Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 61 / 2 / b / Solution

Put $d=\ell /(2\sqrt{3})$ and $a=e^{2\pi i/3}$. Consistent with the actual PDF vertices, parametrize the three oriented sides byThey run counterclockwise, with unit tangent $i{\beta }_j$ and outward normal ${\beta }_j$. Consequently the normal derivative and tangential derivative satisfyLet $\kappa =k{\beta }_j$. On this side the exponential factor becomesPulling back the closed differential one-form therefore givesDefine ${\Phi }_j(\kappa )$ and ${\Psi }_j(\kappa )$ by integrating the Dirichlet terms and normal term in this expression. The Generalized Stokes theorem gives ${\int }_{\partial D}W=0$, soThe factor $2i$ follows by solving $\sum E\Phi +(i/2)\sum E\Psi =0$. The bars on the side-2 rotation are essential; replacing them with $a$ would use the wrong side orientation. The converted TeX also reverses the sign in the upper vertex's exponential, whereas the PDF puts that vertex above the real axis; the parametrization here follows the PDF.