ellint_rg

kyosu::ellint_rg = eve::functor<ellint_rg_t>

inlineconstexpr

Computes the Carlson's elliptic integral \( \mathbf{R}_\mathbf{G}(x, y) = \frac1{4\pi} \int_{0}^{2\pi}\int_{0}^{\pi} \scriptstyle\sqrt{x\sin^2\theta\cos^2\phi +y\sin^2\theta\sin^2\phi +z\cos^2\theta} \scriptstyle\;\mathrm{d}\theta\;\mathrm{d}\phi\).

Header file

#include <kyosu/kyosu.hpp>

Callable Signatures

namespace eve
{
   // Regular overload
   constexpr auto ellint_rg(auto x, auto y, auto z)                 noexcept; // 1
 
   // Lanes masking
   constexpr auto ellint_rg[conditional_expr auto c](/*all previous overloads*/)   noexcept; // 2
   constexpr auto ellint_rg[logical_value auto m](/*all previous overloads*/)      noexcept; // 2
}

Parameters

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

Return value

1. the value of the \(\mathbf{R}_\mathbf{G}\) Carlson elliptic integral: \(\frac14 \int_{0}^{\infty}[(t+x)(t+y)(t+z)]^{-1/2}\left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)t\;\mathrm{d}t\) is returned. All of x, y, z may be 0 and those that are nonzero must lie in the complex plane cut along the nonpositive real axis
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, 4.0, 3.0};
wide_t im2 = { 1.0 , -4.0,  0.0, 0.0};
wide_t re3 = { 0.1, -1.0, 2.0, 4.0};
wide_t im3 = { 2.0 , -4.0,  -3.0, 0.0};
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);
 
int main()
{
  std::cout << "<- p = " << p << "\n";
  std::cout << "<- q = " << q << "\n";
  std::cout << "<- r = " << r << "\n";
 
   std::cout << "-> ellint_rg(p, q, r)                = " << kyosu::ellint_rg(p, q, r) << "\n";
   std::cout << "-> ellint_rg[ignore_last(2)](p, q, r)= " << kyosu::ellint_rg[eve::ignore_last(2)](p, q, r) << "\n";
   std::cout << "-> ellint_rg[q != 4.0](p, q, r)      = " << kyosu::ellint_rg[q != 4.0](p, q, r) << "\n";
}