using Plots using SpecialFunctions # using Elliptic K = ellipk # Computes Complete Elliptic Integral of 1st kind K(m) E = ellipe # Computes Complete Elliptic Integral of 2nd kind E(m) psi0(k) = (K(k) - E(k)) / k @show psi0(1e-8) # psi0(1.0e-8) = 0.7853981909278218 x = range(1e-5, 0.1, length=1000) y = psi0 plot(x, y; label = :none)
The results are the same whether I use
Now, from Wikipedia,
We neglect here the higher terms for
k approaching zero (by cropping screenshot )
Thus, for small k
(K(k) - E(k)) / k ≈ k * π / 4 which tends to zero with
k tending to zero.