  One-dimensional $\delta^{'}$-function potential is discussed in the framework
of Green's function formalism without invoking perturbation expansion. It is
shown that the energy-dependent Green's function for this case is crucially
dependent on the boundary conditions which are provided by self-adjoint
extension method. The most general Green's function which contains four real
self-adjoint extension parameters is constructed. Also the relation between the
bare coupling constant and self-adjoint extension parameter is derived.
