Four properties for complex potentials in power series form

Based on the basic equations of plane and anti-plane problems and the feature of stress fields, four important properties for complex potentials expressed in power series expansion in the plane and anti-plane elastic fields are obtained. First, the complex potentials in the plane or anti-plane elastic fields are both even functions when the stress states are symmetrical with regard to the origin. Next, it is found that the coefficients of complex potentials in the plane elastic field must be real quantity when the stress states are symmetrical with regard to the x-axis. The last conclusion is that the complex potentials in the anti-plane elastic field only have imaginary coefficients when the stress states are anti-symmetric with regard to the x-axis. Then, based on the above conclusions some classic solutions are rearrived at, which indicates the results derived in the paper are right and may simplify the process of constructing and solving the complex potential functions. The present conclusions provide an efficient tool to discover the complex potentials in the sophisticated state.