In this appendix we present the definition and some properties of elliptic integrals which are needed for the derivation of our result. The complete elliptic integrals of the first ( K ) and the second ( E ) kind are defined as For our purpose it is useful to note that In order to study the elliptic integrals near the logarithmic singularity, it is convenient to use the q-series, defined as Inverting, one obtains Now that we have m ( q ) and q ( m ) as given above, we have the following alternative expan- sions. And more importantly, 18 Appendix: Complete Elliptic integrals and the Nome q expansion m = 16( q − 8 q 2 + 44 q 3 − 192 q 4 + · · · ) . (A.7) K ( m ) = π 2 (1 + 4 q + 4 q 2 + 4 q 4 + · · · ) , (A.8) E ( m ) = π 2 (1 − 4 q + 20 q 2 − 64 q 3 + · · · ) . (A.9) K (1 − m ) = − ln q 2 (1 + 4 q + 4 q 2 + 4 q 4 + · · · ) , (A.10) E (1 − m ) = (1 − 4 q + 12 q 2 − 32 q 3 + · · · ) − 4 q ln q (1 − 2 q + 8 q 2 + · · · ) . (A.11) ∫ θ$_{0}$ 0 dθ $^{√}$sin $^{2}$θ$_{0}$ − sin $^{2}$θ = K (sin $^{2}$θ$_{0}$ ) , (A.3) ∫ θ$_{0}$ 0 $^{√}$sin $^{2}$θ$_{0}$ − sin $^{2}$θ dθ = E (sin $^{2}$θ$_{0}$ ) − (1 − sin $^{2}$θ$_{0}$ ) K (sin $^{2}$θ$_{0}$ ) . (A.4) K ( m ) = ∫ π/ 2 0 dφ $^{√}$1 − m sin $^{2}$φ , (A.1) E ( m ) = ∫ π/ 2 0 $^{√}$1 − m sin $^{2}$φ dφ. (A.2) q ≡ exp[ − π K (1 − m ) / K ( m )] (A.5) = m 16 + 8 ( m 16$^{)}$$^{2}$ + 84 ( m 16$^{)}$$^{3}$ + · · · . (A.6)