InverseJacobiNC

InverseJacobiNC[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 .
  • InverseJacobiNC has branch cut discontinuities in the complex 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, InverseJacobiNC automatically evaluates to exact values.
  • InverseJacobiNC can be evaluated to arbitrary numerical precision.
  • InverseJacobiNC automatically threads over lists.

Examples

open allclose all

Basic Examples  (5)

Evaluate numerically:

Check defining property:

Plot the function over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Scope  (20)

Numerical Evaluation  (3)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate InverseJacobiNC efficiently at high precision:

Specific Values  (3)

Simple exact values are generated automatically:

Value at infinity:

Find a real root of the equation TemplateBox[{x, {1, /, 2}}, InverseJacobiNC]=1:

Visualization  (3)

Plot InverseJacobiNC for various values of the second parameter :

Plot InverseJacobiNC as a function of its parameter :

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

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

Differentiation  (4)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Differentiate InverseJacobiNC with respect to the second argument :

Plot higher derivatives for :

Series Expansions  (3)

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

Plot the first three approximations for TemplateBox[{nu, {1, /, 2}}, InverseJacobiNC] around :

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

Plot the first three approximations for TemplateBox[{2, m}, InverseJacobiNC] around :

InverseJacobiNC can be applied to a power series:

Function Identities and Simplifications  (2)

InverseJacobiNC is the inverse function of JacobiNC:

Compose with inverse function:

Use PowerExpand to disregard multivaluedness of the inverse function:

Other Features  (2)

InverseJacobiNC threads elementwise over lists:

TraditionalForm formatting:

Applications  (1)

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

Properties & Relations  (1)

Obtain InverseJacobiNC 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)