This resource function is obsolete. Use the function JacobiEpsilon instead.

Function Repository Resource:

# JacobiEpsilon

Evaluate the Jacobi epsilon function

Contributed by: Jan Mangaldan
 ResourceFunction["JacobiEpsilon"][u,m] gives the Jacobi epsilon function ε(u|m).

## Details and Options

Mathematical function, suitable for both symbolic and numerical manipulation.
For real u, ε(u|m)=E(ϕm), where , E is the EllipticE function and is the JacobiAmplitude.
ResourceFunction["JacobiEpsilon"] is a meromorphic function in both arguments.
ResourceFunction["JacobiEpsilon"] is not periodic and is therefore not strictly an elliptic function.
For certain special arguments, ResourceFunction["JacobiEpsilon"] automatically evaluates to exact values.
ResourceFunction["JacobiEpsilon"] can be evaluated to arbitrary numerical precision.

## Examples

### Basic Examples (3)

Evaluate JacobiEpsilon numerically:

 In[1]:=
 Out[1]=

Plot the Jacobi epsilon function over a subset of the reals:

 In[2]:=
 Out[2]=

 In[3]:=
 Out[3]=

### Scope (5)

Evaluate for complex arguments:

 In[4]:=
 Out[4]=

Evaluate to high precision:

 In[5]:=
 Out[5]=

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

 In[6]:=
 Out[6]=

 In[7]:=
 Out[7]=

Simple exact values are generated automatically:

 In[8]:=
 Out[8]=
 In[9]:=
 Out[9]=

Parity transformation is automatically applied:

 In[10]:=
 Out[10]=

### Applications (3)

Plot the Jacobi epsilon function over the complex plane:

 In[11]:=
 Out[11]=

Motion of a charged particle in a linear magnetic field:

 In[12]:=

Check the solution in Newton's equations of motion with Lorentz force:

 In[13]:=
 Out[13]=

Plot particle trajectories for various initial velocities:

 In[14]:=
 Out[14]=

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

 In[15]:=

Plot the shape of the deformed rod:

 In[16]:=
 Out[16]=

### Properties and Relations (1)

ε(u|m) is a meromorphic extension of the EllipticE function :

 In[17]:=
 Out[17]=
 In[18]:=
 Out[18]=

### Possible Issues (2)

Machine precision is not sufficient to obtain the correct result:

 In[19]:=
 Out[19]=

 In[20]:=
 Out[20]=

## Version History

• 2.0.0 – 11 March 2020
• 1.0.0 – 20 November 2019