The present work uses asymptotic techniques to find analytical solutions for the velocity profile, the turbulent kinetic energy, kappa, and the dissipation rate by unit mass, epsilon, for transpired incompressible boundary layers. The higher-order solutions have an explicit dependence on the transpiration rate and can be used as wall functions for the specification of boundary conditions in the fully turbulent region of the flow. The second-order solutions show that kappa has a (ln y) behaviour whereas epsilon has a (ln y)/y behaviour. All results are validated against experimental data.