# 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(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 :

Plot the imaginary part of :

### Function Properties(10)

Real domain of InverseEllipticNomeQ:

Complex domain of InverseEllipticNomeQ:

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:

### 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 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:

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:

Wolfram Research (1996), InverseEllipticNomeQ, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseEllipticNomeQ.html.

#### Text

Wolfram Research (1996), InverseEllipticNomeQ, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseEllipticNomeQ.html.

#### CMS

Wolfram Language. 1996. "InverseEllipticNomeQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/InverseEllipticNomeQ.html.

#### APA

Wolfram Language. (1996). InverseEllipticNomeQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseEllipticNomeQ.html

#### BibTeX

@misc{reference.wolfram_2024_inverseellipticnomeq, author="Wolfram Research", title="{InverseEllipticNomeQ}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/InverseEllipticNomeQ.html}", note=[Accessed: 19-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_inverseellipticnomeq, organization={Wolfram Research}, title={InverseEllipticNomeQ}, year={1996}, url={https://reference.wolfram.com/language/ref/InverseEllipticNomeQ.html}, note=[Accessed: 19-July-2024 ]}