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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 half‐periods .
ModularLambda gives the parameter for elliptic functions in terms of according to .
ModularLambda is related to EllipticTheta by $TemplateBox[{tau}, ModularLambda]=TemplateBox[{2, 0, q}, EllipticTheta]^4/TemplateBox[{3, 0, q}, EllipticTheta]^4$ 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 can be used with CenteredInterval objects. »
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 (23)

Numerical Evaluation (5)

Evaluate numerically:

Evaluate to high precision:

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

Evaluate efficiently at high precision:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix ModularLambda function using MatrixFunction:

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 threads elementwise over lists:

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:

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 Darboux–Halphen 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:

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