  A single-step Eriksen transformation of~$1S_{1/2}$,~$2P_{1/2}$ and~$2P_{3/2}$
states of the relativistic hydrogen-like atom is performed exactly by
expressing each transformed function (TF) as a linear combination of
eigenstates of the Dirac Hamiltonian. The transformed functions, which are
four-component spinors with vanishing two lower components, are calculated
numerically and have the same symmetries as the initial states. For all nuclear
charges~$Z \in [1\ldots 92]$ a contribution of the initial state to TFs exceeds
86\% of the total probability density. Next large contribution to TFs comes
from continuum states with negative energies close to~$-m_0c^2-E_b$,
where~$E_b$ is the binding energy of initial state. Contribution of other
states to TFs is less than~$0.1\%$ of the total probability density. Other
components of TFs are nearly zero which confirms both validity of the Eriksen
transformation and accuracy of the numerical calculations. The TFs
of~$1S_{1/2}$ and~$2P_{1/2}$ states are close to~$1s$ and~$2p$ states of the
nonrelativistic hydrogen-like atom, respectively, but the TF of~$2P_{3/2}$
state differs qualitatively from the~$2p$ state. Functions calculated with use
of a linearized Eriksen transformation, being equivalent to the second order
Foldy-Wouthuysen transformation, are compared with corresponding functions
obtained by Eriksen transformation. A very good agreement between both results
is obtained.
