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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:
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Click for copyable input
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Evaluate to high precision:
The precision of the output tracks the precision of the input:
ModularLambda threads element-wise over lists:
TraditionalForm formatting:
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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