  We consider in 2D the following special case of the Mumford-Shah functional
$$ J(u, \Gamma)=\int_{B_1\backslash\Gamma} |\nabla u|^2 dx + \lambda^2
\frac{\pi}{2} \mathcal{H}^1(\Gamma). $$ It is known that if the minimizer has a
crack-tip in the ball $B_1$ (assume at the origin), then $u\approx \lambda \Im
\sqrt{z}$ at this point. We calculate higher order terms in the asymptotic
expansion, where the homogeneity orders of those terms appear to be solutions
to a certain trigonometric relation.
