Function Repository Resource:

# BulirschCEL

Evaluate Bulirsch's general complete elliptic integral

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

## Details

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

## Examples

### Basic Examples (1)

Evaluate numerically:

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### Scope (4)

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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### Applications (3)

Evaluate the mutual inductance of two coaxial circles:

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Compare with the result of NIntegrate:

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Visualize the solid angle subtended by a circular disk:

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Evaluate the solid angle:

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Compare with the result of NIntegrate:

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Visualize the intersection of two cylinders:

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Compute the volume of the intersection of two cylinders:

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Compare with the result of Volume:

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### Properties and Relations (3)

Complete Legendre-Jacobi elliptic integrals of all three kinds can be expressed in terms of BulirschCEL:

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BulirschCEL can be used to represent linear combinations of complete elliptic integrals:

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JacobiZeta can be expressed in terms of BulirschCEL:

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### Neat Examples (1)

Magnetic field lines of a cylindrical solenoid:

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## Requirements

Wolfram Language 12.3 (May 2021) or above

## Version History

• 1.0.0 – 14 June 2021