Function Repository Resource:

# BulirschEL2

Evaluate Bulirsch's incomplete elliptic integral of the second kind

Contributed by: Jan Mangaldan
 ResourceFunction["BulirschEL2"][x,m,a,b] gives Bulirsch's incomplete elliptic integral of the second kind .

## Details

Mathematical function, suitable for both symbolic and numerical manipulation.
Bulirsch’s incomplete elliptic integral of the second kind is defined as .
When x=, ResourceFunction["BulirschEL2"] 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["BulirschEL2"] automatically evaluates to exact values.
ResourceFunction["BulirschEL2"] 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 BulirschEL2 at the origin:

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### Applications (3)

Distance along a meridian of the Earth:

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Compare with the result of GeoDistance:

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Calculate the surface area of a triaxial ellipsoid:

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The area of an ellipsoid with semiaxes 3, 2, 1:

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Use RegionMeasure to calculate the surface area of the ellipsoid:

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Parametrization of a Mylar balloon (two flat sheets of plastic sewn together at their circumference and then inflated):

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Plot the resulting balloon:

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### Properties and Relations (3)

Both incomplete and complete cases of EllipticE can be expressed in terms of BulirschEL2:

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EllipticK and EllipticF can be expressed in terms of BulirschEL2:

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BulirschEL2 can be used to represent linear combinations of elliptic integrals of the first and second kinds:

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### Neat Examples (1)

Magnetic field lines of a ring current in cylindrical coordinates:

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## Requirements

Wolfram Language 12.3 (May 2021) or above

## Version History

• 1.0.0 – 22 June 2021