Legendre Elliptic Integral First Kind phi sin(alpha) actual ellf(phi,ak) 0.09 0.03 8.726660e-02 8.726659e-02 0.09 0.50 8.729413e-02 8.729413e-02 0.09 1.00 8.737730e-02 8.737730e-02 0.52 0.03 5.236264e-01 5.236264e-01 0.52 0.50 5.294286e-01 5.294286e-01 0.52 1.00 5.492704e-01 5.492704e-01 1.57 0.03 1.571275e+00 1.571275e+00 1.57 0.50 1.685750e+00 1.685750e+00 1.57 1.00 4.742717e+00 4.742727e+00