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# JacobiSN

 JacobiSN[u, m]gives the Jacobi elliptic function .
• Mathematical function, suitable for both symbolic and numerical manipulation.
• , where .
• is a doubly periodic function in with periods and , where is the elliptic integral EllipticK.
• JacobiSN is a meromorphic function in both arguments.
• For certain special arguments, JacobiSN automatically evaluates to exact values.
• JacobiSN can be evaluated to arbitrary numerical precision.
• JacobiSN automatically threads over lists.
Evaluate numerically:
Series expansions about the origin:
 Scope   (5)
Evaluate for complex arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
JacobiSN threads element-wise over lists:
Simple exact values are generated automatically:
Parity transformation and periodicity relations are automatically applied:
TraditionalForm formatting:
JacobiSN can be applied to power series:
 Applications   (7)
Map a rectangle conformally onto the upper half-plane:
Solution of the pendulum equation:
Check the solution:
Plot solutions:
Cnoidal solution of the KdV equation:
A numerical check of the solution:
Plot the solution:
Closed form of iterates of the Katsura-Fukuda map:
Compare the closed form with explicit iterations:
Plot few hundred iterates:
Implicitly defined periodic maximal surface in Minkowski space:
Calculate partial derivatives:
Check numerically the equation for a maximal surface:
Plot the maximal surface in Euclidean space:
Solution of the Euler top equations for :
Check the solutions numerically:
Plot the solutions:
Define a compacton solution of the nonlinear differential equation :
Verify the solution:
Plot the compacton:
Compose with inverse functions:
Use PowerExpand to disregard multivaluedness of the inverse function:
Evaluate as a result of applying Sin to JacobiAmplitude:
Solve a transcendental equation:
Numerically find a root of a transcendental equation:
Integrals:
Solve the Painlevé-VIII differential equation:
Machine-precision input is insufficient to give the correct answer:
Currently only simple simplification rules are built in for Jacobi functions:
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