Consider a three-dimensional homogeneous isotropic elastic solid containing a flat crack, Ω, subjected to a shear loading. The problem of finding the resulting stress distribution can be reduced to a pair of coupled hypersingular integral equations over Ω for the tangential components of the crack-opening displacement vector. Here, these equations are first written as a single equation for a complex displacement. This equation is then transformed into a similar equation over a circular region D, using a conformal mapping between Ω and D. This new equation is then regularized analytically by using an appropriate expansion method (Fourier series in the azimuthal direction and series of orthogonal polynomials in the radial direction). Analytical results for regions that are approximately circular are also obtained. These include formulae for the crack-opening displacement and the stress-intensity factors in terms of the conformal mapping or the shape of the crack.