NationMaster - Encyclopedia: Lemniscatic elliptic function

In mathematics, and in particular the study of Weierstrass elliptic functions, the lemniscatic case occurs when the Weierstrass invariants satisfy $g 2 = 1$ and $g 3 = 0$ . This page follows the terminology of Abramowitz and Stegun; see also the equianharmonic case. Euclid, detail from The School of Athens by Raphael. ... In mathematics, Weierstrass introduced some particular elliptic functions that have become the basis for the most standard notations used. ... Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ... In mathematics, and in particular the study of Weierstrass elliptic functions, the Equianharmonic case occurs when the Weierstrass invariants satisfy and ; This page follows the terminology of Abramowitz and Stegun; see also the lemniscatic case. ...

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

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

where $Γ$ is the Gamma function. The second smallest half period is pure imaginary and equal to $i ω 1$ . In more algebraic terms, the period lattice is a real multiple of the Gaussian integers. The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ... In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. ... A Gaussian integer is a complex number whose real and imaginary part are both integers. ...

The constants $e 1$ , $e 2$ and $e 3$ are given by A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ...

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

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