InverseJacobiDN

InverseJacobiDN[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 .
  • InverseJacobiDN has branch cut discontinuities in the complex v plane with branch points at and infinity, and in the complex m plane with branch points and infinity.
  • The inverse Jacobi elliptic functions are related to elliptic integrals.
  • For certain special arguments, InverseJacobiDN automatically evaluates to exact values.
  • InverseJacobiDN can be evaluated to arbitrary numerical precision.
  • InverseJacobiDN automatically threads over lists.

Examples

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Basic Examples  (4)

Evaluate numerically:

Plot the real part of the function:

Series expansion at the origin:

Series expansion at Infinity:

Scope  (20)

Numerical Evaluation  (3)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate InverseJacobiDN efficiently at high precision:

Specific Values  (3)

Simple exact results are generated automatically:

Value at infinity:

Find a real root of the equation TemplateBox[{x, {4, /, 5}}, InverseJacobiDN]=1:

Visualization  (3)

Plot InverseJacobiDN for various values of the second parameter :

Plot InverseJacobiDN as a function of its parameter :

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

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

Differentiation  (4)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Differentiate InverseJacobiDN with respect to the second argument :

Higher derivatives:

Series Expansions  (3)

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

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

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

InverseJacobiDN can be applied to a power series:

Function Identities and Simplifications  (2)

InverseJacobiDN is the inverse function of JacobiDN:

Compose with inverse function:

Use PowerExpand to disregard multivaluedness of the inverse function:

Other Features  (2)

InverseJacobiDN threads elementwise over lists:

TraditionalForm formatting:

Applications  (1)

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

Properties & Relations  (1)

Obtain InverseJacobiDN 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)