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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 to 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 (26)

Numerical Evaluation (5)

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:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix JacobiAmplitude function using MatrixFunction:

Specific Values (4)

Simple exact values are generated automatically:

Values at infinity for some special cases:

Find a root of :

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 :

Plot the imaginary part of :

Function Properties (5)

is an analytic function of x:

It has no singularities or discontinuities:

is injective:

is surjective:

JacobiAmplitude is neither non-negative nor non-positive:

JacobiAmplitude is neither convex nor concave:

Differentiation (3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Derivative with respect to :

Series Expansions (3)

Taylor expansion for :

Plot the first three approximations for around :

Taylor expansion for :

Plot the first three approximations for around :

JacobiAmplitude can be applied to a power series:

Function Representations (3)

JacobiAmplitude is the inverse of EllipticF:

Integral representation:

TraditionalForm formatting:

Applications (4)

Solution of the pendulum equation in the overswing mode:

Check with the pendulum equation:

Plot the solution:

Relativistic solution of the sine‐Gordon equation:

Verify the solution by substituting into the sine‐Gordon equation:

Plot the solution for different values of and :

Form and plot generalized Fourier series:

Spherical triangle from an elliptic parameter and triple such that :

Angles at vertices , and and sides opposite to each corresponding vertex:

Verify Cagnoli's equation, which relates the measurements of all the angles and sides of a spherical triangle:

Compute the area of the triangle, also known as the spherical excess:

Compute the excess using L'Huilier's theorem [MathWorld]:

Compute corresponding points on 3D unit sphere:

Check consistency between points found and spherical sides:

Visualize the triangle:

Properties & Relations (5)

Compose with inverse functions:

Use PowerExpand to disregard the multivaluedness of the inverse function:

Apply trigonometric functions to JacobiAmplitude:

Solve a transcendental equation:

Obtain from a differential equation:

JacobiAmplitude has branch cuts from to for any integers and :

Visualize branch cuts for different moduli:

Possible Issues (1)

The branch cuts of for cross the real axis at for integer :

For physical applications where a continuous version of the amplitude is desired for , use the following definition:

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

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