InverseJacobiCD

InverseJacobiCD[v,m]

gives the inverse Jacobi elliptic function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • gives the value of u for which .
  • InverseJacobiCD has branch cut discontinuities in the complex v plane with branch points at and infinity, and in the complex m plane with branch points at and infinity.
  • The inverse Jacobi elliptic functions are related to elliptic integrals.
  • For certain special arguments, InverseJacobiCD automatically evaluates to exact values.
  • InverseJacobiCD can be evaluated to arbitrary numerical precision.
  • InverseJacobiCD automatically threads over lists.

Examples

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Basic Examples  (5)

Evaluate numerically:

Plot the function over a subset of the reals:

Plot over a subset of the complexes:

Series expansions at the origin:

Series expansion at Infinity:

Scope  (20)

Numerical Evaluation  (4)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate InverseJacobiCD efficiently at high precision:

InverseJacobiCD threads elementwise over lists:

Specific Values  (3)

Simple exact results are generated automatically:

Values at infinity:

Find a real root of the equation TemplateBox[{x, {1, /, 3}}, InverseJacobiCD]=1:

Visualization  (3)

Plot InverseJacobiCD for various values of the second parameter :

Plot InverseJacobiCD as a function of its parameter :

Plot the real part of TemplateBox[{{x, +, {ⅈ,  , y}}, {1, /, 3}}, InverseJacobiCD]:

Plot the imaginary part of TemplateBox[{{x, +, {ⅈ,  , y}}, {1, /, 3}}, InverseJacobiCD]:

Differentiation  (4)

First derivative:

Higher derivatives:

Differentiate InverseJacobiCD with respect to the second argument :

Higher derivatives:

Series Expansions  (2)

Taylor expansion for TemplateBox[{nu, m}, InverseJacobiCD]:

Plot the first three approximations for TemplateBox[{nu, {1, /, 3}}, InverseJacobiCD] around :

Taylor expansion for TemplateBox[{nu, m}, InverseJacobiCD]:

Plot the first three approximations for TemplateBox[{{-, {1, /, 2}}, m}, InverseJacobiCD] around :

Function Identities and Simplifications  (2)

InverseJacobiCD is the inverse function of JacobiCD:

Compose with the inverse function:

Use PowerExpand to disregard multivaluedness of the inverse function:

Other Features  (2)

InverseJacobiCD can be applied to a power series:

TraditionalForm formatting:

Generalizations & Extensions  (1)

InverseJacobiCD can be applied to a power series:

Applications  (1)

Plot contours of constant real and imaginary parts in the complex plane:

Properties & Relations  (1)

Obtain InverseJacobiCD from solving equations containing elliptic functions:

Possible Issues  (1)

Machine-precision input is insufficient to get a correct answer:

With exact input, the answer is correct:

Introduced in 1988
 (1.0)