# KleinInvariantJ 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.
• KleinInvariantJ can be used with CenteredInterval objects. »

# 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(22)

### Numerical Evaluation(4)

Evaluate numerically:

Evaluate to high precision:

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

Evaluate efficiently at high precision:

KleinInvariantJ can be used with CenteredInterval objects:

### 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(10)

Complex domain of KleinInvariantJ:

KleinInvariantJ is a periodic function:

KleinInvariantJ is an analytic function on its domain of definition:

It has no singularities or discontinuities there: is neither nondecreasing nor nonincreasing:

KleinInvariantJ is not injective over the complexes: is not surjective: is neither non-negative nor non-positive: is neither convex nor concave:

### 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:

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: