This is documentation for Mathematica 3, which was
based on an earlier version of the Wolfram Language.
 ModularLambda ModularLambda[] gives the modular lambda elliptic function . Mathematical function (see Section A.3.10). ModularLambda is defined only in the upper half of the complex plane. It is not defined for real . The argument is the ratio of Weierstrass half-periods . ModularLambda gives the parameter for elliptic functions in terms of according to . ModularLambda is related to EllipticTheta by where the nome is given by . is invariant under any combination of the modular transformations and . See SectionÂ 3.2.11 for a discussion of argument conventions for elliptic functions. See the Mathematica book: Section 3.2.11. See also: DedekindEta, KleinInvariantJ, WeierstrassHalfPeriods. Further Examples The parameter appearing in the definition of the incomplete elliptic integral is called the modulus. For fixed, can be regarded as a map from (most of) the complex plane into the elliptic curve with half-periods and . Thus from the modulus we obtain the invariant of the curve where the corresponding elliptic functions live. The modular function reverses this correspondence, that is, . In[1]:= Out[1]= The function is invariant under the group of transformations of the complex upper half-plane generated by and . In[2]:= Out[2]=