We present an analytical approach using conformal mapping to the free-boundary problem of the shape of a liquid drop submitted to a strong electrical field, as encountered in electrowetting systems. In agreement with recent numerical calculations, we show that both the curvature of the surface profile and the electric field diverge algebraically close to the three-phase line. The algebraic exponents agree with the numerical results. We show analytically that the local contact angle remains equal to Young's angle, independent of the applied voltage. Furthermore, we present experimental evidence of a curvature increase close to the contact line.