This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
 BUILT-IN MATHEMATICA SYMBOL

# JacobiDS

 JacobiDSgives 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.
• JacobiDS is a meromorphic function in both arguments.
• For certain special arguments, JacobiDS automatically evaluates to exact values.
• JacobiDS can be evaluated to arbitrary numerical precision.
• JacobiDS automatically threads over lists.
Evaluate numerically:
Series expansions:
Evaluate numerically:
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Series expansions:
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 Out[2]=
 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 transformation and periodicity relations are automatically applied:
JacobiDS can be applied to a power series:
 Applications   (4)
Conformal map from a rectangle to the unit disk:
Visualize the map:
Generator for the hierarchy of solutions of the nonlinear diffusion equation :
Numerical check of the solutions:
Conformal map from an ellipse to the unit disk:
Visualize the map:
Cartesian coordinates of a pendulum:
Plot the time-dependence of the coordinates:
Plot the trajectory:
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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