# JacobiEpsilon

JacobiEpsilon[u,m]

gives the Jacobi epsilon function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• .
• Argument conventions for elliptic integrals are discussed in "Elliptic Integrals and Elliptic Functions".
• JacobiEpsilon is a meromorphic function in both arguments.
• For certain special arguments, JacobiEpsilon automatically evaluates to exact values.
• JacobiEpsilon can be evaluated to arbitrary numerical precision.
• JacobiEpsilon automatically threads over lists.

# Examples

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## Basic Examples(3)

Evaluate numerically:

## Scope(23)

### Numerical Evaluation(4)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate JacobiEpsilon efficiently at high precision:

### Specific Values(3)

Simple exact values are generated automatically:

JacobiEpsilon has poles coinciding with the poles of JacobiDN:

Find a root of JacobiEpsilon[u, ]=2:

### Visualization(3)

Plot the JacobiEpsilon functions for various values of parameter m:

Plot JacobiEpsilon as a function of its parameter m:

Plot the real part of JacobiEpsilon[x+y, ]:

Plot the imaginary part of JacobiEpsilon[x+y, ]:

### Function Properties(2)

JacobiEpsilon is additive quasiperiodic with quasiperiod :

JacobiEpsilon is additive quasiperiodic with quasiperiod :

JacobiEpsilon is an odd function:

### Differentiation(3)

First derivative:

Higher-order derivatives:

Plot derivatives for parameter :

Derivative with respect to parameter m:

### Integration(1)

Indefinite integral of JacobiEpsilon:

### Series Expansions(3)

Series expansion for JacobiEpsilon[u, ]:

Plot the first three approximations for JacobiEpsilon[u, ] around :

Taylor expansion for JacobiEpsilon[2,m]:

Plot the first three series approximations for JacobiEpsilon[2,m] around :

JacobiEpsilon can be applied to power series:

### Function Identities and Simplifications(2)

Parity transformation and quasiperiodicity relations are automatically applied:

Automatic argument simplifications:

### Function Representations(2)

JacobiEpsilon is related to elliptic integral of the second kind:

## Applications(5)

JacobiEpsilon arises in derivatives of Jacobi elliptic functions with respect to parameter :

Plot JacobiEpsilon over the complex plane:

Motion of the charged particle in the magnetic field:

Verify that it solves Newton's equation of motion with Lorentz force:

Plot particle trajectories for several different initial velocities:

Parameterization of a rotating elastic rod (fixed at the origin):

Plot the shape of the deformed rod:

The parameterization parameter is the length of the rod:

Parameterization of Costa's minimal surface [MathWorld]:

## Properties & Relations(3)

JacobiEpsilon is defined as a definite integral of :

JacobiEpsilon[u,m] is a meromorphic extension of :

JacobiEpsilon is related to JacobiZN: