# InverseEllipticNomeQ

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

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• yields the unique value of the parameter m which makes equal to q.
• The nome q must always satisfy .
• InverseEllipticNomeQ can be evaluated to arbitrary numerical precision.
• InverseEllipticNomeQ automatically threads over lists.

# Examples

open allclose all

## Basic Examples(4)

Evaluate numerically:

Plot over a subset of the reals:

Series expansion at the origin:

Asymptotic expansion at a singular point:

## Scope(18)

### 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 :

Plot the imaginary part of :

### Function Properties(4)

Real domain of InverseEllipticNomeQ:

Complex domain of InverseEllipticNomeQ:

### 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(3)

Convert between elliptic modulus and nome in elliptic function identities:

Partition function for a oneatom 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:

## 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: