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 Built-in Mathematica Symbol

# JacobiDN

 JacobiDN[u, m]gives the Jacobi elliptic function .
• Mathematical function, suitable for both symbolic and numerical manipulation.
• , where .
• a is doubly periodic function in u with periods and , where is the elliptic integral EllipticK.
• JacobiDN is a meromorphic function in both arguments.
• For certain special arguments, JacobiDN automatically evaluates to exact values.
• JacobiDN can be evaluated to arbitrary numerical precision.
• JacobiDN automatically threads over lists.
Evaluate numerically:
 Scope   (6)
Evaluate for complex arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Simple exact values are generated automatically:
Parity transformations and periodicity relations are automatically applied:
JacobiDN can be applied to a power series:
 Applications   (8)
Cartesian coordinates of a pendulum:
Plot the time dependence of the coordinates:
Plot the trajectory:
Uniformization of a Fermat cubic :
Check:
Conformal map from a unit triangle to the unit disk:
Show points before and after the map:
Solution of Nahm equations:
Check that the solutions fulfill the Nahm equations:
Periodic solution of the nonlinear Schrödinger equation :
Plot the solution:
Show arc length parametrization and classical parametrization:
Zero modes of the periodic supersymmetric partner potentials:
Plot the zero modes:
Complex parametrization of a "sphere":
Plot real and imaginary parts:
Compose with inverse functions:
Use PowerExpand to disregard multivaluedness of the inverse function:
Evaluate as a result of applying D to JacobiAmplitude:
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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