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 Built-in Mathematica Symbol

# JacobiNS

 JacobiNS[u, m]gives the Jacobi elliptic function .
• Mathematical function, suitable for both symbolic and numerical manipulation.
• , where .
• is a doubly periodic function in u with periods and , where is the elliptic integral EllipticK.
• JacobiNS is a meromorphic function in both arguments.
• For certain special arguments, JacobiNS automatically evaluates to exact values.
• JacobiNS can be evaluated to arbitrary numerical precision.
• JacobiNS automatically threads over lists.
Evaluate numerically:
Series expansions about the origin:
 Scope   (6)
Evaluate for complex arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
JacobiNS threads element-wise over lists:
Simple exact values are generated automatically:
Parity transformation and periodicity relations are automatically applied:
JacobiNS can be applied to power series:
 Applications   (4)
Map a rectangle conformally onto the lower half-plane:
Solution of the pendulum equation:
Check the solution:
Plot solutions:
Closed form of iterates of the Katsura-Fukuda map:
Compare the closed form with explicit iterations:
Plot a few hundred iterates:
Solution of the sinh-Gordon equation :
Check the solution:
Plot the solution:
Compose with inverse functions:
Use PowerExpand to disregard multivaluedness of the inverse function:
Solve a transcendental equation:
Integrals:
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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