Function Repository Resource:

# BulirschEL1

Evaluate Bulirsch's incomplete elliptic integral of the first kind

Contributed by: Jan Mangaldan
 ResourceFunction["BulirschEL1"][x,m] gives Bulirsch's incomplete elliptic integral of the first kind .

## Details

Mathematical function, suitable for both symbolic and numerical manipulation.
Bulirsch’s incomplete elliptic integral of the first kind is defined as .
When x=, ResourceFunction["BulirschEL1"] 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["BulirschEL1"] automatically evaluates to exact values.
ResourceFunction["BulirschEL1"] 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 BulirschEL1 at the origin:

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### Applications (2)

Arc length of a lemniscate of Bernoulli:

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Compare with the result of ArcLength:

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Calculate the reaction time needed for an autocatalytic termolecular reaction system:

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### Properties and Relations (3)

EllipticF can be expressed in terms of BulirschEL1:

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EllipticK can be expressed in terms of BulirschEL1:

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InverseJacobiSN can be expressed in terms of BulirschEL1:

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## Requirements

Wolfram Language 12.3 (May 2021) or above

## Version History

• 1.0.0 – 22 June 2021