# 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:

## 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(4)

Solution of the pendulum equation in the overswing mode:

Check with the pendulum equation:

Plot the solution:

Relativistic solution of the sineGordon equation:

Verify the solution by substituting into the sineGordon 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 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_2024_jacobiamplitude, author="Wolfram Research", title="{JacobiAmplitude}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiAmplitude.html}", note=[Accessed: 24-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_jacobiamplitude, organization={Wolfram Research}, title={JacobiAmplitude}, year={2020}, url={https://reference.wolfram.com/language/ref/JacobiAmplitude.html}, note=[Accessed: 24-June-2024 ]}