Lemniscate elliptic functions

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 ${\displaystyle \omega _{1}}$ is real and equal to

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

where ${\displaystyle \Gamma }$ is the Gamma function. The second smallest half period is pure imaginary and equal to ${\displaystyle i\omega _{1}}$. In more algebraic terms, the period lattice is a real multiple of the Gaussian integers.

The constants ${\displaystyle e_{1}}$, ${\displaystyle e_{2}}$ and ${\displaystyle e_{3}}$ are given by

${\displaystyle e_{1}={\tfrac {1}{2}},\qquad e_{2}=0,\qquad e_{3}=-{\tfrac {1}{2}}.}$

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.

References

• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 18". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 627. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.