In this paper the inverse scattering problem is considered for a version of the one-dimensional Schrödinger equation with turning point on the half-line (0, ∞). The scattering data of the problem is defined and the fundamental equation is derived. With the help of the derived fundamental equation, in terms of the scattering data, the potential is recovered uniquely.MSC: 58C40; 34L25; 34B05; 47A40

In the present paper, Muskhelishvili's complex variable method of solving two-dimensional elasticity problems has been applied to derive exact expressions for Goursat's functions for the boundary value problems of the infinite plate weakened by a hole having many poles and arbitrary shape which is conformally mapped n the dorhain outside a unit circle by means of general rational mapping function. Also the components of stress are obtained, when a stationary heat is flowing uniformly in the perpendicular direction of the hole. Some applications are considered. The interesting cases when the shape of the hole takes different shapes are included as special cases.

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