ellint_rj

kyosu::ellint_rj = eve::functor<ellint_rj_t>

inlineconstexpr

Computes the Carlson's elliptic integral \( \mathbf{R}_\mathbf{J}(x, y) = \frac32 \int_{0}^{\infty} \scriptstyle(t+p)^{-1}[(t+x)(t+y)(t+z)]^{-1/2}\scriptstyle\;\mathrm{d}t\).

Header file

#include <eve/module/elliptic.hpp>

Callable Signatures

namespace eve
{
   // Regular overload
   constexpr auto ellint_rj(auto x, auto y, auto z, auto p)                          noexcept; // 1
 
   // semantic modifier
   constexpr auto ellint_rj[threshold = tol](auto x, auto y, auto z, auto p)         noexcept; // 1
 
   // Lanes masking
   constexpr auto ellint_rj[conditional_expr auto c](auto x, auto y, auto z, auto p) noexcept; // 2
   constexpr auto ellint_rj[logical_value auto m](auto x, auto y, auto z, auto p)    noexcept; // 2
}

Parameters

• x, y, z: floating real arguments.
• p: floating real arguments.
• c: Conditional expression masking the operation.
• m: Logical value masking the operation.

Return value

1. the value of the \(\mathbf{R}_\mathbf{J}\) Carlson elliptic integral: \( \frac32 \int_{0}^{\infty} \scriptstyle(t+p)^{-1}[(t+x)(t+y)(t+z)]^{-1/2}\;\mathrm{d}t\) is returned with relative error less in magnitude than tol (default to eps), The integral is well defined if x, y, z lie in the complex plane cut along the nonpositive real axis, with the exception that at at most one of x, y, z can be 0.
2. The operation is performed conditionnaly

External references

• DLMF: Elliptic Integrals
• Wolfram MathWorld: Elliptic Integral

Example

// revision 1
#include <eve/module/elliptic.hpp>
#include <iostream>
 
using wide_t = eve::wide <double, eve::fixed<4>>;
using r_t = double;
wide_t re1 = { 3.0, 2.0, 1.0, 0.5};
wide_t im1 = { 2.0, -1.0,  -5.0, 0.0};
wide_t re2 = { 0.0, 1.0, 2.0, 3.0};
wide_t im2 = { 1.0 , -4.0,  -2.0, 0.0};
wide_t re3 = { 0.1, -1.0, 2.0, 4.0};
wide_t im3 = { 2.0 , -4.0,  -3.0, 0.0};
wide_t re4 = { 0.1, -1.5, 2.3, 0.5};
wide_t im4 = { 0.0 , -1.0,  -2.0, 0.0};
auto p = kyosu::complex_t<wide_t>(re1, im1);
auto q = kyosu::complex_t<wide_t>(re2, im2);
auto r = kyosu::complex_t<wide_t>(re3, im3);
auto s = kyosu::complex_t<wide_t>(re4, im4);
 
int main()
{
  std::cout << "<- p = " << p << "\n";
  std::cout << "<- q = " << q << "\n";
  std::cout << "<- r = " << r << "\n";
  std::cout << "<- s = " << s << "\n";
 
 
  std::cout << "-> ellint_rj(p, q, r, s)                = " << kyosu::ellint_rj(p, q, r, s) << "\n";
  std::cout << "-> ellint_rj[ignore_last(2)](p, q, r, s)= " << kyosu::ellint_rj[eve::ignore_last(2)](p, q, r, s) << "\n";
  std::cout << "-> ellint_rj[q != 4.0](p, q, r, s)    = " << kyosu::ellint_rj[q != 4.0](p, q, r, s) << "\n";
}