JacobiAmplitude

JacobiAmplitude[u,m]

gives the amplitude for Jacobi elliptic functions.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • JacobiAmplitude[u,m] converts from the argument u for an elliptic function to the amplitude ϕ.
  • JacobiAmplitude is the inverse of the elliptic integral of the first kind. If , then .
  • JacobiAmplitude[u,m] has a branch cut discontinuity in the complex u plane running from 2 s TemplateBox[{m}, EllipticK]+ⅈ (4 t+1) TemplateBox[{{1, -, m}}, EllipticK] to 2 s TemplateBox[{m}, EllipticK]+ⅈ (4 t+3) TemplateBox[{{1, -, m}}, EllipticK] for every pair of integers and .
  • 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  (4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansions about the origin:

Scope  (20)

Numerical Evaluation  (4)

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate JacobiAmplitude efficiently at high precision:

JacobiAmplitude threads elementwise over lists:

Specific Values  (4)

Simple exact values are generated automatically:

Values at infinity for some special cases:

Find a root of TemplateBox[{x, {-, 3}}, JacobiAmplitude]⩵1:

Parity transformation is automatically applied:

Visualization  (3)

Plot the JacobiAmplitude functions for various values of parameter :

Plot JacobiAmplitude as a function of its parameter :

Plot the real part of TemplateBox[{{x, +, {ⅈ,  , y}}, 2}, JacobiAmplitude]:

Plot the imaginary part of TemplateBox[{{x, +, {ⅈ,  , y}}, 2}, JacobiAmplitude]:

Differentiation  (3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

Series Expansions  (3)

Taylor expansion for TemplateBox[{x, {2, /, 3}}, JacobiAmplitude]:

Plot the first three approximations for TemplateBox[{x, {2, /, 3}}, JacobiAmplitude] around :

Taylor expansion for TemplateBox[{1, m}, JacobiAmplitude]:

Plot the first three approximations for TemplateBox[{1, m}, JacobiAmplitude] around :

JacobiAmplitude can be applied to a power series:

Function Representations  (3)

JacobiAmplitude is the inverse of EllipticF:

Integral representation:

TraditionalForm formatting:

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 sineGordon equation:

Plot the solution for different values of :

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  (5)

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:

JacobiAmplitude has branch cuts from 2 s TemplateBox[{m}, EllipticK]+ⅈ (4 t+1) TemplateBox[{{1, -, m}}, EllipticK] to 2 s TemplateBox[{m}, EllipticK]+ⅈ (4 t+3) TemplateBox[{{1, -, m}}, EllipticK] for any integers and :

Visualize branch cuts for different moduli:

Possible Issues  (1)

Branch cuts of TemplateBox[{u, m}, JacobiAmplitude] for cross the real axis at 2(2 s+1) TemplateBox[{{1, /, m}}, EllipticK]/sqrt(m) for integer :

For physical applications where a continuous version is desired for , use:

The preceding function coincides with sin^(-1)(TemplateBox[{u, m}, JacobiSN]) for real and :

Introduced in 1988
 (1.0)
 |
Updated in 2020
 (12.1)