With these studies, the properties of the general damping and enhancing functions are well understood, which provides a solid physical and mathematical foundation for developing the complete analytical formulations for law-of-the-wall, which are expected to include the inner layer, the transition layer, the semi-log linear layer and the wake layer. type-A turbulent boundary layer, semi-finite zero-pressure gradient flat-plate flow, law-of-the-wall, inner-layer law, direct numerical simulation