Function Repository Resource:

# BulirschEL

Evaluate Bulirsch's general incomplete elliptic integral

Contributed by: Jan Mangaldan
 ResourceFunction["BulirschEL"][x,m,p,a,b] gives Bulirsch's general incomplete elliptic integral .

## Details

Mathematical function, suitable for both symbolic and numerical manipulation.
Bulirsch’s general incomplete elliptic integral is defined as .
When x=, ResourceFunction["BulirschEL"] is referred to as a complete integral.
Argument conventions for elliptic integrals are discussed in "Elliptic Integrals and Elliptic Functions".
For certain special arguments, ResourceFunction["BulirschEL"] automatically evaluates to exact values.
ResourceFunction["BulirschEL"] can be evaluated to arbitrary precision.

## Examples

### Basic Examples (1)

Evaluate numerically:

 In[1]:=
 Out[1]=
 In[2]:=
 Out[2]=

### Scope (5)

Evaluate numerically for complex arguments:

 In[3]:=
 Out[3]=

Evaluate to high precision:

 In[4]:=
 Out[4]=

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

 In[5]:=
 Out[5]=

Simple exact results are generated automatically:

 In[6]:=
 Out[6]=
 In[7]:=
 Out[7]=

 In[8]:=
 Out[8]=

Series expansion of BulirschEL at the origin:

 In[9]:=
 Out[9]=

### Properties and Relations (2)

All incomplete elliptic integrals can be expressed in terms of BulirschEL:

 In[10]:=
 Out[10]=
 In[11]:=
 Out[11]=
 In[12]:=
 Out[12]=

Linear combinations of incomplete elliptic integrals can be expressed in terms of BulirschEL:

 In[13]:=
 Out[13]=
 In[14]:=
 Out[14]=

## Requirements

Wolfram Language 12.3 (May 2021) or above

## Version History

• 1.0.0 – 06 July 2021