A new form of an exact linear elastic solution is obtained for the problem of a crack in a semi-infinite plate subjected, at infinity, to antiplane stress (Mode III) loading. The use of the conformal mapping technique results in a convenient stress and displacement solution for each point of the semi-infinite domain. For completeness, the solution technique is extended to Mode I, II problems with center-cracks as well as the Mode III V-notch problem. It is shown that the limiting case of the V-notch collapses to the stress functions independently derived for the edge-cut. At distances equivalent to 10% of the crack length away from the crack tip, the exact solutions give stresses about 7.5% greater than the one-term results. Ramifications of the exact solutions to finite-element solutions, elastic-plastic and diffusion problems are discussed.