1.90 1.95 2.00 2.05 3.45 3.50 3.55 3.60 3.65 1 2 3 4 2 3 4 5 6 FIG. 2: Matching the expansions (C23) and (C24). The region near x = 1 . 95 is enlarged in the right figure. and finally we obtain For an T × R rectangle with T ≫ R , when resulting in we reproduce the well-known result for the L¨ uscher term by the substitution a = T/π , b = R/ 2. For an outstretched ellipse we have from the expansion (C9b) for x < x$_{0}$ ≈ 1 . 95, where the series converges, and for x > x$_{0}$ ≈ 1 . 95. They match each other pretty well, as is depicted in Fig. 2. The numerical f ( x ) x>x 0 = 1 . 600095 x + 0 . 960520 x − 1 . 041899 x 3 + 2 . 418227 x 5 − 7 . 138640 x 7 + 23 . 73152 x 9 − 84 . 70585 x 11 + . . . (C24) f ( x ) x<x 0 = 1 + x + x 2 4 − x 3 16 − x 4 128 + 15 x 5 1024 − 9 x 6 2048 + . . . (C23) f ( x ) = √$_{1 +}$$_{x}$ 2 rectangle , (C22) λ$_{n,m}$ = π $^{2}$n2 R 2 + π $^{2}$m2 T 2 rectangle (C21) ∑ n,m ln ( λ − 1 / 2 n,m ) = − ∑ n πna 2 b ∫ ∞ 0 d x [1 − xf ′ ( x ) /f ( x )] = π 24 a b ∫ ∞ 0 d x [1 − xf ′ ( x ) /f ( x )] . (C20) 19