ModularLambda

ModularLambda[τ]

gives the modular lambda elliptic function .

Details

  • 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 halfperiods .
  • 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 . »
  • For certain special arguments, ModularLambda automatically evaluates to exact values.
  • ModularLambda can be evaluated to arbitrary numerical precision.
  • ModularLambda automatically threads over lists.

Examples

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Basic Examples  (3)

Evaluate numerically:

Plot over a subset of the reals:

Series expansion at the origin:

Scope  (16)

Numerical Evaluation  (3)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate efficiently at high precision:

Specific Values  (2)

Value at a fixed point:

Find the first positive minimum of ModularLambda[x+I]:

Visualization  (3)

Plot the real part of ModularLambda:

Plot the absolute value of ModularLambda:

Plot the real part of ModularLambda function:

Plot the imaginary part of ModularLambda function:

Function Properties  (4)

Complex domain of ModularLambda:

ModularLambda is a periodic function:

ModularLambda threads elementwise over lists:

TraditionalForm formatting:

Differentiation  (2)

First derivative with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z:

Series Expansions  (2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

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 DarbouxHalphen system:

Check:

Plot the real part in the complex plane:

Properties & Relations  (2)

Find derivatives:

Find a numerical root:

Possible Issues  (2)

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:

Introduced in 1996
 (3.0)