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

# JacobiSD

 JacobiSDgives 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.
• JacobiSD is a meromorphic function in both arguments.
• For certain special arguments, JacobiSD automatically evaluates to exact values.
• JacobiSD can be evaluated to arbitrary numerical precision.
• JacobiSD automatically threads over lists.
Evaluate numerically:
Series expansions:
Evaluate numerically:
 Out[1]=

 Out[1]=

Series expansions:
 Out[1]=
 Out[2]=
 Scope   (6)
Evaluate for complex arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
JacobiSD threads element-wise over lists:
Simple exact values are generated automatically:
Parity transformation and periodicity relations are automatically applied:
JacobiSD 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:
New in 1