Lemniscatic elliptic function

In mathematics, a lemniscatic elliptic function is an elliptic function related to the arc length of a lemniscate of Bernoulli studied by Giulio Carlo de' Toschi di Fagnano in 1718. It has a square period lattice and is closely related to the Weierstrass elliptic function when the Weierstrass invariants satisfy $g 2 = 1$ and $g 3 = 0$ .

In the lemniscatic case, the minimal half period $ω 1$ is real and equal to

where $Γ$ is the gamma function. The second smallest half period is pure imaginary and equal to $iω 1$ . In more algebraic terms, the period lattice is a real multiple of the Gaussian integers.

The constants $e 1$ , $e 2$ , and $e 3$ are given by

The case $g 2 = a$ , $g 3 = 0$ may be handled by a scaling transformation. However, this may involve complex numbers. If it is desired to remain within real numbers, there are two cases to consider: $a > 0$ and $a < 0$ . The period parallelogram is either a square or a rhombus.

The lemniscate sine (Latin: sinus lemniscatus) and lemniscate cosine (Latin: cosinus lemniscatus) functions $sinlemn$ aka $sl$ and $coslemn$ aka $cl$ are analogues of the usual sine and cosine functions, with a circle replaced by a lemniscate. They are defined by

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