InverseJacobiSN

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

Examples

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Basic Examples  (5)

Evaluate numerically:

Plot the function at different values of the modulus m:

Plot over a subset of the complexes:

Series expansions at the origin:

Series expansion at Infinity:

Scope  (21)

Numerical Evaluation  (3)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate InverseJacobiSN efficiently at high precision:

Specific Values  (4)

Simple exact values are generated automatically:

Value at infinity:

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

Parity transformation is automatically applied:

Visualization  (3)

Plot InverseJacobiSN for various values of the second parameter :

Plot InverseJacobiSN as a function of its parameter :

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

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

Differentiation and Integration  (4)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Differentiate InverseJacobiSN with respect to the second argument :

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

Series Expansions  (2)

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

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

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

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

Function Identities and Simplifications  (2)

InverseJacobiSN is the inverse function of JacobiSN:

Compose with inverse function:

Use PowerExpand to disregard multivaluedness of the inverse function:

Other Features  (3)

InverseJacobiSN threads elementwise over lists:

InverseJacobiSN can be applied to a power series:

TraditionalForm formatting:

Generalizations & Extensions  (1)

InverseJacobiSN 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 InverseJacobiSN 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)