Function gauss2F1SeriesC

src/compute-engine/numerics/numeric-complex.ts:575–595

(
  a: Complex,
  b: Complex,
  c: Complex,
  z: Complex,
  maxTerms = 10_000
)

  }
  const K = new Complex(Math.PI / 2, 0).div(a);
  return K.mul(C_ONE.sub(sum));
}

//
// ---------------- Carlson symmetric elliptic integrals (complex) ----------------
//
// Duplication-theorem algorithms (Carlson 1995, same series tails as the
// machine-real kernels in numerics/special-functions.ts and as mpmath).
// With principal-branch square roots the duplication theorem is valid for
// arguments in the cut plane C ∖ (−∞, 0); arguments ON the negative real
// axis are evaluated as their boundary value from above (Im → 0⁺), which
// matches the mpmath/Mathematica convention for the incomplete elliptic
// integrals built on them.
//

const CARLSON_TOL_C = 1e-24;

/** Carlson R_C(x, y), complex, principal value for y on (−∞, 0). */
export function carlsonRCComplex(x: Complex, y: Complex): Complex {
  if (x.isNaN() || y.isNaN()) return C_NAN;
  if (y.isZero()) return new Complex(Infinity, 0);
  if (x.isZero()) return new Complex(Math.PI / 2, 0).div(y.sqrt());
  // Cauchy principal value for real y < 0 (DLMF 19.2.20)
  if (y.im === 0 && y.re < 0)