Lemniscatic elliptic function

In mathematics, and in particular the study of Weierstrass elliptic functions, the lemniscatic case occurs when the Weierstrass invariants satisfy g₂=1 and g₃=0. This page follows the terminology of Abramowitz and Stegun; see also the equianharmonic case.

In the lemniscatic case, the minimal half period $\omega_1$ is real and equal to

$\frac{\Gamma^2(\tfrac{1}{4})}{4\sqrt{\pi}}$

where $\Gamma$ is the Gamma function. The second smallest half period is pure imaginary and equal to $i\omega_1$. 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

$e_1=\tfrac{1}{2},\qquad e_2=0,\qquad e_3=-\tfrac{1}{2}.$

The case g₂=a, g₃=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.

See also

• Gauss's constant

References

• Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 18", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, pp. 658, ISBN 978-0486612720, MR0167642, http://www.math.sfu.ca/~cbm/aands/page_658.htm .