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

# JacobiAmplitude

 JacobiAmplitudegives the amplitude for Jacobi elliptic functions.
• Mathematical function, suitable for both symbolic and numerical manipulation.
• JacobiAmplitude converts from the argument u for an elliptic function to the amplitude .
• JacobiAmplitude is the inverse of the elliptic integral of the first kind. If , then .
• For certain special arguments, JacobiAmplitude automatically evaluates to exact values.
Evaluate numerically:
Evaluate numerically:
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 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 is automatically applied:
JacobiAmplitude can be applied to a power series:
 Applications   (5)
Solution of the pendulum equation in the overswing mode:
Check:
Plot the solution:
Motion of a charged particle in a linear magnetic field:
Check the solution in Newton's equations of motion with Lorentz force:
Plot particle trajectories for various initial velocities:
Relativistic solution of the sine-Gordon equation:
Plot the solution:
Parametrization of a rotating elastic rod (fixed at the origin):
Plot the shape of the deformed rod:
Form and plot generalized Fourier series:
Compose with inverse functions:
Use PowerExpand to disregard multivaluedness of the inverse function:
Apply trigonometric functions to JacobiAmplitude:
Solve a transcendental equation:
Obtain from a differential equation:
MachinePrecision is not sufficient to obtain the correct result: