In this paper, some problems of boundary values of the elastic plane are treated, which can be reduced to the Hilbert-Riemann problem. In the classic procedure, we work on determining two functions $\Phi(z)$ and $\Psi(z)$, while this paper shows the procedure when we stay with the determination of the original functions $\varphi(z)$ and $\psi(z)$, for which we are looking for such a solution, which is within the interval $[\alpha_k.\beta_k]$ finite and at the point $z=\infty$ bounded.