Evaluate numerically to high precision:

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

Evaluate for complex arguments:

Evaluate JacobiNS efficiently at high precision:

JacobiNS threads elementwise over lists:

Simple exact values are generated automatically:

Some poles of JacobiNS:

Find a local minimum of JacobiNS as a root of :

Plot the JacobiNS functions for various parameter values:

Plot JacobiNS as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

JacobiNS is -periodic along the real axis:

JacobiNS is -periodic along the imaginary axis:

JacobiNS is an odd function in its first argument:

JacobiNS is not an analytic function:

It has both singularities and discontinuities:

is neither nondecreasing nor nonincreasing:

is not injective for any fixed :

It is injective for :

JacobiNS is not surjective for any fixed :

JacobiNS is neither non-negative nor non-positive:

JacobiNS is neither convex nor concave:

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

Indefinite integral of JacobiNS:

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

More integrals:

Series expansion for :

Plot the first three approximations for around :

Taylor expansion for :

Plot the first three approximations for around :

JacobiNS can be applied to power series:

Parity transformation and periodicity relations are automatically applied:

Identity involving JacobiCS:

Argument simplifications:

Representation in terms of Csc of JacobiAmplitude:

Relation to other Jacobi elliptic functions:

TraditionalForm formatting: