ϕ + 1/4 m (ϕ - sin(ϕ) cos(ϕ)) + 3/256 m^2 (12 ϕ - 8 sin(2 ϕ) + sin(4 ϕ)) - (5 m^3 (-60 ϕ + 45 sin(2 ϕ) - 9 sin(4 ϕ) + sin(6 ϕ)))/3072 + (35 m^4 (840 ϕ - 672 sin(2 ϕ) + 168 sin(4 ϕ) - 32 sin(6 ϕ) + 3 sin(8 ϕ)))/393216 + O(m^5) (Taylor series)

d/dm(F(ϕ|m)) = -(-(m sin(2 ϕ))/sqrt(1 - m sin^2(ϕ)) + 2 (m - 1) F(ϕ|m) + 2 E(ϕ|m))/(4 (m - 1) m)

integral F(ϕ|m) dm = 2 (m - 1) F(ϕ|m) + 2 E(ϕ|m) + cot(ϕ) sqrt(2 m cos(2 ϕ) - 2 m + 4) + constant

F(ϕ|m) = Π(0;ϕ|m)

F(ϕ|m) = -EllipticLog[{cot^2(ϕ), cos(ϕ) csc^3(ϕ) sqrt(1 - m + m cos^2(ϕ))}, {2 - m, 1 - m}]

F(ϕ|m) = ((sqrt(1 - n) cot(ϕ) + E(ϕ|m)) K(m) - sqrt(1 - n) cot(ϕ) Π(n|m))/E(m) for (ϕ = sin^(-1)(sqrt(n/m)) and 0

F(ϕ|m) = sum_(k=0)^∞ ((m - m_0)^k F^(0, k)(ϕ|m_0))/(k!)

F(ϕ|m) = sum_(k=0)^∞ ((ϕ - z_0)^k F^(k, 0)(z_0|m))/(k!)

F(ϕ|m) = (2 ϕ K(m))/π + sum_(k=1)^∞ ((-1/4)^k m^k binomial(-1/2 + k, k) 2F1(1/2 + k, 1/2 + k, 1 + 2 k, m) sin(2 k ϕ))/k

integral_(-∞)^∞ integral_(-∞)^∞ F(ϕ|m) dϕ dm = 0