This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# ModularLambda

 ModularLambda[]gives the modular lambda elliptic function .
• Mathematical function, suitable for both symbolic and numerical manipulation.
• 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 .
• is invariant under any combination of the modular transformations and . »
• For certain special arguments, ModularLambda automatically evaluates to exact values.
Evaluate numerically:
Evaluate numerically:
 Out[1]=

 Out[1]=
 Scope   (4)
Evaluate to high precision:
The precision of the output tracks the precision of the input:
 Applications   (4)
Some modular properties of ModularLambda are automatically applied:
Verify a more complicated identity numerically:
ModularLambda is a modular function. Make an ansatz for a modular equation:
Form an overdetermined system of equations and solve it:
This is the modular equation of order 2:
Solution of the Darboux-Halphen system:
Check:
Plot the real part in the complex plane:
Find derivatives:
Find a numerical root:
Machine-precision input is insufficient to give a correct answer:
With exact input, the answer is correct:
ModularLambda remains unevaluated outside of its domain of analyticity:
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