Function Repository Resource:

# EllipticRationalR

Evaluate the elliptic rational function

Contributed by: Jan Mangaldan
 ResourceFunction["EllipticRationalR"][n,ξ,x] gives the elliptic rational function Rn(ξ,x).

## Details

Mathematical function, suitable for both symbolic and numerical manipulation.
The elliptic rational function with degree n and selectivity factor ξ is defined as , where cd(um) is the Jacobi elliptic function JacobiCD, cd-1(vm) is InverseJacobiCD, K(m) is the complete elliptic integral of the first kind EllipticK and n(ξ)=Rn(ξ,ξ) is the largest value of Rn(ξ,x) for Abs[x]>1, and is referred to as the discrimination factor.
ResourceFunction["EllipticRationalR"] can be evaluated to arbitrary numerical precision.

## Examples

### Basic Examples (2)

Evaluate numerically:

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Plot R5(1.1,x):

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### Scope (3)

Evaluate symbolically:

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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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### Applications (2)

Compare an elliptic rational function and a Chebyshev polynomial:

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Use the elliptic rational function to construct the best minimax approximation of a unit square pulse:

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Compare with an approximation using ChebyshevT:

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### Properties and Relations (2)

Compare EllipticRationalR with the definition:

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Verify the inversion identity:

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## Version History

• 1.0.0 – 24 March 2021