  We study a Kitaev chain model, which is the simplest model of topological
superconductors hosting Majorana fermion, appearing as a zero-energy state at
the edge. We analytically calculate the Green's function of the semi-infinite
Kitaev chain with a delta-function-type impurity potential within the
quasi-classical regime to obtain the spatial dependence of the induced
odd-frequency pairing. It is found that if the position of the impurity is not
far from the edge, the spatial profile of the local density of states (LDOS)
and the odd-frequency spin-triplet $s$-wave pair amplitude is tunable as a
function of the strength of the impurity potential. Moreover, the zero-energy
LDOS and low-frequency odd-frequency pair amplitude are found to have the same
spatial dependence. The spatial profile of the zero-energy LDOS is analyzed
based on the wave function of Majorana fermions.
