We investigate the existence and properties of the Jost solution associated with the differential equation $-y''+q(x)y=\lambda y$, $x\geq0$, for a class of real- or complex-valued slowly decaying potentials $q$. In particular, it is shown how the traditional condition $q\in L(\mathbb{R}^{+})$ for the existence of the Jost solution can be replaced by $q'\in L(\mathbb{R}^{+})$ for a class of potentials considered here. We also examine the asymptotics of the Titchmarsh–Weyl function for a class of real- or complex-valued slowly decaying potentials and the form of the spectral density for a class of real-valued slowly decaying potentials.