Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate JacobiDN efficiently at high precision:

Compute average case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix JacobiDN function using MatrixFunction:

Simple exact values are generated automatically:

Some poles of JacobiDN:

Find a local maximum of JacobiDN as a root of :

Plot the JacobiDN functions for various parameter values:

Plot JacobiDN as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

JacobiDN is -periodic along the real axis:

JacobiDN is -periodic along the imaginary axis:

JacobiDN is an even function in its first argument:

JacobiDN is an analytic function:

It has no singularities or discontinuities:

is neither nondecreasing nor nonincreasing:

is not injective for any fixed :

is not surjective for any fixed :

is non-negative for :

In general, it is neither non-negative nor non-positive:

JacobiDN is neither convex nor concave:

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

Indefinite integral of JacobiDN:

Definite integral of JacobiDN 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 :

JacobiDN can be applied to a power series:

Parity transformations and periodicity relations are automatically applied:

Identity involving JacobiSN:

Argument simplifications:

Differential representation:

Relation to other Jacobi elliptic functions:

TraditionalForm formatting: