Abstract: By the theoretical method for complex functions, dynamic propagation problems concerning asymmetrical mode Ⅲ cracks were researched. The analytical solutions can be attained by the approaches of self-similar functions. The problems dealt with can be readily transformed into Riemann-Hilbert problems by this technique, and general expressions of analytical solutions are given considering the action of pulse loads and unit-step loads located at the origin of the coordinates respectively. Utilizing these solutions obtained and superposition principle, the solutions of discretionarily intricate problems can be acquired.