This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.2)


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.
  • JacobiCN is a meromorphic function in both arguments.
  • For certain special arguments, JacobiCN automatically evaluates to exact values.
  • JacobiCN can be evaluated to arbitrary numerical precision.
  • JacobiCN automatically threads over lists.
Evaluate numerically:
Series expansions about the origin:
Evaluate numerically:
Click for copyable input
Click for copyable input
Series expansions about the origin:
Click for copyable input
Click for copyable input
Evaluate for complex arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
JacobiCN threads element-wise over lists:
Simple exact values are generated automatically:
Parity transformations and periodicity relations are automatically applied:
TraditionalForm formatting:
JacobiCN can be applied to power series:
Cnoidal solution of the KdV equation:
Verify the solution:
Plot the solution:
Conformal map from a unit triangle to the unit disk:
Show points before and after the map:
Solution of an anharmonic oscillator :
Plot various solutions:
Elliptic parametrization of an ellipse:
Plot using elliptic parametrization and circular parametrization:
Solution of Nahm equations:
Check that the solutions fulfill the Nahm equations:
Parametrization of a mylar balloon (two flat sheets of plastic sewn together at their circumference and then inflated):
Plot the inflated balloon:
Compose with inverse functions:
Use PowerExpand to disregard multivaluedness of the inverse function:
Evaluate as a result of applying Cos to JacobiAmplitude:
Solve a transcendental equation:
Machine-precision input is insufficient to give the correct answer:
Currently only simple simplification rules are built in for Jacobi functions:
New in 1