Elliptic alpha functions relate the complete elliptic integrals of the first K(k_r) and second kinds E(k_r) at elliptic integral singular values k_r according to α(r) | = | (E'(k_r))/(K(k_r)) - π/(4[K(k_r)]^2) | = | π/(4[K(k_r)]^2) + sqrt(r) - (E(k_r) sqrt(r))/(K(k_r)) | = | (π^(-1) - 4sqrt(r)q(d ϑ_4(q))/(d q) 1/(ϑ_4(q)))/(ϑ_3^4(q)), where ϑ_3(q) is a Jacobi theta function and k_r | = | λ^*(r) q | = | e^(-π sqrt(r)),

elliptic integral of the first kind | elliptic integral of the second kind | elliptic integral singular value | elliptic lambda function