15 FIG. 5: The three contours over which integrations are per- formed. The dashed circle is the unit circle, and the circular contour actually lies right on the unit circle. FIG. 4: Contours for the integral on the right hand side of (B2). Therefore, the right hand side of (B1) becomes points on the real axis are associated with the values of z for which the argument of the natural logarithm goes through zero. This occurs when cosh M 2 $^{√}$4 τ 2 + φ 2 = 0 (B4) 1 2 πi ∫ ln cosh M 2 γ ( z ) N z dz (B10) − 1 2 πi ∫ ln cosh M 2 γ ( z ) N z 1 z N + 1 dz (B11) and that of (B7) becomes 1 2 πi ∫ ln cosh M 2 γ ( z ) N z dz − 1 2 πi ∫ ln cosh M 2 γ ( z ) N z 1 z N − 1 dz (B12) cosh M 2 $^{√}$4 τ 2 + φ 2 → cos M 2 $^{√}$Φ 2 − 4 τ 2 (B5) M 2 $^{√}$Φ 2 − 4 τ 2 = π 2 (B6) Φ = √ ( π M$^{)}$$^{2}$ + 4 τ 2 (B7) We cause the hyperbolic cosine to go through zero by making its argument imaginary so that it becomes the trigonometric cosine. To do this, we have φ → ± i Φ. When Φ > 4 τ , The right hand side of (B5) goes to zero when Solving for Φ, we find The first terms in (B11) and (B12) are integrated around the unit circle and become the dominant contribution to the sum we are performing. The second terms are deformed to wrap around the branch cut that is outside the unit circle. Of course, the cosine goes to zero at an infinite set of points on the real axis, corresponding to ( M/ 2) √ Φ 2 − 4 τ 2 = (2 n + 1) π/ 2. That is to say, there is an infinite number of branch points on the real axis. However, we are interested in the one closest to the unit circle, corresponding to the conditions in (B6) and (B7). The two branch points of interest are just inside and just outsize the unit circle. The contribution of the integral around the branch cut ending at those branch points will be dominated by a multiplicative exponential factor. The portion of the distorted contour that lies just inside the unit circle can be pushed towards the origin so that it wraps around the branch cut. The general strategy for deforming the contours is illustrated in Fig. 5. Note that the branch cuts are along the real axis. To assess the con- tribution of the contour lying just outside the unit circle, we rewrite the last portions of the integrands as follows In the above, we have neglected the difference between factors going as (1 ± Φ) and 1. This is because Φ is going to be small ( O (1 /M )). Looking at the contour around the branch cut outside the unit circle, we see that the magnitude and the sign of the contribution is fundamentally the same for both of the ln cosh sums. On the other hand, because of the sign differences in the second terms on the right hand sides of (B8) and (B9), we see that the two sums will differ by exponentially small amounts, the size of the difference being dominated by e − $_{N}$√$_{(}$$_{π/M}$$_{)}$$_{2}$$_{+4}$$_{τ}$$_{2}$ . In the case of the branch cut just inside the unit circle, z = 1 − Φ, and we z − N = (1 + Φ) − N → e − $_{N}$√$_{(}$$_{π/M}$$_{)}$$_{2}$$_{+Δ}$$_{2}$ (B13) For the branch cut just outside the unit circle, z = 1 + Φ, and the factor of interest goes as Nz N − 1 z N + 1 = N z − N z 1 z N + 1 (B8) Nz N − 1 z N − 1 = N z + N z 1 z N − 1 (B9)