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

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot over a subset of the reals:

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:

ModularLambda can be used with CenteredInterval objects:

### 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(9)

ModularLambda is defined in the upper half-plane:

ModularLambda is a periodic function:

ModularLambda is an analytic function on its domain:

Therefore it has neither singularities nor discontinuities there: is neither nondecreasing nor nonincreasing: is not surjective: is neither non-negative nor non-positive: is neither convex nor concave:

### 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: