# EllipticNomeQ

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

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• EllipticNomeQ is related to EllipticK by .
• has a branch cut discontinuity in the complex m plane running from to .
• For certain special arguments, EllipticNomeQ automatically evaluates to exact values.
• EllipticNomeQ can be evaluated to arbitrary numerical precision.
• EllipticNomeQ automatically threads over lists.

# Examples

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## Basic Examples(6)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Asymptotic expansion at Infinity:

Asymptotic expansion at a singular point:

## Scope(29)

### Numerical Evaluation(6)

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:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix EllipticNomeQ function using MatrixFunction:

### Specific Values(5)

Values at fixed points:

Evaluate symbolically:

Value at zero:

Simple arguments evaluate automatically:

Find a value of x for which EllipticNomeQ[x]=0.1:

### Visualization(2)

Plot the EllipticNomeQ function for various parameters:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(10)

Real and complex domains of EllipticNomeQ:

Approximate function range of EllpiticNomeQ:

EllipticNomeQ is not an analytic function:

Has both singularities and discontinuities for x1:

EllipticNomeQ is nondecreasing over its real domain:

EllipticNomeQ is injective:

EllipticNomeQ is not surjective:

EllipticNomeQ is neither non-negative nor non-positive:

EllipticNomeQ is convex over its real domain:

### Differentiation(2)

First derivative with respect to m:

Higher derivatives with respect to m:

Plot the higher derivatives with respect to m:

### Series Expansions(4)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find the series expansion at:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

## Generalizations & Extensions(1)

EllipticNomeQ can be applied to power series:

## Applications(3)

Define the Halphen constant [MathWorld]:

Find the extended precision value:

Verify that it is zero of the function :

Plot EllipticNomeQ over the complex plane:

Closed form of the iteration steps for calculating the arithmeticgeometric mean:

Show convergence speed using arbitraryprecision arithmetic:

Compute a thousand digits of :

## Properties & Relations(6)

Use FullSimplify to simplify expressions containing EllipticNomeQ:

Compose with inverse functions:

Find the derivative:

Symbolically solve a transcendental equation:

Numerically find a root of a transcendental equation:

Special values of Neville theta functions involve EllipticNomeQ:

## Possible Issues(1)

For most named special functions, the direct function is singlevalued and the inverse is multivalued. EllipticNomeQ is a multivalued function and the inverse function, InverseEllipticNomeQ, is single-valued. As a result, the following is correct everywhere:

## Neat Examples(1)

Riemann surface of EllipticNomeQ:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_ellipticnomeq, author="Wolfram Research", title="{EllipticNomeQ}", year="1996", howpublished="\url{https://reference.wolfram.com/language/ref/EllipticNomeQ.html}", note=[Accessed: 16-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_ellipticnomeq, organization={Wolfram Research}, title={EllipticNomeQ}, year={1996}, url={https://reference.wolfram.com/language/ref/EllipticNomeQ.html}, note=[Accessed: 16-September-2024 ]}