content_image/344771.jpg content_image/344770.jpg content_image/344769.jpg content_image/344768.jpg content_image/344767.jpg content_image/344766.jpg content_image/344765.jpg
In the limit T/R → ∞ , when µ → 0, this equation simplifies to 

<img src='content_image/344771.jpg'>

When z = s runs along the real axis, the variable ω runs along the boundary of the rectangle with s = − 1 / √ µ, − √ µ, + √ µ, +1 / √ µ ( µ < 1) mapped, respectively, onto the vertices of the rectangle: ( − R/ 2 , T/ 2) , ( − R/ 2 , − T/ 2) , ( R/ 2 , − T/ 2) , ( R/ 2 , T/ 2). The given choice of the argument of the mapping preserves the symmetry s → 1 /s . 

When z has positive imaginary part, the coordinates 

<img src='content_image/344770.jpg'>

take their values inside the rectangle. These coordinates are conformal. For this reason we have 

<img src='content_image/344769.jpg'>

whose implementation for the function t ∗ ( s ) is discussed below. 

The boundary contour given by Eq. (23) satisfies Douglas’ minimization (see Appendix B, Eq. (B3)). Correspondingly, the Douglas integral 

<img src='content_image/344768.jpg'>

as it should. We have verified these two equations numerically. 

A natural parametrization of the boundary of a rectangle is through τ ∈ S $^{1}$: 

and analogously for negative τ . Introducing 

we rewrite Eq. (25) as 

<img src='content_image/344767.jpg'>

<img src='content_image/344766.jpg'>

<img src='content_image/344765.jpg'>

