Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 InverseEllipticE

Source Notebook

Evaluate the inverse function of EllipticE

Contributed by: Jan Mangaldan

ResourceFunction["InverseEllipticE"][u,m]

gives the inverse of the elliptic integral of the second kind.

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
ResourceFunction["InverseEllipticE"][u,m] is the solution for z in u=E(z❘m), where E(ϕ❘m) is the elliptic integral of the second kind EllipticE.
For certain special arguments, ResourceFunction["InverseEllipticE"] automatically evaluates to exact values.
ResourceFunction["InverseEllipticE"] can be evaluated to arbitrary numerical precision.
ResourceFunction["InverseEllipticE"] automatically threads over lists.

Examples

Basic Examples (2) 

Evaluate numerically:

In[1]:=
ResourceFunction["InverseEllipticE"][4., 2/3]
Out[1]=

Plot over a subset of the reals:

In[2]:=
Plot[ResourceFunction["InverseEllipticE"][x, 2/3], {x, -4, 4}]
Out[2]=

Scope (3) 

Evaluate to high precision:

In[3]:=
N[ResourceFunction["InverseEllipticE"][Pi/3, 1/5], 50]
Out[3]=

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

In[4]:=
ResourceFunction["InverseEllipticE"][ Pi/3, 0.200000000000000000000000000]
Out[4]=

InverseEllipticE threads element‐wise over lists:

In[5]:=
ResourceFunction["InverseEllipticE"][{\[Alpha], \[Beta]}, m]
Out[5]=

Simple exact values are generated automatically:

In[6]:=
{ResourceFunction["InverseEllipticE"][u, 0], ResourceFunction["InverseEllipticE"][0, m]}
Out[6]=

Parity transformation is automatically applied:

In[7]:=
ResourceFunction["InverseEllipticE"][-u, m]
Out[7]=

Properties and Relations (1) 

InverseEllipticE is the inverse of EllipticE:

In[8]:=
ResourceFunction["InverseEllipticE"][3.5, 0.66]
Out[8]=
In[9]:=
EllipticE[%, 0.66]
Out[9]=

Version History

  • 1.0.0 – 08 June 2021

Source Metadata

  • Citation:
    • Boyd J.P., "Numerical, perturbative and Chebyshev inversion of the incomplete elliptic integral of the second kind." Applied Mathematics and Computation, vol. 218, no. 13, 7005–7013, 2012. DOI: 10.1016/j.amc.2011.12.021

Related Resources

Author Notes

Currently, InverseEllipticE can only be numerically evaluated for real arguments, with the additional restriction m<1. The function uses Boyd's starting value, followed by Newton-Raphson or Chebyshev iteration.

License Information