Function Repository Resource:

# BulirschEL3

Evaluate Bulirsch's incomplete elliptic integral of the third kind

Contributed by: Jan Mangaldan
 ResourceFunction["BulirschEL3"][x,m,p] gives Bulirsch's incomplete elliptic integral of the third kind .

## Details

Mathematical function, suitable for both symbolic and numerical manipulation.
Bulirsch’s incomplete elliptic integral of the third kind is defined as .
When x=, ResourceFunction["BulirschEL3"] 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["BulirschEL3"] automatically evaluates to exact values.
ResourceFunction["BulirschEL3"] can be evaluated to arbitrary precision.

## Examples

### Basic Examples (1)

Evaluate numerically:

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### Scope (5)

Evaluate numerically for complex arguments:

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Evaluate to high precision:

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

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Simple exact results are generated automatically:

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Series expansion of BulirschEL3 at the origin:

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### Applications (2)

Total arc length of a cornoid:

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Compare with the result of ArcLength:

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Visualize the solid angle subtended by a cylinder:

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Evaluate the solid angle:

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Compare with the result of NIntegrate:

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### Properties and Relations (2)

Both incomplete and complete cases of EllipticPi can be expressed in terms of BulirschEL3:

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Express EllipticF and EllipticE in terms of BulirschEL3:

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## Requirements

Wolfram Language 12.3 (May 2021) or above

## Version History

• 1.0.0 – 29 June 2021