Evaluate numerically to high precision:

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

Evaluate for complex arguments:

Evaluate JacobiDS efficiently at high precision:

JacobiDS threads elementwise over lists:

Simple exact values are generated automatically:

Some poles of JacobiDS:

Find a local minimum of JacobiDS as a root of :

Plot the JacobiDS functions for various parameter values:

Plot JacobiDS as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

JacobiDS is -periodic along the real axis:

JacobiDS is -periodic along the imaginary axis:

JacobiDS is an odd function in its first argument:

JacobiDS is not an analytic function:

It has both singularities and discontinuities:

is neither nondecreasing nor nonincreasing:

is not injective for any fixed :

is not surjective for fixed :

It is surjective for :

JacobiDS neither non-negative nor non-positive:

JacobiDS is neither convex nor concave:

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

Indefinite integral of JacobiDS:

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 :

JacobiDS can be applied to a 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: