Arc Length and Elliptic Functions
The determination of the arc length of arcs of the lemniscate leads to elliptic integrals, as was discovered in the eighteenth century. Around 1800, the elliptic functions inverting those integrals were studied by C. F. Gauss (largely unpublished at the time, but allusions in the notes to his Disquisitiones Arithmeticae).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 lattices are of a very special form, being proportional to the Gaussian integers. For this reason the case of elliptic functions with complex multiplication by the square root of minus one is called the lemniscatic case in some sources.
Read more about this topic: Lemniscate Of Bernoulli
Famous quotes containing the words arc, length and/or functions:
“Male urination really is a kind of accomplishment, an arc of transcendance. A woman merely waters the ground she stands on.”
—Camille Paglia (b. 1947)
“Oft have I mused, but now at length I find,
Why those that die, men say they do depart.”
—Sir Philip Sidney (1554–1586)
“Mark the babe
Not long accustomed to this breathing world;
One that hath barely learned to shape a smile,
Though yet irrational of soul, to grasp
With tiny finger—to let fall a tear;
And, as the heavy cloud of sleep dissolves,
To stretch his limbs, bemocking, as might seem,
The outward functions of intelligent man.”
—William Wordsworth (1770–1850)