Evaluate numerically to high precision:

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

Evaluate for complex arguments:

Evaluate JacobiND efficiently at high precision:

JacobiND threads elementwise over lists:

Simple exact values are generated automatically:

Some poles of JacobiND:

Find a local minimum of JacobiND as a root of :

Plot the JacobiND functions for various parameter values:

Plot JacobiND as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

JacobiND is -periodic along the real axis:

JacobiND is -periodic along the imaginary axis:

JacobiND is an even function in its first argument:

is an analytic function of for :

It is not, in general, an analytic function:

It has both singularities and discontinuities:

is neither nondecreasing nor nonincreasing:

is not injective for :

is not surjective for any fixed :

is non-negative for :

In general, it has indeterminate sign:

JacobiND is neither convex nor concave:

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

Indefinite integral of JacobiND:

Definite integral of the even function over the interval centered at the origin:

This is twice the integral over half the interval:

More integrals:

Taylor expansion for :

Plot the first three approximations for around :

Taylor expansion for :

Plot the first three approximations for around :

JacobiND can be applied to a power series:

Parity transformations and periodicity relations are automatically applied:

Identity involving JacobiSD:

Argument simplifications:

Primary definition:

Relation to other Jacobi elliptic functions:

TraditionalForm formatting: