Biorthogonal expansions in the first fundamental problem of elasticity theory

The first fundamental boundary-value problem of elasticity theory is considered for a rectangular semi-infinite strip whose long sides are free of stress. Separation of variables is used to reduce the solution to a series expansion of two functions defined in a closed interval (the “end” of the half-strip), in terms of homogeneous solutions. The system of homogeneous solutions over an interval of the real axis is proved to be complete in L₂. Systems of functions biorthogonal to the systems of homogeneous solutions are constructed on a certain contour on the Riemann surface of the logarithm. This biorthogonality concept is a natural generalization of biorthogonality over a closed interval. The biorthogonal systems constructed are used to find explicit expressions for the expansion coefficients.