InverseJacobiCS

InverseJacobiCS[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 .
  • InverseJacobiCS 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, InverseJacobiCS automatically evaluates to exact values.
  • InverseJacobiCS can be evaluated to arbitrary numerical precision.
  • InverseJacobiCS automatically threads over lists.

Examples

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Basic Examples  (4)

Evaluate numerically:

Plot the function over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Scope  (22)

Numerical Evaluation  (4)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate InverseJacobiCS efficiently at high precision:

InverseJacobiCS threads elementwise over lists:

Specific Values  (4)

Simple exact results are generated automatically:

Values at infinity:

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

Parity transformation is automatically applied:

Visualization  (3)

Plot InverseJacobiCS for various values of the second parameter :

Plot InverseJacobiCS as a function of its parameter :

Plot the real part of TemplateBox[{{1, /, 2}, {x, +, {ⅈ,  , y}}}, InverseJacobiCS]:

Plot the imaginary part of TemplateBox[{{1, /, 2}, {x, +, {ⅈ,  , y}}}, InverseJacobiCS]:

Differentiation and Integration  (5)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Differentiate InverseJacobiCS with respect to the second argument :

Higher derivatives:

Definite integral of an odd function over an interval centered at the origin is 0:

Series Expansions  (2)

Series expansion for TemplateBox[{nu, m}, InverseJacobiCS]:

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

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

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

Function Identities and Simplifications  (2)

InverseJacobiCS is the inverse function of JacobiCS:

Compose with inverse function:

Use PowerExpand to disregard multivaluedness of the inverse function:

Other Features  (2)

InverseJacobiCS can be applied to a power series:

TraditionalForm formatting:

Generalizations & Extensions  (1)

InverseJacobiCS 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 InverseJacobiCS from solving equations containing elliptic functions:

Introduced in 1988
 (1.0)