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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 Section 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]=