<math>\begin{align} L &= 4a \int_{0}^1\frac{dt}{\sqrt{1-t^4}} = 4a\,\operatorname{arcsl}1 = 2\varpi a \\[6pt] &= \frac{\Gamma (1/4)^2}{\sqrt\pi}\,c =\frac{2\pi}{\operatorname{M}(1,1/\sqrt{2})}c\approx 7{.}416 \cdot c \end{align}</math>
where <math>c</math> and <math>a = \sqrt{2}c</math> are defined as above, <math>\varpi = 2 \operatorname{arcsl}{1}</math> is the [[lemniscate constant]], <math>\Gamma</math> is the [[gamma function]] and <math>\operatorname{M}</math> is the [[arithmetic–geometric mean]].