content_image/328959.jpg content_image/328962.jpg content_image/328957.jpg content_image/328958.jpg content_image/328955.jpg content_image/328956.jpg
value of the integral on the right-hand side of Eq. (C20) is then 2.84 to be compared with π as is expected from Eq. (51). 

To clarify the calculation, let us keep only three terms on the right-hand side of Eq. (C9b), when we get a quadratic equation and analytical formulas are available. We then find 

and expanding it either in x or in 1 /x we obtain analogs of Eqs. (C23) and (C24). The two expansions now match at x$_{0}$ = √$_{2 = 1}$$_{.}$$_{41 to be compared with}$ x$_{0}$ = 1 . 95 for all nine terms left on the right-hand side of Eq. (C9b). The value of the integral is now √$_{2 log}$ [ $^{$_{3 + 2}$√$_{2}$}$] = 2 . 49 which is smaller than 2 . 84, which may characterize to which extend the approximation of α$_{m}$ by the eight term in (C9b) is better than by the tree terms. In Appendix D we confirm the value of π by another method. 

## Appendix D: L¨ uscher term from conformal anomaly 

A starting point is the formula [7, 19] 

<img src='content_image/328959.jpg'>

<img src='content_image/328962.jpg'>

that relates the L¨ uscher term to the conformal anomaly. Here the metric ρ is 

where 

<img src='content_image/328957.jpg'>

<img src='content_image/328958.jpg'>

runs inside the ellipse with τ and σ being elliptic coordinates as in Eq. (C2). The function (inverse to (B23) from [6]) 

<img src='content_image/328955.jpg'>

conformally maps ellipse onto a unit disk. 

To calculate (D2) for a/b ≫ 1, we substitute 

<img src='content_image/328956.jpg'>

