# 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

open allclose all

## Basic Examples(4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

## Scope(25)

### 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:

### 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:

## Applications(7)

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 :

Supersymmetric zeroenergy solution of the Schrödinger equation in a periodic potential:

Check the Schrödinger equation:

Plot the superpotential, the potential and the wavefunction:

Define a conformal map:

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:

Compute area of the triangle, also known as 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 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)

Branch cuts of for cross the real axis at for integer :

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

The preceding function coincides with for real and :

Wolfram Research (1988), JacobiAmplitude, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiAmplitude.html (updated 2020).

#### Text

Wolfram Research (1988), JacobiAmplitude, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiAmplitude.html (updated 2020).

#### CMS

Wolfram Language. 1988. "JacobiAmplitude." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/JacobiAmplitude.html.

#### APA

Wolfram Language. (1988). JacobiAmplitude. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JacobiAmplitude.html

#### BibTeX

@misc{reference.wolfram_2022_jacobiamplitude, author="Wolfram Research", title="{JacobiAmplitude}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiAmplitude.html}", note=[Accessed: 21-March-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_jacobiamplitude, organization={Wolfram Research}, title={JacobiAmplitude}, year={2020}, url={https://reference.wolfram.com/language/ref/JacobiAmplitude.html}, note=[Accessed: 21-March-2023 ]}