KleinInvariantJ

KleinInvariantJ[τ]

gives the Klein invariant modular elliptic function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The argument is the ratio of Weierstrass halfperiods .
  • KleinInvariantJ is given in terms of Weierstrass invariants by .
  • is invariant under any combination of the modular transformations and .
  • For certain special arguments, KleinInvariantJ automatically evaluates to exact values.
  • KleinInvariantJ can be evaluated to arbitrary numerical precision.
  • KleinInvariantJ automatically threads over lists.

Examples

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Basic Examples  (4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Scope  (15)

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)

Values at fixed points:

Find the first positive maximum of the real part of KleinInvariantJ:

Visualization  (2)

Plot the real part of KleinInvariantJ:

Plot the real part of J(τ) function:

Plot the imaginary part of J(τ) function:

Function Properties  (4)

Complex domain of KleinInvariantJ:

KleinInvariantJ is a periodic function:

KleinInvariantJ threads elementwise over lists:

TraditionalForm formatting:

Differentiation  (2)

First derivative with respect to τ:

The first and second derivatives with respect to τ:

Plot the first and second derivatives with respect to τ:

Series Expansions  (2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

Applications  (6)

Some modular properties of KleinInvariantJ are automatically applied:

Verify a more complicated identity numerically:

Find values at quadratic irrationals:

KleinInvariantJ 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 Chazy equation :

Plot the solution:

Plot the absolute value in the complex plane:

Plot the imaginary part in the complex plane:

Properties & Relations  (2)

Find derivatives:

Find a numerical root:

Possible Issues  (2)

Machine-precision input may be insufficient to give the correct answer:

With exact input, the answer is correct:

KleinInvariantJ remains unevaluated outside of its domain of analyticity:

Introduced in 1996
 (3.0)