Figure 1. Log plots showing the convergence over n in the mode sum expression (5.1). The blue lines show log ∣ ∣κgnλ(ζ)− 12D + 00Ψnλ(0, 0|r) ∣ ∣, which is O(n−3) for large n. The red lines show log ∣ ∣ ∣ κgnλ(ζ)− 12 ∑1 i=0 ∑i j=0 D+ijΨnλ(i, j|r) ∣ ∣ ∣ , which is O(n−5) for large n. The orange lines show log ∣ ∣ ∣ κgnλ(ζ)− 12 ∑2 i=0 ∑i j=0 D+ijΨnλ(i, j|r) ∣ ∣ ∣ , which is O(n−7) for large n. The top plot is for k = 1, the middle plot for k = 0, and the bottom plot for k = −1. In each case we have set α = 1, L = 1, λ = 25 and ζ = 1+1/ √ 10. Similar results are found for other values of the parameters.
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