InverseEllipticNomeQ

InverseEllipticNomeQ[q]

gives the parameter m corresponding to the nome q in an elliptic function.

Details

Examples

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Basic Examples  (4)

Evaluate numerically:

Plot over a subset of the reals:

Series expansion at the origin:

Asymptotic expansion at a singular point:

Scope  (24)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

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

Complex number input:

Evaluate efficiently at high precision:

Specific Values  (4)

Value at a fixed point:

Evaluate symbolically:

Value at zero:

Find a value of for which InverseEllipticNomeQ[x]=0.9:

Visualization  (2)

Plot the InverseEllipticNomeQ function for various parameters:

Plot the real part of TemplateBox[{z}, InverseEllipticNomeQ]:

Plot the imaginary part of TemplateBox[{z}, InverseEllipticNomeQ]:

Function Properties  (10)

Real domain of InverseEllipticNomeQ:

Complex domain of InverseEllipticNomeQ:

InverseEllipticNomeQ threads element-wise over lists:

InverseEllipticNomeQ is an analytic function over its real domain:

In general, it has both singularities and discontinuities:

InverseEllipticNomeQ is nondecreasing over its real domain:

InverseEllipticNomeQ is injective:

InverseEllipticNomeQ is not surjective:

InverseEllipticNomeQ is neither non-negative nor non-positive:

InverseEllipticNomeQ is concave over its real domain:

TraditionalForm formatting:

Differentiation  (2)

First derivative with respect to :

Higher derivatives with respect to :

Plot the higher derivatives with respect to :

Series Expansions  (2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

Generalizations & Extensions  (1)

InverseEllipticNomeQ can be applied to power series:

Applications  (4)

Convert between elliptic modulus and nome in elliptic function identities:

Partition function for a one‐atom monatomic gas in a finite container of unit length:

Form partition functions for n bosonic particles:

Calculate and plot mean energies:

InverseEllipticNomeQ 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:

Verify using Series:

Find the modulus corresponding to the elliptic curve, specified by Weierstrass invariants:

Compute the modulus alternatively using InverseEllipticNomeQ:

Properties & Relations  (5)

Compose with inverse functions:

Find derivatives:

Symbolically solve a transcendental equation:

Numerically find a root of a transcendental equation:

Relation to q-series:

Possible Issues  (2)

InverseEllipticNomeQ remains unevaluated outside its domain of analyticity:

InverseEllipticNomeQ is single valued, and EllipticNomeQ is multivalued:

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