BUILT-IN MATHEMATICA SYMBOL

JacobiAmplitude

gives the amplitude

for Jacobi elliptic functions.

MORE INFORMATION

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.

JacobiAmplitude can be evaluated to arbitrary numerical precision.

JacobiAmplitude automatically threads over lists.

EXAMPLES

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Basic Examples (3)

Evaluate numerically:

Series expansion about the origin:

Evaluate numerically:

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Series expansion about the origin:

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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:

JacobiAmplitude threads element-wise over lists:

Simple exact values are generated automatically:

Parity transformation is automatically applied:

TraditionalForm formatting:

Generalizations & Extensions (1)

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:

Properties & Relations (4)

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:

Possible Issues (1)

MachinePrecision is not sufficient to obtain the correct result:

Use arbitrary-precision arithmetic instead: