Evaluate numerically to high precision:

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

Evaluate for complex arguments:

Evaluate JacobiSD efficiently at high precision:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix JacobiSD function using MatrixFunction:

Simple exact values are generated automatically:

Some poles of JacobiSD:

Find a local maximum of JacobiSD as a root of :

Plot the JacobiSD functions for various values of parameter:

Plot JacobiSD as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

JacobiSD is -periodic along the real axis:

JacobiSD is -periodic along the imaginary axis:

JacobiSD is an odd function in its first argument:

is an analytic function for :

It is not in general analytic:

It has both singularities and discontinuities for :

is neither nondecreasing nor nonincreasing:

JacobiSD is not injective for any fixed :

It is injective for :

is not surjective for :

It is surjective for :

JacobiSD is neither non-negative nor non-positive:

JacobiSD is neither convex nor concave:

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

Indefinite integral of JacobiSD:

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

More integrals:

Taylor expansion for :

Plot the first three approximations for around :

Taylor expansion for :

Plot the first three approximations for around :

JacobiSD can be applied to a power series:

Parity transformation and periodicity relations are automatically applied:

Identity involving JacobiCD:

Automatic argument simplifications:

Representation in terms of Csc and JacobiAmplitude:

Relation to other Jacobi elliptic functions:

TraditionalForm formatting: