e^(-(π K(1 - m))/K(m))

(no roots exist)

m/16 + m^2/32 + (21 m^3)/1024 + (31 m^4)/2048 + (6257 m^5)/524288 + O(m^6) (Taylor series)

exp(-(π (log(m) + log(16)))/(-i log(m) + π - i log(16))) (1 + π^2/(2 (-i log(m) - i log(16) + π)^2 m) + (π^2 (-13 log^2(m) - 26 log(16) log(m) - 26 i π log(m) - 16 log(m) - 13 log^2(16) - 26 i π log(16) - 16 log(16) + 21 π^2 - 16 i π))/(64 (-i log(m) - i log(16) + π)^4 m^2) + O((1/m)^13))

d/dm(q(m)) = -(π^2 q(m))/(4 (m - 1) m K(m)^2)

max{q(m)} = 1 at m = 1

lim_(m-> ± ∞) q(m) = -1

q(m) = exp(-(π K(1 - m))/K(m))

q(m) = exp((π (i WeierstrassHalfPeriods[{g_2, g_3}][[2]]))/(WeierstrassHalfPeriods[{g_2, g_3}][[1]])) for (m = {λ((WeierstrassHalfPeriods[{g_2, g_3}][[2]])/g_2), λ((WeierstrassHalfPeriods[{g_2, g_3}][[2]])/g_3)} and {{g_2, g_3}, WeierstrassHalfPeriods[{g_2, g_3}][[2]]} = WeierstrassHalfPeriods[{g_2, g_3}])

q(m) = exp(-(2 sum_(k=0)^∞ (m^k ((1/2)_k)^2 (-ψ(1/2 + k) + ψ(1 + k)))/(k!)^2)/( sum_(k=0)^∞ (m^k ((1/2)_k)^2)/(k!)^2)) m

q(m) = e^(π^2 w) sum_(k=0)^∞ ((-1 + A/B)^k π^(2 k) w^k)/(k!) for (A = sum_(k=0)^∞ ((1 - m)^k ((1/2)_k)^2)/(k!)^2 and B = sum_(k=0)^∞ ((1 - m)^k ((1/2)_k)^2 (1 + 2 w (log(4) + ψ(1/2 + k) - ψ(1 + k))))/(k!)^2 and w = 1/log((1 - m)/16))

q(m) = q(m_0) sum_(k=0)^∞ ((m - m_0)^k sum_(p=0)^k (π^p sum_(j=0)^p (-1)^(j + p) binomial(p, j) (K(1 - m_0)/(K(m_0)))^j (u function (K(1 - u)/K(u))^(p - j))^(k) (m_0))/(p!))/(k!)