EllipticNomeQ

EllipticNomeQ[m]

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 .
  • EllipticNomeQ[m] 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  (21)

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  (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 TemplateBox[{{x, +, {i,  , y}}}, EllipticNomeQ]:

Plot the imaginary part of TemplateBox[{{x, +, {i,  , y}}}, EllipticNomeQ]:

Function Properties  (4)

Real and complex domains of EllipticNomeQ:

Approximate function range of EllpiticNomeQ:

EllipticNomeQ threads element-wise over lists:

TraditionalForm formatting:

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 Infinity:

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  (1)

Plot EllipticNomeQ over the complex plane:

Properties & Relations  (5)

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:

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:

Introduced in 1996
 (3.0)