Incomplete elliptic integral of the second kind with negative parameter

Question

Abramowitz & Stegun 17.4.18 gives the following formula for $E(u, -m)$:

$$ E(u,-m) = (1+m)^{1/2} \{E(u(1+m)^{1/2},m(m+1)^{-1}) - m(1+m)^{-1/2} \mathrm{sn}(u(1+m)^{1/2}, m(1+m)^{-1})\; \mathrm{cd}(u(1+m)^{1/2}, m(1+m)^{-1}) \}. $$

However, a quick Mathematica test does not seem to verify this relationship:

u = RandomReal[]; m = RandomReal[]; a = u Sqrt[1 + m]; b = m / (m + 1);

Sqrt[1 + m] (EllipticE[a, b] - m / Sqrt[1 + m] JacobiSN[a, b] JacobiCD[a, b])
EllipticE[u, -m]

>>> 0.136459
>>> 0.214315 + 0. I

Does anyone know the correct equation, or have I made an error somewhere?

Answer 1

Try

test[u_,m_] := Module[{s, a, b}, s = Sqrt[1 + m]; a = u s; b = m/(1 + m);
{ s EllipticE[JacobiAmplitude[a, b], b] - 
    m / s JacobiSN[a, b] JacobiCD[a, b],
  EllipticE[JacobiAmplitude[u, -m], -m] }];